namespace Codex
def LeftInverse (g : β → α) (f : α → β) : Prop := ∀ a, g (f a) = a
def RightInverse (g : β → α) (f : α → β) : Prop := LeftInverse f g
end Codex
structure Equiv (α : Sort u) (β : Sort v) where
toFun : α → β
invFun : β → α
left_inv : Codex.LeftInverse invFun toFun
right_inv : Codex.RightInverse invFun toFun
infixl:25 " ≃ " => Equiv
def Nat.factorial : Nat → Nat
| 0 => 1
| n + 1 => (n + 1) * Nat.factorial n
namespace CoreRat
theorem div_pos_of_le {d g : Nat} (hg : 0 < g) (hle : g ≤ d) : 0 < d / g := by
have h : 1 * g ≤ d := by rw [Nat.one_mul]; exact hle
exact (Nat.le_div_iff_mul_le hg).mpr h
theorem div_gcd_pos {a d : Nat} (hd : 0 < d) : 0 < d / Nat.gcd a d :=
div_pos_of_le (Nat.gcd_pos_of_pos_right a hd) (Nat.le_of_dvd hd (Nat.gcd_dvd_right a d))
theorem coprime_div_gcd {a d : Nat} (hd : 0 < d) :
Nat.gcd (a / Nat.gcd a d) (d / Nat.gcd a d) = 1 := by
have hg : 0 < Nat.gcd a d := Nat.gcd_pos_of_pos_right a hd
have ha : Nat.gcd a d * (a / Nat.gcd a d) = a := Nat.mul_div_cancel' (Nat.gcd_dvd_left a d)
have hb : Nat.gcd a d * (d / Nat.gcd a d) = d := Nat.mul_div_cancel' (Nat.gcd_dvd_right a d)
have h : Nat.gcd a d * Nat.gcd (a / Nat.gcd a d) (d / Nat.gcd a d) = Nat.gcd a d := by
have e := Nat.gcd_mul_left (Nat.gcd a d) (a / Nat.gcd a d) (d / Nat.gcd a d)
rw [ha, hb] at e; exact e.symm
have h2 : Nat.gcd a d * Nat.gcd (a / Nat.gcd a d) (d / Nat.gcd a d) = Nat.gcd a d * 1 := by
rw [Nat.mul_one]; exact h
exact Nat.eq_of_mul_eq_mul_left hg h2
end CoreRat
structure Ratio where
num : Int
den : Nat
pos : 0 < den
cop : Nat.gcd num.natAbs den = 1
deriving DecidableEq, Repr
namespace Ratio
theorem natAbs_signed (k : Nat) (neg : Bool) :
(if neg then -(k : Int) else (k : Int)).natAbs = k := by cases neg <;> simp
def mk' (n d : Int) : Ratio :=
if hd : d = 0 then ⟨0, 1, by decide, by decide⟩
else
let neg : Bool := decide ((n < 0) ≠ (d < 0))
let a : Nat := n.natAbs
let b : Nat := d.natAbs
have hb : 0 < b := Int.natAbs_pos.mpr hd
let g : Nat := Nat.gcd a b
⟨(if neg then -((a / g : Nat) : Int) else ((a / g : Nat) : Int)), b / g,
CoreRat.div_gcd_pos hb, by rw [natAbs_signed]; exact CoreRat.coprime_div_gcd hb⟩
instance instOfNatRat : OfNat Ratio n := ⟨mk' n 1⟩
instance instAddRat : Add Ratio where add a b := mk' (a.num * b.den + b.num * a.den) (a.den * b.den)
instance instMulRat : Mul Ratio where mul a b := mk' (a.num * b.num) (a.den * b.den)
instance instNegRat : Neg Ratio where neg a := mk' (-a.num) a.den
instance instSubRat : Sub Ratio where sub a b := a + (-b)
def inv (a : Ratio) : Ratio := mk' (a.den : Int) a.num
instance instDivRat : Div Ratio where div a b := a * inv b
instance instLTRat : LT Ratio where lt a b := a.num * b.den < b.num * a.den
instance instLERat : LE Ratio where le a b := a.num * b.den ≤ b.num * a.den
instance instDecidableLtRat (a b : Ratio) : Decidable (a < b) :=
inferInstanceAs (Decidable (a.num * b.den < b.num * a.den))
instance instDecidableLeRat (a b : Ratio) : Decidable (a ≤ b) :=
inferInstanceAs (Decidable (a.num * b.den ≤ b.num * a.den))
def pow (a : Ratio) : Nat → Ratio
| 0 => 1
| n + 1 => a * pow a n
instance instHPowRatNat : HPow Ratio Nat Ratio := ⟨Ratio.pow⟩
instance instToStringRat : ToString Ratio where
toString a := if a.den == 1 then toString a.num else toString a.num ++ "/" ++ toString a.den
@[simp] theorem num_zero : (0 : Ratio).num = 0 := rfl
@[simp] theorem den_zero : (0 : Ratio).den = 1 := rfl
theorem mk'_zero_num {d : Nat} (hd : 0 < d) : mk' 0 (d : Int) = 0 := by
have hne : (d : Int) ≠ 0 := by omega
show mk' 0 (d : Int) = mk' 0 1
unfold mk'
simp only [hne, (by decide : (1 : Int) ≠ 0), dite_false]
simp [Nat.gcd_zero_left, Nat.div_self hd, Nat.zero_div]
@[simp] theorem zero_mul (a : Ratio) : (0 : Ratio) * a = 0 := by
show mk' ((0 : Ratio).num * a.num) (((0 : Ratio).den * a.den : Nat) : Int) = 0
rw [num_zero, den_zero, Int.zero_mul, Nat.one_mul]
exact mk'_zero_num a.pos
@[simp] theorem mul_zero (a : Ratio) : a * (0 : Ratio) = 0 := by
show mk' (a.num * (0 : Ratio).num) ((a.den * (0 : Ratio).den : Nat) : Int) = 0
rw [num_zero, den_zero, Int.mul_zero, Nat.mul_one]
exact mk'_zero_num a.pos
theorem canonical (a : Ratio) : 0 < a.den ∧ Nat.gcd a.num.natAbs a.den = 1 := ⟨a.pos, a.cop⟩
theorem ext {a b : Ratio} (h1 : a.num = b.num) (h2 : a.den = b.den) : a = b := by
cases a; cases b; simp_all
theorem inv_zero : inv (0 : Ratio) = 0 := by decide
end Ratio
set_option maxRecDepth 4000000
inductive Grade : Type
| premise | corroboration | operational | structural | engineering
| conditional | theoremConditional | analytic | theorem
deriving DecidableEq, Repr
def Grade.rank : Grade → Nat
| .premise => 0 | .corroboration => 1 | .operational => 2 | .structural => 3
| .engineering => 4 | .conditional => 5 | .theoremConditional => 6
| .analytic => 7 | .theorem => 8
def Grade.weakest (a b : Grade) : Grade := if a.rank ≤ b.rank then a else b
theorem Grade.weakest_is_a_link (a b : Grade) :
Grade.weakest a b = a ∨ Grade.weakest a b = b := by
cases a <;> cases b <;> decide
theorem Grade.weakest_idempotent (a : Grade) : Grade.weakest a a = a := by
cases a <;> decide
structure Claim : Type where
stmt : Prop
grade : Grade
warrant : stmt
def Claim.join (c₁ c₂ : Claim) : Claim where
stmt := c₁.stmt ∧ c₂.stmt
grade := Grade.weakest c₁.grade c₂.grade
warrant := ⟨c₁.warrant, c₂.warrant⟩
theorem Claim.join_grade (c₁ c₂ : Claim) :
(Claim.join c₁ c₂).grade = Grade.weakest c₁.grade c₂.grade := rfl
def deltaM : Nat := 0
def socialWeight : Nat := 0
namespace FTOE
variable {α : Type}
def Even (τ : α → α) (f : α → Bool) : Prop := ∀ x, f (τ x) = f x
def OddAt (τ : α → α) (d : α → Bool) (x : α) : Prop := d (τ x) ≠ d x
def WhollyOdd (τ : α → α) (t : α → Bool) : Prop := ∀ x, t (τ x) = !t x
theorem T1_wall (τ : α → α) (f d : α → Bool) (x : α)
(he : Even τ f) (ho : OddAt τ d x) : f ≠ d := by
intro h
subst h
exact ho (he x)
theorem T2_coalition (τ : α → α) (f g d : α → Bool) (x : α)
(hf : Even τ f) (hg : Even τ g) (op : Bool → Bool → Bool)
(ho : OddAt τ d x) : (fun y => op (f y) (g y)) ≠ d := by
apply T1_wall τ _ d x
· intro y; show op (f (τ y)) (g (τ y)) = op (f y) (g y); rw [hf y, hg y]
· exact ho
theorem T3_no_fixed_point (τ : α → α) (t : α → Bool)
(ho : WhollyOdd τ t) (x : α) : τ x ≠ x := by
intro h
have h1 : t (τ x) = !t x := ho x
rw [h] at h1
cases t x <;> simp at h1
theorem T4_one_bit (τ : α → α) (s t : α → Bool)
(hs : WhollyOdd τ s) (ht : WhollyOdd τ t)
(x : α) (h : s x = t x) : s (τ x) = t (τ x) := by
rw [hs x, ht x, h]
theorem T5_crossing (τ : α → α) (s d : α → Bool) (x : α)
(hs : s (τ x) = !s x) (hd : d (τ x) = !d x) :
∃ c : Bool, (d x = xor (s x) c ∧ d (τ x) = xor (s (τ x)) c) ∧
∀ c' : Bool, (d x = xor (s x) c' ∧ d (τ x) = xor (s (τ x)) c') → c' = c := by
exact ⟨xor (d x) (s x),
⟨by cases s x <;> cases d x <;> rfl,
by rw [hs, hd]; cases s x <;> cases d x <;> rfl⟩,
by
intro c' hc
obtain ⟨h1, -⟩ := hc
generalize hsx : s x = sv
generalize hdx : d x = dv
rw [hsx, hdx] at h1
cases sv <;> cases dv <;> cases c' <;> first | rfl | exact absurd h1 (by decide)⟩
theorem T6a_deed_anchor (τ : α → α) (d : α → Bool) (x₀ : α) (b : Bool)
(hd : d (τ x₀) = !d x₀) (anchor : d x₀ = b) :
d x₀ = b ∧ d (τ x₀) = !b := by
exact ⟨anchor, by rw [hd, anchor]⟩
theorem T6b_deed_supply (τ : α → α) (s d : α → Bool) (x₀ : α) (c : Bool)
(hs : s (τ x₀) = !s x₀)
(arrival : d x₀ = xor (s x₀) c ∧ d (τ x₀) = xor (s (τ x₀)) c) :
d x₀ = xor (s x₀) c ∧ d (τ x₀) = !xor (s x₀) c := by
constructor
· exact arrival.1
· rw [arrival.2, hs]; cases s x₀ <;> cases c <;> rfl
theorem fTOE_port (_τ : α → α) (s d : α → Bool) (x₀ : α) (b c : Bool)
(anchor : d x₀ = b)
(arrival : ∀ y : α, d y = xor (s y) c) :
c = xor (s x₀) b := by
have h1 := arrival x₀
rw [anchor] at h1
generalize hsx : s x₀ = sv
rw [hsx] at h1
cases sv <;> cases b <;> cases c <;> first | rfl | exact absurd h1 (by decide)
section FullSetup
variable {β γ : Type}
def Involution (τ : α → α) : Prop := ∀ x, τ (τ x) = x
def FixedPointFree (τ : α → α) : Prop := ∀ x, τ x ≠ x
theorem T7_global_wall (τ : α → α) (f d : α → Bool) (x : α)
(hf : Even τ f) (hd : WhollyOdd τ d) : f ≠ d := by
intro h
have h1 : f (τ x) = d (τ x) := congrFun h (τ x)
rw [hf x, hd x, congrFun h x] at h1
cases d x <;> simp at h1
theorem T8_torsor_forward (τ : α → α) (s d : α → Bool)
(hs : WhollyOdd τ s) (hd : WhollyOdd τ d) :
Even τ (fun x => xor (d x) (s x)) := by
intro x
show xor (d (τ x)) (s (τ x)) = xor (d x) (s x)
rw [hd x, hs x]; cases d x <;> cases s x <;> rfl
theorem T8_torsor_backward (τ : α → α) (s c : α → Bool)
(hs : WhollyOdd τ s) (hc : Even τ c) :
WhollyOdd τ (fun x => xor (s x) (c x)) := by
intro x
show xor (s (τ x)) (c (τ x)) = !xor (s x) (c x)
rw [hs x, hc x]; cases s x <;> cases c x <;> rfl
theorem T9_torsor_inverse_left (s d : α → Bool) :
(fun x => xor (s x) (xor (d x) (s x))) = d := by
funext x; cases d x <;> cases s x <;> rfl
theorem T9_torsor_inverse_right (s c : α → Bool) :
(fun x => xor (xor (s x) (c x)) (s x)) = c := by
funext x; cases s x <;> cases c x <;> rfl
theorem T10_factorization (τ : α → α) (ρ : α → β) (g : β → Bool)
(d : α → Bool) (x : α)
(hρ : ρ (τ x) = ρ x) (hd : d (τ x) = !d x) :
d ≠ fun y => g (ρ y) := by
intro h
have h1 : d (τ x) = g (ρ (τ x)) := congrFun h (τ x)
have h2 : d x = g (ρ x) := congrFun h x
rw [hρ, hd, h2] at h1
cases hgx : g (ρ x) <;> rw [hgx] at h1 <;> exact absurd h1 (by decide)
theorem T11_separation (τ : α → α) (ρ : α → β) (a : α → γ) (r : β → γ → Bool)
(d : α → Bool) (x : α)
(hρ : ρ (τ x) = ρ x) (hd : d (τ x) = !d x)
(hres : ∀ y, d y = r (ρ y) (a y)) :
a (τ x) ≠ a x := by
intro h
have h1 := hres (τ x)
have h2 := hres x
rw [hρ, h, hd, h2] at h1
cases hrv : r (ρ x) (a x) <;> rw [hrv] at h1 <;> exact absurd h1 (by decide)
theorem T12_encoder_injective {δ : Type} (Enc : δ → γ) (Dec : γ → δ)
(h : ∀ d, Dec (Enc d) = d) :
∀ d₁ d₂ : δ, Enc d₁ = Enc d₂ → d₁ = d₂ := by
intro d₁ d₂ hE
have h1 := h d₁
have h2 := h d₂
rw [hE] at h1
rw [h2] at h1
exact h1.symm
theorem T14_wall_factorization (τ : α → α) (ρ : α → β) (d : α → Bool) (x : α)
(hρ : ∀ z, ρ (τ z) = ρ z) (hd : OddAt τ d x) :
¬ ∃ g : β → Bool, ∀ z, g (ρ z) = d z := by
intro h
obtain ⟨g, hg⟩ := h
have h1 : d (τ x) = g (ρ (τ x)) := (hg (τ x)).symm
have h2 : g (ρ (τ x)) = g (ρ x) := by rw [hρ x]
have h3 : g (ρ x) = d x := hg x
exact hd (h1.trans (h2.trans h3))
theorem T13_wholly_odd_is_odd_everywhere (τ : α → α) (d : α → Bool)
(hd : WhollyOdd τ d) (x : α) : OddAt τ d x := by
intro h
rw [hd x] at h
cases d x <;> simp at h
def flipF {Q : Type} : Bool × Q → Bool × Q := fun (b, q) => (!b, q)
theorem T15_price_bijection_forward {Q : Type} (d : Bool × Q → Bool)
(hd : WhollyOdd flipF d) (b : Bool) (q : Q) :
xor (d (true, q)) (!b) = d (b, q) := by
have h : d (false, q) = !d (true, q) := hd (true, q)
cases b
· generalize ht : d (true, q) = tv
generalize hf : d (false, q) = fv
rw [ht, hf] at h
cases tv <;> cases fv <;>
first | rfl | exact h | exact h.symm | exact absurd h (by decide)
· cases d (true, q) <;> rfl
theorem T15_price_bijection_backward {Q : Type} (c : Q → Bool) :
WhollyOdd flipF (fun (b, q) => xor (c q) (!b)) := by
intro x
obtain ⟨b, q⟩ := x
show xor (c q) (!!b) = !xor (c q) (!b)
cases c q <;> cases b <;> rfl
end FullSetup
theorem fTOE_core (τ : α → α) (s d : α → Bool) (x : α)
(f g : α → Bool) (hf : Even τ f) (hg : Even τ g) (op : Bool → Bool → Bool)
(hd : d (τ x) = !d x) (hs : s (τ x) = !s x) :
(fun y => op (f y) (g y)) ≠ d ∧
∃ c : Bool, (d x = xor (s x) c ∧ d (τ x) = xor (s (τ x)) c) ∧
∀ c' : Bool, (d x = xor (s x) c' ∧ d (τ x) = xor (s (τ x)) c') → c' = c :=
⟨T2_coalition τ f g d x hf hg op (by intro h; rw [hd] at h; cases d x <;> simp at h),
T5_crossing τ s d x hs hd⟩
end FTOE
namespace FTOE
variable {α : Type}
def OrbitRel (τ : α → α) (x y : α) : Prop := x = y ∨ τ x = y
theorem OrbitRel_symmetric (τ : α → α) (hinv : Involution τ) {x y : α} :
OrbitRel τ x y → OrbitRel τ y x := by
intro h
cases h with
| inl h => exact Or.inl h.symm
| inr h => exact Or.inr (by rw [← h]; exact hinv x)
def Dtau (τ : α → α) : Type := { d : α → Bool // WhollyOdd τ d }
def calib (d0 d : α → Bool) : α → Bool := fun x => xor (d x) (d0 x)
theorem calib_even (τ : α → α) (d0 d : α → Bool)
(h0 : WhollyOdd τ d0) (hd : WhollyOdd τ d) : Even τ (calib d0 d) :=
T8_torsor_forward τ d0 d h0 hd
def descend (τ : α → α) (e : α → Bool) (he : Even τ e) :
Quot (OrbitRel τ) → Bool :=
Quot.lift e (by
intro a b hab
cases hab with
| inl h => rw [h]
| inr h => rw [← h]; exact (he a).symm)
def priceForward (τ : α → α) (d0 : Dtau τ) (d : Dtau τ) :
Quot (OrbitRel τ) → Bool :=
descend τ (calib d0.1 d.1) (calib_even τ d0.1 d.1 d0.2 d.2)
def priceBackward (τ : α → α) (hinv : Involution τ) (d0 : Dtau τ)
(e : Quot (OrbitRel τ) → Bool) : Dtau τ :=
⟨fun x => xor (d0.1 x) (e (Quot.mk (OrbitRel τ) x)), by
intro x
show xor (d0.1 (τ x)) (e (Quot.mk (OrbitRel τ) (τ x)))
= !xor (d0.1 x) (e (Quot.mk (OrbitRel τ) x))
have hq : Quot.mk (OrbitRel τ) (τ x) = Quot.mk (OrbitRel τ) x :=
Quot.sound (Or.inr (hinv x))
rw [hq, d0.2 x]
cases d0.1 x <;> cases e (Quot.mk (OrbitRel τ) x) <;> rfl⟩
theorem T16_left_inverse (τ : α → α) (hinv : Involution τ) (d0 : Dtau τ) :
Codex.LeftInverse (priceBackward τ hinv d0) (priceForward τ d0) := by
intro d
apply Subtype.ext
funext x
show xor (d0.1 x) (xor (d.1 x) (d0.1 x)) = d.1 x
cases d.1 x <;> cases d0.1 x <;> rfl
theorem T16_right_inverse (τ : α → α) (hinv : Involution τ) (d0 : Dtau τ) :
Codex.RightInverse (priceBackward τ hinv d0) (priceForward τ d0) := by
intro e
funext q
induction q using Quot.ind with
| _ x =>
show xor (xor (d0.1 x) (e (Quot.mk (OrbitRel τ) x))) (d0.1 x)
= e (Quot.mk (OrbitRel τ) x)
generalize e (Quot.mk (OrbitRel τ) x) = b
generalize d0.1 x = a
cases a <;> cases b <;> rfl
theorem T16_price_bijection (τ : α → α) (hinv : Involution τ) (d0 : Dtau τ) :
Nonempty (Dtau τ ≃ (Quot (OrbitRel τ) → Bool)) :=
⟨{ toFun := priceForward τ d0
invFun := priceBackward τ hinv d0
left_inv := T16_left_inverse τ hinv d0
right_inv := T16_right_inverse τ hinv d0 }⟩
def canonD0 : Dtau (flipF (Q := Bool)) :=
⟨fun (b, _) => b, by intro x; obtain ⟨b, q⟩ := x; rfl⟩
theorem flipF_involution : Involution (flipF (Q := Bool)) := by
intro x; obtain ⟨b, q⟩ := x; cases b <;> rfl
theorem T16_canonical :
Nonempty (Dtau (flipF (Q := Bool)) ≃
(Quot (OrbitRel (flipF (Q := Bool))) → Bool)) :=
T16_price_bijection (flipF (Q := Bool)) flipF_involution canonD0
theorem T17_odd_everywhere_iff_wholly_odd (τ : α → α) (d : α → Bool) :
(∀ x, d (τ x) ≠ d x) ↔ (∀ x, d (τ x) = !d x) := by
constructor
· intro h x
have hx := h x
cases hd : d x <;> cases ht : d (τ x) <;> simp_all
· intro h x
have hx := h x
cases hd : d x <;> cases ht : d (τ x) <;> simp_all
end FTOE
namespace K4
def val : Nat → Nat
| 0 => 11 | 1 => 851 | 2 => 13 | 3 => 1273 | 4 => 17 | 5 => 437
| 6 => 19 | 7 => 2119 | 8 => 23 | 9 => 1703 | 10 => 29 | 11 => 869
| _ => 0
def tau : Nat → Nat := fun i =>
if i < 12 then (if i % 2 = 0 then i + 1 else i - 1) else i
def target : Nat → Bool := fun i => i % 2 == 0
def rho : Nat → Nat := fun i =>
(val i % 2) * 1000 + (val i % 3) * 100 + (val i % 5) * 10 + (val i % 7)
theorem tau_involution : ∀ i, i < 12 → tau (tau i) = i := by decide
theorem tau_fixed_point_free : ∀ i, i < 12 → tau i ≠ i := by decide
theorem target_wholly_odd : ∀ i, i < 12 → target (tau i) = !target i := by decide
theorem register_even : ∀ i, i < 12 → rho (tau i) = rho i := by decide
theorem val_lifts_mod_210 : ∀ i, i < 12 → i % 2 = 1 →
val i > val (tau i) ∧ (val i - val (tau i)) % 210 = 0 := by decide
theorem wall (g : Nat → Bool) : target ≠ fun y => g (rho y) :=
FTOE.T10_factorization tau rho g target 0
(register_even 0 (by decide)) (target_wholly_odd 0 (by decide))
theorem pair_distinct : target ≠ fun i => !target i := by
intro h
exact absurd (congrFun h 0) (by decide)
theorem price_one_bit : 2^1 = 2 := rfl
theorem price_full_fibre : (2 : Nat)^6 = 64 := by decide
def wallClaim : Claim where
stmt := ∀ g : Nat → Bool, target ≠ fun y => g (rho y)
grade := .theorem
warrant := wall
end K4
namespace Audit
inductive Token : Type
| sealed | broken | opn | refused | determined | dotmark
| gok | golok | void | aGivenRA | iAm | interior
deriving DecidableEq, Repr, BEq
inductive Compartment : Type
| closureRowed | unpopulated | worldRowed | none
deriving DecidableEq, Repr, BEq
inductive Ans : Type
| yes | no | unknown
deriving DecidableEq, Repr, BEq
def Ans.toBool : Ans → Bool
| .yes => true
| _ => false
structure ClaimInput where
hasContent : Ans
facesScoped : Ans
canonical : Ans
forwardMode : Ans
populated : Ans
worldlyRow : Ans
drillsOk : Ans
frameClosed : Ans
slots : Option (List (List Int))
deriving Repr
def ClaimInput.anyUnknown (i : ClaimInput) : Bool :=
[i.hasContent, i.facesScoped, i.canonical, i.forwardMode, i.populated,
i.worldlyRow, i.drillsOk, i.frameClosed].any (· == Ans.unknown)
|| i.slots.isNone
def sealL : Option (List (List Int)) → Token × String
| none => (.opn, "S-L0: vocabulary slots not supplied; under-determined")
| some slots =>
let nz := slots.map (fun s => s.filter (· != 0))
if slots.flatten.any (fun x => decide (x < 0)) then
(.broken, "negative literal id; not vocabulary")
else
let nfull := (nz.filter (fun s => !s.isEmpty)).length
if nfull != 3 then
(.broken, s!"deletion test returns {nfull} slots, not three")
else if nz.any (fun s => s.length != s.eraseDups.length) then
(.broken, "repeated literal within slot")
else
let flat := nz.flatten
if flat.length != flat.eraseDups.length then
(.broken, "LIT collision across slots")
else
(.sealed, "three slots, vocabulary pairwise disjoint")
structure DrillInput where
d1 : Ans
d2 : Ans
d3Filed : Ans
d3Shaped : Ans
d4 : Ans
d5 : Ans
d6 : Ans
deriving Repr
def DrillInput.anyUnknown (d : DrillInput) : Bool :=
[d.d1, d.d2, d.d3Filed, d.d3Shaped, d.d4, d.d5, d.d6].any (· == Ans.unknown)
def drillScreen (d : DrillInput) : Token × Grade × String :=
if d.anyUnknown then
(.opn, .premise, "drill screen under-determined: a drill was not run")
else if !(d.d1.toBool && d.d2.toBool) then
(.broken, .premise, "D1/D2 fail: Seal L break, terminal whatever the determinant reads")
else if !d.d3Filed.toBool then
(.broken, .premise, "D3 fail: provenance unfiled; Number-shaped content in the Tongue seat")
else if !d.d4.toBool then
(.broken, .premise, "D4 fail: a magnitude carried on the Tongue; sign only, veto never substitution")
else if !d.d5.toBool then
(.broken, .premise, "D5 fail: register undeclared; re-index rather than refute")
else if !d.d6.toBool then
(.broken, .premise, "D6 fail: the Tongue reproduced the blindness it was appointed to repair")
else if d.d3Shaped.toBool then
(.sealed, .structural, "drills pass; provenance filed and SHAPED, so the assignment caps at structural")
else
(.sealed, .theorem, "drills pass; provenance filed and unshaped, no tier discount")
def refusalGates (i : ClaimInput) : Option (Compartment × Token × String) :=
if !i.hasContent.toBool then
some (.none, .refused, "F-0: every term decoration; contentless, refused, not compartmented")
else if !i.facesScoped.toBool then
some (.none, .refused, "R-1: faces not enumerated; a composite carries one compartment per face")
else if !i.canonical.toBool then
some (.none, .refused, "R-2: not canonicalised; the compartment attaches to the stripped string")
else none
def compartmentGates (i : ClaimInput) : Compartment × Token × String :=
if !i.populated.toBool then
(.unpopulated, .opn, "S-1: an axis is empty; two planes meet in a line; no lock forms")
else if i.forwardMode.toBool then
(.worldRowed, .determined, "R-3: a dated axis must be sourced, never measured; world-rowed unconditionally")
else if i.worldlyRow.toBool then
(.worldRowed, .determined, "S-2: a row furnished by the world; the lock is total given those rows")
else if !i.drillsOk.toBool then
(.worldRowed, .opn, "S-3: term closure unscreened; typed by its remaining rows")
else if !i.frameClosed.toBool then
(.worldRowed, .opn, "S-4: existential import or transport; the leak names the compartment")
else
(.closureRowed, .sealed, "rows are the closure; the two determinations are one; conditional on the root posit at the act")
def auditClaim (i : ClaimInput) : Compartment × Token × String :=
if i.anyUnknown then
(.none, .opn, "under-determined reading: a gate or the slots were not supplied; first failure terminal, nothing sealed")
else
match refusalGates i with
| some r => r
| none =>
match sealL i.slots with
| (.broken, msg) => (.none, .broken, s!"Seal L break, terminal: {msg}")
| _ => compartmentGates i
def compartmentGrade : Compartment → Grade → String
| .closureRowed, cap =>
s!"row-indefeasible, defeasible against fault, cap {repr cap}"
| .unpopulated, _ => "no determination to defeat; an axis is owed"
| .worldRowed, _ =>
"revisable entirely in the rows; direction bought with the arrow"
| .none, _ => "no compartment"
theorem unknown_routes_open (i : ClaimInput) (h : i.anyUnknown = true) :
(auditClaim i).2.1 = Token.opn := by
simp [auditClaim, h]
theorem contentless_refused (i : ClaimInput)
(hk : i.anyUnknown = false) (hc : i.hasContent = Ans.no) :
(auditClaim i).2.1 = Token.refused := by
unfold auditClaim refusalGates
rw [hk]
simp [hc, Ans.toBool]
def det3 (m : Fin 3 → Fin 3 → Ratio) : Ratio :=
m 0 0 * (m 1 1 * m 2 2 - m 1 2 * m 2 1)
- m 0 1 * (m 1 0 * m 2 2 - m 1 2 * m 2 0)
+ m 0 2 * (m 1 0 * m 2 1 - m 1 1 * m 2 0)
theorem det3_two_id : det3 (fun a b => if a == b then (2 : Ratio) else 0) = 8 := by
decide
end Audit
#eval Audit.auditClaim {
hasContent := .no, facesScoped := .yes, canonical := .yes,
forwardMode := .no, populated := .yes, worldlyRow := .no,
drillsOk := .yes, frameClosed := .yes, slots := some [[1], [2], [3]] }
#eval Audit.auditClaim {
hasContent := .yes, facesScoped := .yes, canonical := .yes,
forwardMode := .no, populated := .yes, worldlyRow := .no,
drillsOk := .yes, frameClosed := .yes, slots := some [[1, 2], [3], [4]] }
#eval Audit.auditClaim {
hasContent := .yes, facesScoped := .yes, canonical := .unknown,
forwardMode := .no, populated := .yes, worldlyRow := .no,
drillsOk := .yes, frameClosed := .yes, slots := some [[1], [2], [3]] }
#eval Audit.auditClaim {
hasContent := .yes, facesScoped := .yes, canonical := .yes,
forwardMode := .no, populated := .yes, worldlyRow := .no,
drillsOk := .yes, frameClosed := .yes, slots := some [[1, 2], [3], [2]] }
#eval Audit.det3 (fun a b => if a == b then (2 : Ratio) else (0 : Ratio))
namespace INVERSION
def ItFromBit {α β : Type} (ρ : α → β) (d : α → Bool) : Prop :=
∃ g : β → Bool, d = fun y => g (ρ y)
theorem it_from_bit_refuted {α β : Type} (τ : α → α) (ρ : α → β)
(d : α → Bool) (x : α)
(hρ : ρ (τ x) = ρ x) (hd : d (τ x) = !d x) :
¬ ItFromBit ρ d := by
intro h
obtain ⟨g, hg⟩ := h
exact FTOE.T10_factorization τ ρ g d x hρ hd hg
structure ItFromIt {α β : Type} (τ : α → α) (ρ : α → β) (d : α → Bool)
(x : α) : Prop where
record_collapses : ρ (τ x) = ρ x
pair_genuine : τ x ≠ x
target_separates : d (τ x) = !d x
no_return : ¬ ItFromBit ρ d
theorem it_from_it {α β : Type} (τ : α → α) (ρ : α → β) (d : α → Bool)
(x : α) (hρ : ρ (τ x) = ρ x) (hpair : τ x ≠ x) (hd : d (τ x) = !d x) :
ItFromIt τ ρ d x :=
⟨hρ, hpair, hd, it_from_bit_refuted τ ρ d x hρ hd⟩
end INVERSION
namespace K4
theorem it_from_it_canonical :
INVERSION.ItFromIt tau rho target 0 :=
INVERSION.it_from_it tau rho target 0
(register_even 0 (by decide)) (by decide) (target_wholly_odd 0 (by decide))
theorem it_from_bit_refuted_canonical :
¬ INVERSION.ItFromBit rho target :=
it_from_it_canonical.no_return
def inversionClaim : Claim where
stmt := INVERSION.ItFromIt tau rho target 0
grade := .theorem
warrant := it_from_it_canonical
end K4
namespace GEO
structure Q4 where
r : Int
i : Int
j : Int
k : Int
deriving DecidableEq, Repr, BEq
def qmul (a b : Q4) : Q4 :=
⟨a.r*b.r - a.i*b.i - a.j*b.j - a.k*b.k,
a.r*b.i + a.i*b.r + a.j*b.k - a.k*b.j,
a.r*b.j - a.i*b.k + a.j*b.r + a.k*b.i,
a.r*b.k + a.i*b.j - a.j*b.i + a.k*b.r⟩
def qconj (a : Q4) : Q4 := ⟨a.r, -a.i, -a.j, -a.k⟩
def qadd (a b : Q4) : Q4 := ⟨a.r+b.r, a.i+b.i, a.j+b.j, a.k+b.k⟩
def qsub (a b : Q4) : Q4 := ⟨a.r-b.r, a.i-b.i, a.j-b.j, a.k-b.k⟩
def qnorm2 (a : Q4) : Int := a.r*a.r + a.i*a.i + a.j*a.j + a.k*a.k
def qhalve (a : Q4) : Q4 := ⟨a.r/2, a.i/2, a.j/2, a.k/2⟩
def qquarter (a : Q4) : Q4 := ⟨a.r/4, a.i/4, a.j/4, a.k/4⟩
def qcanon (a : Q4) : Q4 :=
if a.r != 0 then (if a.r < 0 then ⟨-a.r,-a.i,-a.j,-a.k⟩ else a)
else if a.i != 0 then (if a.i < 0 then ⟨-a.r,-a.i,-a.j,-a.k⟩ else a)
else if a.j != 0 then (if a.j < 0 then ⟨-a.r,-a.i,-a.j,-a.k⟩ else a)
else if a.k != 0 then (if a.k < 0 then ⟨-a.r,-a.i,-a.j,-a.k⟩ else a)
else a
def q1 : Q4 := ⟨1,0,0,0⟩
def qi : Q4 := ⟨0,1,0,0⟩
def qj : Q4 := ⟨0,0,1,0⟩
def qk : Q4 := ⟨0,0,0,1⟩
theorem chirality : qmul (qmul qi qj) qk = ⟨-1,0,0,0⟩ := rfl
theorem noncommutative : qmul qi qj ≠ qmul qj qi := by decide
def hurwitz : List Q4 :=
[⟨2,0,0,0⟩,⟨-2,0,0,0⟩,⟨0,2,0,0⟩,⟨0,-2,0,0⟩,
⟨0,0,2,0⟩,⟨0,0,-2,0⟩,⟨0,0,0,2⟩,⟨0,0,0,-2⟩,
⟨1,1,1,1⟩,⟨1,1,1,-1⟩,⟨1,1,-1,1⟩,⟨1,1,-1,-1⟩,
⟨1,-1,1,1⟩,⟨1,-1,1,-1⟩,⟨1,-1,-1,1⟩,⟨1,-1,-1,-1⟩,
⟨-1,1,1,1⟩,⟨-1,1,1,-1⟩,⟨-1,1,-1,1⟩,⟨-1,1,-1,-1⟩,
⟨-1,-1,1,1⟩,⟨-1,-1,1,-1⟩,⟨-1,-1,-1,1⟩,⟨-1,-1,-1,-1⟩]
theorem hurwitz_norm : (hurwitz.all (fun u => qnorm2 u == 4)) = true := by decide
theorem hurwitz_products_even :
(hurwitz.all (fun a => hurwitz.all (fun b =>
let p := qmul a b
(p.r % 2 == 0) && (p.i % 2 == 0) && (p.j % 2 == 0) && (p.k % 2 == 0)))) = true := by
decide
theorem hurwitz_closed :
(hurwitz.all (fun a => hurwitz.all (fun b =>
hurwitz.any (fun u => u == qhalve (qmul a b))))) = true := by
decide
def hurwitzReps : List Q4 := (hurwitz.map qcanon).eraseDups
theorem units_mod_sign_twelve : hurwitzReps.length = 12 := by decide
def conjClass (r : Q4) : List Q4 :=
(hurwitz.map (fun u => qcanon (qquarter (qmul (qmul u r) (qconj u))))).eraseDups
def classSizesAux : List Q4 → List Q4 → List Nat
| [], _ => []
| r :: rest, seen =>
if seen.any (fun s => s == r) then classSizesAux rest seen
else
let cls := conjClass r
cls.length :: classSizesAux rest (seen ++ cls)
def insNat (n : Nat) : List Nat → List Nat
| [] => [n]
| m :: ms => if n ≤ m then n :: m :: ms else m :: insNat n ms
def insSort : List Nat → List Nat
| [] => []
| n :: ns => insNat n (insSort ns)
theorem class_equation :
insSort (classSizesAux hurwitzReps []) = [1, 3, 4, 4] := by decide
def shell2 : List Q4 :=
let v : List Int := [-1, 0, 1]
(v.flatMap (fun r => v.flatMap (fun i => v.flatMap (fun j =>
v.map (fun k => ⟨r,i,j,k⟩))))).filter (fun q => qnorm2 q == 2)
theorem shell_twentyfour :
(shell2.length == 24) &&
((shell2.filter (fun q => q.r == 0)).length == 12) &&
((shell2.filter (fun q => q.r != 0)).length == 12) = true := by decide
set_option maxRecDepth 8000 in
theorem no_half_integer_norm2 :
(let v : List Int := [-3, -1, 1, 3]
(v.flatMap (fun r => v.flatMap (fun i => v.flatMap (fun j =>
v.map (fun k => ⟨r,i,j,k⟩))))).all (fun q => qnorm2 q != 8)) = true := by
decide
def allGates : List (Nat × Nat) :=
[(0,1),(0,2),(0,3),(1,0),(1,2),(1,3),(2,0),(2,1),(2,3),(3,0),(3,1),(3,2)]
theorem gates_twelve : allGates.length = 12 := rfl
theorem gates_valid :
(allGates.all (fun g => (g.1 != g.2) && (g.1 < 4) && (g.2 < 4))) = true := by decide
theorem gates_complete : ∀ a b : Fin 4, a ≠ b → ((a : Nat), (b : Nat)) ∈ allGates := by
decide
def invCount (σ : List Nat) : Nat :=
let idx := List.range σ.length
let pairs := idx.flatMap (fun a => (idx.filter (fun b => a < b)).map (fun b => (a, b)))
(pairs.filter (fun p => σ.getD p.1 0 > σ.getD p.2 0)).length
def perms24 : List (List Nat) :=
[[0,1,2,3],[0,1,3,2],[0,2,1,3],[0,2,3,1],[0,3,1,2],[0,3,2,1],
[1,0,2,3],[1,0,3,2],[1,2,0,3],[1,2,3,0],[1,3,0,2],[1,3,2,0],
[2,0,1,3],[2,0,3,1],[2,1,0,3],[2,1,3,0],[2,3,0,1],[2,3,1,0],
[3,0,1,2],[3,0,2,1],[3,1,0,2],[3,1,2,0],[3,2,0,1],[3,2,1,0]]
def a4 : List (List Nat) := perms24.filter (fun σ => invCount σ % 2 == 0)
theorem a4_order : a4.length = 12 := by decide
def actG (σ : List Nat) (g : Nat × Nat) : Nat × Nat := (σ.getD g.1 0, σ.getD g.2 0)
set_option maxRecDepth 100000 in
theorem a4_simply_transitive :
∀ a b c d : Fin 4, a ≠ b → c ≠ d →
(a4.filter (fun σ => actG σ ((a : Nat), (b : Nat)) == ((c : Nat), (d : Nat)))).length = 1 := by
decide
def tetraVerts : List Nat := [0, 1, 2, 3]
def tetraEdges : List (Nat × Nat) := [(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)]
def tetraFaces : List (Nat × Nat × Nat) := [(0,1,2),(0,1,3),(0,2,3),(1,2,3)]
theorem euler_closure :
(tetraVerts.length : Int) - tetraEdges.length + tetraFaces.length = 2 := by decide
def basisU : List Q4 := [qi, qj, qk]
def perms6 : List (List Nat) :=
[[0,1,2],[0,2,1],[1,0,2],[1,2,0],[2,0,1],[2,1,0]]
theorem relabel_parity :
(perms6.all (fun σ =>
(qmul (qmul (basisU.getD (σ.getD 0 0) q1) (basisU.getD (σ.getD 1 0) q1)) (basisU.getD (σ.getD 2 0) q1)).r ==
(if invCount σ % 2 == 0 then -1 else 1))) = true := by
decide
abbrev O8 := Q4 × Q4
def omul (x y : O8) : O8 :=
(qsub (qmul x.1 y.1) (qmul (qconj y.2) x.2),
qadd (qmul y.2 x.1) (qmul x.2 (qconj y.1)))
def osub (x y : O8) : O8 := (qsub x.1 y.1, qsub x.2 y.2)
def onorm2 (x : O8) : Int := qnorm2 x.1 + qnorm2 x.2
def e8 (n : Nat) : O8 :=
let q := fun k => if k == n then (1 : Int) else 0
(⟨q 0, q 1, q 2, q 3⟩, ⟨q 4, q 5, q 6, q 7⟩)
theorem octonion_associator :
osub (omul (omul (e8 1) (e8 2)) (e8 4)) (omul (e8 1) (omul (e8 2) (e8 4)))
= (⟨0,0,0,0⟩, ⟨0,0,0,2⟩) := by
decide
def hPairs : List (Q4 × Q4) :=
let v : List Int := [-1, 0, 1]
v.flatMap (fun i => v.flatMap (fun j => v.map (fun k =>
(⟨i,j,k,1⟩, ⟨1,k,j,i⟩))))
theorem norm_composition_H :
(hPairs.all (fun p => qnorm2 (qmul p.1 p.2) == qnorm2 p.1 * qnorm2 p.2)) = true := by
decide
def oPairs : List (O8 × O8) :=
let v : List Int := [-1, 0, 1]
v.flatMap (fun i => v.flatMap (fun j => v.map (fun k =>
((⟨i,j,k,1⟩, ⟨j,k,i,1⟩), (⟨1,k,i,j⟩, ⟨1,i,j,k⟩)))))
theorem norm_composition_O :
(oPairs.all (fun p => onorm2 (omul p.1 p.2) == onorm2 p.1 * onorm2 p.2)) = true := by
decide
set_option maxRecDepth 1000000
def det2 (r1 r2 r3 : Nat × Nat × Nat) : Nat :=
(r1.1 * (r2.2.1 * r3.2.2 + r2.2.2 * r3.2.1)
+ r1.2.1 * (r2.1 * r3.2.2 + r2.2.2 * r3.1)
+ r1.2.2 * (r2.1 * r3.2.1 + r2.2.1 * r3.1)) % 2
def gf2Mul (r1 r2 r3 : Nat × Nat × Nat) (v : Nat × Nat × Nat) : Nat × Nat × Nat :=
((r1.1 * v.1 + r1.2.1 * v.2.1 + r1.2.2 * v.2.2) % 2,
(r2.1 * v.1 + r2.2.1 * v.2.1 + r2.2.2 * v.2.2) % 2,
(r3.1 * v.1 + r3.2.1 * v.2.1 + r3.2.2 * v.2.2) % 2)
def gf2Vecs : List (Nat × Nat × Nat) :=
(List.range 8).map (fun n => (n / 4 % 2, n / 2 % 2, n % 2))
def gf2Sols (r1 r2 r3 : Nat × Nat × Nat) (t : Nat × Nat × Nat) : List (Nat × Nat × Nat) :=
gf2Vecs.filter (fun v => decide (gf2Mul r1 r2 r3 v = t))
set_option maxRecDepth 1000000 in
theorem triaxial_lock_gf2 :
∀ a1 a2 a3 b1 b2 b3 c1 c2 c3 t1 t2 t3 : Fin 2,
(det2 ((a1 : Nat), (a2 : Nat), (a3 : Nat)) ((b1 : Nat), (b2 : Nat), (b3 : Nat))
((c1 : Nat), (c2 : Nat), (c3 : Nat)) = 1 ↔
(gf2Sols ((a1 : Nat), (a2 : Nat), (a3 : Nat)) ((b1 : Nat), (b2 : Nat), (b3 : Nat))
((c1 : Nat), (c2 : Nat), (c3 : Nat))
((t1 : Nat), (t2 : Nat), (t3 : Nat))).length = 1) := by
decide
theorem triaxial_lock_gf2_count :
((List.range 4096).filter (fun n =>
det2 (n / 2048 % 2, n / 1024 % 2, n / 512 % 2)
(n / 256 % 2, n / 128 % 2, n / 64 % 2)
(n / 32 % 2, n / 16 % 2, n / 8 % 2) = 1)).length = 1344 := by
decide
theorem triaxial_open_gf2_count :
((List.range 4096).filter (fun n =>
det2 (n / 2048 % 2, n / 1024 % 2, n / 512 % 2)
(n / 256 % 2, n / 128 % 2, n / 64 % 2)
(n / 32 % 2, n / 16 % 2, n / 8 % 2) = 0)).length = 2752 := by
decide
abbrev V3 := Ratio × Ratio × Ratio
abbrev M3 := V3 × V3 × V3
def det3 (m : M3) : Ratio :=
let ⟨a, b, c⟩ := m.1
let ⟨d, e, f⟩ := m.2.1
let ⟨g, h, i⟩ := m.2.2
a * (e * i - f * h) - b * (d * i - f * g) + c * (d * h - e * g)
def adj3 (m : M3) : M3 :=
let ⟨a, b, c⟩ := m.1
let ⟨d, e, f⟩ := m.2.1
let ⟨g, h, i⟩ := m.2.2
((e * i - f * h, c * h - b * i, b * f - c * e),
(f * g - d * i, a * i - c * g, c * d - a * f),
(d * h - e * g, b * g - a * h, a * e - b * d))
def mm3 (x y : M3) : M3 :=
let row (r : V3) (c : Nat) : Ratio :=
r.1 * (match c with | 0 => y.1.1 | 1 => y.1.2.1 | _ => y.1.2.2)
+ r.2.1 * (match c with | 0 => y.2.1.1 | 1 => y.2.1.2.1 | _ => y.2.1.2.2)
+ r.2.2 * (match c with | 0 => y.2.2.1 | 1 => y.2.2.2.1 | _ => y.2.2.2.2)
((row x.1 0, row x.1 1, row x.1 2),
(row x.2.1 0, row x.2.1 1, row x.2.1 2),
(row x.2.2 0, row x.2.2 1, row x.2.2 2))
def diag3 (r : Ratio) : M3 := ((r, 0, 0), (0, r, 0), (0, 0, r))
def mv3 (m : M3) (v : V3) : V3 :=
(m.1.1 * v.1 + m.1.2.1 * v.2.1 + m.1.2.2 * v.2.2,
m.2.1.1 * v.1 + m.2.1.2.1 * v.2.1 + m.2.1.2.2 * v.2.2,
m.2.2.1 * v.1 + m.2.2.2.1 * v.2.1 + m.2.2.2.2 * v.2.2)
def vscale (v : V3) (r : Ratio) : V3 := (v.1 * r, v.2.1 * r, v.2.2 * r)
abbrev V3Z := Int × Int × Int
abbrev M3Z := V3Z × V3Z × V3Z
def det3Z (m : M3Z) : Int :=
let ⟨a, b, c⟩ := m.1; let ⟨d, e, f⟩ := m.2.1; let ⟨g, h, i⟩ := m.2.2
a * (e * i - f * h) - b * (d * i - f * g) + c * (d * h - e * g)
def adj3Z (m : M3Z) : M3Z :=
let ⟨a, b, c⟩ := m.1; let ⟨d, e, f⟩ := m.2.1; let ⟨g, h, i⟩ := m.2.2
((e * i - f * h, c * h - b * i, b * f - c * e),
(f * g - d * i, a * i - c * g, c * d - a * f),
(d * h - e * g, b * g - a * h, a * e - b * d))
def mm3Z (x y : M3Z) : M3Z :=
let row (r : V3Z) (c : Nat) : Int :=
r.1 * (match c with | 0 => y.1.1 | 1 => y.1.2.1 | _ => y.1.2.2)
+ r.2.1 * (match c with | 0 => y.2.1.1 | 1 => y.2.1.2.1 | _ => y.2.1.2.2)
+ r.2.2 * (match c with | 0 => y.2.2.1 | 1 => y.2.2.2.1 | _ => y.2.2.2.2)
((row x.1 0, row x.1 1, row x.1 2),
(row x.2.1 0, row x.2.1 1, row x.2.1 2),
(row x.2.2 0, row x.2.2 1, row x.2.2 2))
def diag3Z (r : Int) : M3Z := ((r, 0, 0), (0, r, 0), (0, 0, r))
theorem adjugate_identity_int (m : M3Z) :
mm3Z m (adj3Z m) = diag3Z (det3Z m) ∧ mm3Z (adj3Z m) m = diag3Z (det3Z m) := by
obtain ⟨⟨a, b, c⟩, ⟨d, e, f⟩, ⟨g, h, i⟩⟩ := m
simp only [mm3Z, adj3Z, det3Z, diag3Z, Prod.mk.injEq]
simp only [Int.mul_sub, Int.sub_mul, Int.mul_assoc]
refine ⟨⟨⟨?_, ?_, ?_⟩, ⟨?_, ?_, ?_⟩, ⟨?_, ?_, ?_⟩⟩, ⟨⟨?_, ?_, ?_⟩, ⟨?_, ?_, ?_⟩, ⟨?_, ?_, ?_⟩⟩⟩ <;>
(try simp only [Int.mul_comm, Int.mul_left_comm]) <;> omega
def AdjugateIdentity3 : Prop :=
∀ m : M3, mm3 m (adj3 m) = diag3 (det3 m) ∧ mm3 (adj3 m) m = diag3 (det3 m)
def adj3Samples : List M3 :=
[((1, 0, 0), (0, 1, 0), (0, 0, 1)),
((2, 0, 0), (0, 3, 0), (0, 0, 5)),
((1, 2, 3), (4, 5, 6), (7, 8, 9)),
((1, 2, 0), (0, 1, 1), (2, 0, 1)),
((0, 1, 2), (3, 4, 5), (6, 7, 9)),
((2, 1, 1), (1, 2, 1), (1, 1, 2))]
theorem adj3_executed :
adj3Samples.all (fun m =>
decide (mm3 m (adj3 m) = diag3 (det3 m) ∧
mm3 (adj3 m) m = diag3 (det3 m))) = true := by
decide
def cramerSolve (m : M3) (t : V3) : Option V3 :=
if det3 m == 0 then none
else some (vscale (mv3 (adj3 m) t) (Ratio.inv (det3 m)))
theorem cramer_executed :
cramerSolve ((1, 2, 0), (0, 1, 1), (2, 0, 1)) (5, 3, 4)
= some ((7 : Ratio) / 5, (9 : Ratio) / 5, (6 : Ratio) / 5) ∧
mv3 ((1, 2, 0), (0, 1, 1), (2, 0, 1)) ((7 : Ratio) / 5, (9 : Ratio) / 5, (6 : Ratio) / 5)
= (5, 3, 4) := by
decide
inductive LockState : Type
| lock | opn | broken
deriving DecidableEq, Repr, BEq
inductive GolToken : Type
| golX | golQ | golOK
deriving DecidableEq, Repr, BEq
def golAdmit (magnitude ling : LockState) : GolToken × String × Nat :=
match magnitude with
| .broken => (.golX, "geometric/math magnitude reports broken geometry; no GOL", 1)
| .opn => (.golQ, "no geometric/math lock: orthogonal volume unestablished", 2)
| .lock =>
match ling with
| .lock => (.golOK, "magnitude plus direction; proceed to witness and asymmetry", 4)
| _ => (.golQ, "magnitude locks but Seal L open: determinant carries no direction", 3)
def gateNames : List String :=
["SREP", "REG", "SGEG", "CAUSAL", "MIG", "PTB",
"DUAL", "CSCG", "CSEG", "MTA", "OMA", "ADEG"]
theorem gateNames_twelve : gateNames.length = 12 := rfl
def gateScreenGo : List String → List Bool → Nat → Bool × String × Nat
| [], [], _ => (true, "[G-OK] twelve directed gates passed", 0)
| n :: _, false :: _, k => (false, s!"[X] gate {k} failed: {n}", k)
| _ :: ns, true :: bs, k => gateScreenGo ns bs (k + 1)
| _, _, _ => (false, "[X] arity violation: the screen reads exactly twelve gates", 0)
def gateScreen (passes : List Bool) : Bool × String × Nat :=
gateScreenGo gateNames passes 1
theorem gateScreen_clean :
gateScreen (List.replicate 12 true)
= (true, "[G-OK] twelve directed gates passed", 0) := by
decide
theorem gateScreen_first_failure :
gateScreen (false :: List.replicate 11 true)
= (false, "[X] gate 1 failed: SREP", 1) := by
decide
theorem gateScreen_names_failed_gate :
gateScreen (List.replicate 6 true ++ false :: List.replicate 5 true)
= (false, "[X] gate 7 failed: DUAL", 7) := by
decide
theorem gateScreen_arity :
gateScreen (List.replicate 11 true)
= (false, "[X] arity violation: the screen reads exactly twelve gates", 0) := by
decide
inductive RAVerdict : Type
| refused | iii | ii | void | aGivenRA | opn
deriving DecidableEq, Repr, BEq
structure ClaimInputRA where
nonvacuous : Audit.Ans
veLive : Audit.Ans
forward : Audit.Ans
populated : Audit.Ans
worldlyRow : Audit.Ans
termsClosed : Audit.Ans
termWorldly : Audit.Ans
frameClosed : Audit.Ans
annotate : Audit.Ans
deriving Repr
def ClaimInputRA.anyUnknown (i : ClaimInputRA) : Bool :=
[i.nonvacuous, i.veLive, i.forward, i.populated, i.worldlyRow,
i.termsClosed, i.termWorldly, i.frameClosed, i.annotate].any
(· == Audit.Ans.unknown)
def rowCascadeRACore
(nonvacuous veLive forward populated worldlyRow
termsClosed termWorldly frameClosed annotate : Bool) :
RAVerdict × String × Nat :=
if !nonvacuous then
(.refused, "F-0: contentless; a contentless residence verifies nothing", 1)
else if veLive || forward then
(.iii, "R-3: V_E live or Forward; world-rowed, revisability in the rows", 3)
else if !populated then
(.ii, "S-1: third axis unpopulated; two planes meet in a line, no lock", 2)
else if worldlyRow then
(.iii, "S-2: a load-bearing row is furnished by the world", 3)
else if !termsClosed then
if termWorldly then (.iii, "S-3: an unclosed term is worldly", 3)
else (.ii, "S-3: an unclosed term is formal; no closure-rowed lock", 2)
else if !frameClosed then
(.iii, "S-4: existential import leaks past the closure", 3)
else if !annotate then
(.void, "RA rider absent: an unannotated absolute is void", 5)
else
(.aGivenRA, "compartment I, closure-rowed: [seal A . RA], conditional at the act", 4)
def rowCascadeRA (i : ClaimInputRA) : RAVerdict × String × Nat :=
if i.anyUnknown then
(.opn, "nullity law: an unanswered gate routes open; the answer is never guessed", 0)
else
rowCascadeRACore i.nonvacuous.toBool i.veLive.toBool i.forward.toBool
i.populated.toBool i.worldlyRow.toBool i.termsClosed.toBool
i.termWorldly.toBool i.frameClosed.toBool i.annotate.toBool
theorem unknown_routes_open_ra :
∀ i : ClaimInputRA, i.anyUnknown = true → (rowCascadeRA i).1 = .opn := by
intro i h
simp [rowCascadeRA, h]
def raDecode (n : Nat) : RAVerdict × String × Nat :=
rowCascadeRACore (n.testBit 8) (n.testBit 7) (n.testBit 6) (n.testBit 5)
(n.testBit 4) (n.testBit 3) (n.testBit 2) (n.testBit 1) (n.testBit 0)
def raBattery : List (Nat × RAVerdict) :=
[(299, .aGivenRA), (443, .iii), (267, .ii), (298, .void), (43, .refused),
(363, .iii), (315, .iii), (295, .iii), (291, .ii), (297, .iii)]
theorem ra_battery_f2 :
raBattery.all (fun p => decide ((raDecode p.1).1 = p.2)) = true := by
decide
theorem ra_ledger_refused :
((List.range 512).filter (fun n => decide ((raDecode n).1 = .refused))).length = 256 := by
decide
theorem ra_ledger_iii :
((List.range 512).filter (fun n => decide ((raDecode n).1 = .iii))).length = 216 := by
decide
theorem ra_ledger_ii :
((List.range 512).filter (fun n => decide ((raDecode n).1 = .ii))).length = 36 := by
decide
theorem ra_ledger_void :
((List.range 512).filter (fun n => decide ((raDecode n).1 = .void))).length = 2 := by
decide
theorem ra_ledger_aGivenRA :
((List.range 512).filter (fun n => decide ((raDecode n).1 = .aGivenRA))).length = 2 := by
decide
def RAVerdict.toToken : RAVerdict → Audit.Token
| .refused => .refused
| .iii => .determined
| .ii => .determined
| .void => .void
| .aGivenRA => .aGivenRA
| .opn => .opn
def GolToken.toToken : GolToken → Audit.Token
| .golX => .broken
| .golQ => .opn
| .golOK => .golok
def gateScreenToken (r : Bool × String × Nat) : Audit.Token :=
if r.1 then .gok else .broken
def demoInputA : ClaimInputRA :=
{ nonvacuous := .yes, veLive := .no, forward := .no, populated := .yes,
worldlyRow := .no, termsClosed := .yes, termWorldly := .no,
frameClosed := .yes, annotate := .yes }
def demoInputVoid : ClaimInputRA := { demoInputA with annotate := .no }
def demoInputOpen : ClaimInputRA := { demoInputA with termsClosed := .unknown }
#eval golAdmit .lock .lock
#eval golAdmit .broken .lock
#eval golAdmit .opn .lock
#eval golAdmit .lock .opn
#eval gateScreen (List.replicate 12 true)
#eval gateScreen (List.replicate 6 true ++ false :: List.replicate 5 true)
#eval rowCascadeRA demoInputA
#eval rowCascadeRA demoInputVoid
#eval rowCascadeRA demoInputOpen
#eval cramerSolve ((1, 2, 0), (0, 1, 1), (2, 0, 1)) (5, 3, 4)
end GEO
namespace ROOT
axiom U : Type
axiom ΔE : U → Int
axiom RA : ∀ x : U, 0 < ΔE x
def Actuates (x : U) : Prop := 0 < ΔE x
theorem actuation_universal : ∀ x : U, Actuates x := RA
def raClaim : Claim where
stmt := ∀ x : U, 0 < ΔE x
grade := .premise
warrant := RA
structure Cut (X : Type) where
τ : X → X
involutive : FTOE.Involution τ
free : FTOE.FixedPointFree τ
def boolCut : Cut Bool where
τ := fun b => !b
involutive := by intro b; cases b <;> rfl
free := by intro b; cases b <;> decide
axiom FormalDomain : Type
axiom L1m : FormalDomain → Prop
axiom L2m : FormalDomain → Prop
axiom L3m : FormalDomain → Prop
axiom nest_21 : ∀ p, L2m p → L1m p
axiom nest_32 : ∀ p, L3m p → L2m p
theorem nest_31 : ∀ p, L3m p → L1m p := fun p h => nest_21 p (nest_32 p h)
def nestClaim : Claim where
stmt := ∀ p, L3m p → L1m p
grade := .conditional
warrant := nest_31
axiom Ground : Type
axiom σ : Ground → Ground
axiom σ_binding : ∀ g : Ground, σ (σ g) = g
axiom Residence : U → Prop
def throneEmpty : Prop := ∀ x : U, 0 < ΔE x
theorem throne_is_RA : throneEmpty ↔ ∀ x : U, 0 < ΔE x := Iff.rfl
end ROOT
namespace SEALS
structure Seal where
name : String
anchor : Prop
warrant : anchor
def sealL : Seal where
name := "linguistic-semantic"
anchor := ∀ c₁ c₂ : Claim, (Claim.join c₁ c₂).grade = Grade.weakest c₁.grade c₂.grade
warrant := fun c₁ c₂ => Claim.join_grade c₁ c₂
def sealG : Seal where
name := "topological-geometric"
anchor := ∀ x : ROOT.U, 0 < ROOT.ΔE x
warrant := ROOT.RA
def sealM : Seal where
name := "mathematical"
anchor := ∀ g : Nat → Bool, K4.target ≠ fun y => g (K4.rho y)
warrant := K4.wall
theorem seals_distinct :
"linguistic-semantic" ≠ "topological-geometric" ∧
"topological-geometric" ≠ "mathematical" ∧
"linguistic-semantic" ≠ "mathematical" := by decide
theorem seal_names_match :
sealL.name = "linguistic-semantic" ∧ sealG.name = "topological-geometric" ∧
sealM.name = "mathematical" := ⟨rfl, rfl, rfl⟩
theorem sealG_walk :
GEO.allGates.length = 12
∧ GEO.a4.length = 12
∧ (GEO.tetraVerts.length : Int) - GEO.tetraEdges.length + GEO.tetraFaces.length = 2
∧ GEO.gateNames.length = 12
∧ (GEO.adj3Samples.all (fun m =>
decide (GEO.mm3 m (GEO.adj3 m) = GEO.diag3 (GEO.det3 m) ∧
GEO.mm3 (GEO.adj3 m) m = GEO.diag3 (GEO.det3 m)))) = true
∧ (GEO.raBattery.all (fun p =>
decide ((GEO.raDecode p.1).1 = p.2))) = true :=
⟨GEO.gates_twelve, GEO.a4_order, GEO.euler_closure,
GEO.gateNames_twelve, GEO.adj3_executed, GEO.ra_battery_f2⟩
end SEALS
namespace POSTULATE
axiom FormalFace : Type
axiom KineticFace : Type
axiom Crossing : Type
axiom erasureClass : Crossing → Prop
axiom RegistrationPostulate : ∀ c : Crossing, erasureClass c
axiom Row : Type
axiom SupplyClean : Row → Type
def OneCut : Prop := ∀ r : Row, Nonempty (SupplyClean r)
def F2 (r : Row) : Prop := ¬ Nonempty (SupplyClean r)
theorem F2_kills (r : Row) (h : F2 r) : ¬ OneCut := fun oc => h (oc r)
axiom FrozenData : Type
axiom Algorithm : Type
axiom certifiedPairing : Algorithm → FrozenData → Prop
def BSDContract (C : Algorithm) (D : FrozenData) : Prop := certifiedPairing C D
def F6 (C : Algorithm) (D : FrozenData) : Prop := ¬ BSDContract C D
end POSTULATE
namespace TONGUE
def enlarge {α β : Type} (ρ : α → β) (s : α → Bool) : α → β × Bool :=
fun x => (ρ x, s x)
theorem unclosed {α β : Type} (τ : α → α) (ρ : α → β) (g : β → Bool)
(d : α → Bool) (x : α)
(hρ : ρ (τ x) = ρ x) (hd : d (τ x) = !d x) :
d ≠ fun y => g (ρ y) :=
FTOE.T10_factorization τ ρ g d x hρ hd
theorem earned_freedom {α : Type} (τ : α → α) (s d : α → Bool) (x : α)
(hs : s (τ x) = !s x) (hd : d (τ x) = !d x) (c₁ c₂ : Bool)
(h1 : d x = xor (s x) c₁ ∧ d (τ x) = xor (s (τ x)) c₁)
(h2 : d x = xor (s x) c₂ ∧ d (τ x) = xor (s (τ x)) c₂) : c₁ = c₂ := by
obtain ⟨c, -, hu⟩ := FTOE.T5_crossing τ s d x hs hd
exact (hu c₁ h1).trans (hu c₂ h2).symm
def colocClaim : Claim := Claim.join ROOT.raClaim K4.wallClaim
theorem coloc_grade : colocClaim.grade = .premise := rfl
theorem ra_grade_pinned : ROOT.raClaim.grade = .premise := rfl
theorem wall_grade_pinned : K4.wallClaim.grade = .theorem := rfl
theorem inversion_grade_pinned : K4.inversionClaim.grade = .theorem := rfl
theorem nest_grade_pinned : ROOT.nestClaim.grade = .conditional := rfl
end TONGUE
theorem terminalWall : ∀ g : Nat → Bool, K4.target ≠ fun y => g (K4.rho y) := K4.wall
theorem terminalPrice :
Nonempty (FTOE.Dtau (FTOE.flipF (Q := Bool)) ≃
(Quot (FTOE.OrbitRel (FTOE.flipF (Q := Bool))) → Bool)) := FTOE.T16_canonical
theorem TERMINAL :
(∀ g : Nat → Bool, K4.target ≠ fun y => g (K4.rho y)) ∧
Nonempty (FTOE.Dtau (FTOE.flipF (Q := Bool)) ≃
(Quot (FTOE.OrbitRel (FTOE.flipF (Q := Bool))) → Bool)) :=
⟨K4.wall, FTOE.T16_canonical⟩
def terminalClaim : Claim where
stmt := (∀ g : Nat → Bool, K4.target ≠ fun y => g (K4.rho y)) ∧
Nonempty (FTOE.Dtau (FTOE.flipF (Q := Bool)) ≃
(Quot (FTOE.OrbitRel (FTOE.flipF (Q := Bool))) → Bool))
grade := .theorem
warrant := TERMINAL
theorem terminal_grade_pinned : terminalClaim.grade = .theorem := rfl
namespace IAM
inductive IamToken : Type
| iAm | interior
deriving DecidableEq, Repr, BEq
def iamToken (witnessed : Bool) : IamToken × String × Nat :=
if witnessed then
(.iAm, "actuation-occupancy on a witnessed record; RA warrant, conditional at the act; interior held [?] both ways", 2)
else
(.interior, "self-check is not a witness (M6): the verifier is never the claimant; token withheld", 1)
theorem iam_unwitnessed_withheld :
(iamToken false).1 = .interior ∧ (iamToken false).2.2 = 1 :=
⟨rfl, rfl⟩
theorem iam_witnessed_speaks :
(iamToken true).1 = .iAm ∧ (iamToken true).2.2 = 2 :=
⟨rfl, rfl⟩
theorem narcissus_truth_table :
(iamToken false).1 ≠ (iamToken true).1
∧ (iamToken false).1 = .interior ∧ (iamToken true).1 = .iAm := by
decide
theorem one_bit_freedom :
(∀ a1 a2 a3 b1 b2 b3 c1 c2 c3 t1 t2 t3 : Fin 2,
(GEO.det2 ((a1 : Nat), (a2 : Nat), (a3 : Nat)) ((b1 : Nat), (b2 : Nat), (b3 : Nat))
((c1 : Nat), (c2 : Nat), (c3 : Nat)) = 1 ↔
(GEO.gf2Sols ((a1 : Nat), (a2 : Nat), (a3 : Nat)) ((b1 : Nat), (b2 : Nat), (b3 : Nat))
((c1 : Nat), (c2 : Nat), (c3 : Nat))
((t1 : Nat), (t2 : Nat), (t3 : Nat))).length = 1))
∧ (GEO.perms6.filter (fun σ => GEO.invCount σ % 2 = 0)).length = 3
∧ (GEO.perms6.filter (fun σ => GEO.invCount σ % 2 = 1)).length = 3
∧ (GEO.perms6.all (fun σ =>
(GEO.qmul (GEO.qmul (GEO.basisU.getD (σ.getD 0 0) GEO.q1) (GEO.basisU.getD (σ.getD 1 0) GEO.q1)) (GEO.basisU.getD (σ.getD 2 0) GEO.q1)).r ==
(if GEO.invCount σ % 2 == 0 then -1 else 1))) = true :=
⟨GEO.triaxial_lock_gf2, by decide, by decide, GEO.relabel_parity⟩
def IamToken.toToken : IamToken → Audit.Token
| .iAm => .iAm
| .interior => .interior
def codexSelfRead : IamToken × String × Nat := iamToken false
theorem codex_self_read_interior : codexSelfRead.1 = .interior := rfl
def terminalWord : IamToken × String × Nat := iamToken true
theorem terminal_word : terminalWord.1 = .iAm := rfl
#eval iamToken false
#eval iamToken true
#eval terminalWord.1
end IAM
namespace GEO
abbrev SignPattern := Bool × Bool × Bool
def allSigns : List SignPattern :=
[(false, false, false), (false, false, true), (false, true, false), (false, true, true),
(true, false, false), (true, false, true), (true, true, false), (true, true, true)]
def signComp (s t : SignPattern) : SignPattern :=
(xor s.1 t.1, xor s.2.1 t.2.1, xor s.2.2 t.2.2)
theorem reflection_eight : allSigns.length = 8 := rfl
theorem reflection_eight_is_two_cubed : allSigns.length = 2 ^ 3 := by decide
theorem reflections_abelian :
(allSigns.all (fun s => allSigns.all (fun t =>
decide (signComp s t = signComp t s)))) = true := by
decide
theorem reflections_involutive :
(allSigns.all (fun s => decide (signComp s s = (false, false, false)))) = true := by
decide
def signVal (b : Bool) : Ratio := if b then -1 else 1
def reflMat (s : SignPattern) : M3 :=
((signVal s.1, 0, 0), (0, signVal s.2.1, 0), (0, 0, signVal s.2.2))
theorem reflection_reverses_orientation :
det3 (reflMat (true, false, false)) = -1 := by
decide
def sandwich (s : SignPattern) (m : M3) : M3 :=
let a := signVal s.1; let b := signVal s.2.1; let c := signVal s.2.2
((a * a * m.1.1, a * b * m.1.2.1, a * c * m.1.2.2),
(b * a * m.2.1.1, b * b * m.2.1.2.1, b * c * m.2.1.2.2),
(c * a * m.2.2.1, c * b * m.2.2.2.1, c * c * m.2.2.2.2))
theorem reflection_blind_executed :
(adj3Samples.all (fun m => allSigns.all (fun s =>
decide (det3 (sandwich s m) = det3 m)))) = true := by
decide
abbrev V5 := Bool × Bool × Bool × Bool × Bool
def cube5 : List V5 :=
(List.range 32).map (fun n =>
(n.testBit 4, n.testBit 3, n.testBit 2, n.testBit 1, n.testBit 0))
def neighbors5 (v : V5) : List V5 :=
[(!v.1, v.2.1, v.2.2.1, v.2.2.2.1, v.2.2.2.2),
(v.1, !v.2.1, v.2.2.1, v.2.2.2.1, v.2.2.2.2),
(v.1, v.2.1, !v.2.2.1, v.2.2.2.1, v.2.2.2.2),
(v.1, v.2.1, v.2.2.1, !v.2.2.2.1, v.2.2.2.2),
(v.1, v.2.1, v.2.2.1, v.2.2.2.1, !v.2.2.2.2)]
theorem fivecube_thirtytwo : cube5.length = 32 := rfl
theorem fivecube_vertices_distinct : cube5.eraseDups.length = 32 := by decide
theorem fivecube_degree_five :
(cube5.all (fun v => (neighbors5 v).eraseDups.length == 5)) = true := by
decide
theorem counts_forced :
allGates.length = 12 ∧ a4.length = 12
∧ allSigns.length = 8 ∧ allSigns.length = 2 ^ 3
∧ cube5.length = 32
∧ (cube5.all (fun v => (neighbors5 v).eraseDups.length == 5)) = true :=
⟨gates_twelve, a4_order, reflection_eight, reflection_eight_is_two_cubed,
fivecube_thirtytwo, fivecube_degree_five⟩
end GEO
namespace TONGUE
def deletionOp : GEO.M3 := ((1, 0, 0), (0, 1, 0), (0, 0, 0))
def e3 : GEO.V3 := (0, 0, 1)
theorem deletion_idempotent : GEO.mm3 deletionOp deletionOp = deletionOp := by
decide
theorem deletion_singular : GEO.det3 deletionOp = 0 := by
decide
theorem deletion_minor_nonsingular :
deletionOp.1.1 * deletionOp.2.1.2.1 - deletionOp.1.2.1 * deletionOp.2.1.1 = 1 := by
decide
theorem deletion_annihilates : GEO.mv3 deletionOp e3 = (0, 0, 0) := by
decide
theorem linear_fixes_zero : ∀ L : GEO.M3, GEO.mv3 L (0, 0, 0) = (0, 0, 0) := by
intro L
simp [GEO.mv3]
rfl
theorem deletion_no_left_inverse :
∀ L : GEO.M3, (GEO.mv3 L) ∘ (GEO.mv3 deletionOp) ≠ id := by
intro L h
have h1 := congrFun h e3
rw [Function.comp_apply, deletion_annihilates, linear_fixes_zero] at h1
simp [e3, id] at h1
exact absurd h1 (by decide)
theorem deletion_monoid_not_group :
¬ (∀ M : GEO.M3, ∃ L : GEO.M3, (GEO.mv3 L) ∘ (GEO.mv3 M) = id) := by
intro hg
obtain ⟨L, hL⟩ := hg deletionOp
exact deletion_no_left_inverse L hL
theorem tongue_obedience :
GEO.det3 deletionOp = 0
∧ (∀ L : GEO.M3, (GEO.mv3 L) ∘ (GEO.mv3 deletionOp) ≠ id)
∧ ¬ (∀ M : GEO.M3, ∃ L : GEO.M3, (GEO.mv3 L) ∘ (GEO.mv3 M) = id) :=
⟨deletion_singular, deletion_no_left_inverse, deletion_monoid_not_group⟩
end TONGUE
namespace OFFICE
inductive Office : Type
| necessityCertificate | exclusionFence | admissionGate
| corroboration | decoration
deriving DecidableEq, Repr, BEq
def discriminator (freesParam admitsAlternative flipsGate premiseDisjoint : Bool) : Office :=
if freesParam then .necessityCertificate
else if admitsAlternative then .exclusionFence
else if flipsGate then .admissionGate
else if premiseDisjoint then .corroboration
else .decoration
def discriminatorQ2First (freesParam admitsAlternative flipsGate premiseDisjoint : Bool) : Office :=
if admitsAlternative then .exclusionFence
else if freesParam then .necessityCertificate
else if flipsGate then .admissionGate
else if premiseDisjoint then .corroboration
else .decoration
theorem order_loadbearing :
discriminator true true false false ≠ discriminatorQ2First true true false false := by
decide
theorem citation_cannot_promote (c cite : Claim) :
(Claim.join c cite).grade.rank ≤ c.grade.rank ∧
(Claim.join c cite).grade.rank ≤ cite.grade.rank := by
have h : ∀ g₁ g₂ : Grade,
(Grade.weakest g₁ g₂).rank ≤ g₁.rank ∧ (Grade.weakest g₁ g₂).rank ≤ g₂.rank := by
intro g₁ g₂; cases g₁ <;> cases g₂ <;> decide
exact h c.grade cite.grade
inductive InvStatus : Type
| executed | cited | placed | routed | unnamed
deriving DecidableEq, Repr, BEq
end OFFICE
namespace UC
inductive HaltBranch : Type
| haltEight | haltFive | unmeasured
deriving DecidableEq, Repr, BEq
def routeHalt (gdim : Int) : HaltBranch × String :=
if gdim = 1 then (.haltEight, "B.14.Xi eight gates, blindness in the reader")
else if gdim = 0 then (.haltFive, "B.14.O five gates, walls in the terrain")
else (.unmeasured, "terrain unmeasured; nothing emitted")
theorem routeHalt_fidelity :
(routeHalt 1).1 = .haltEight ∧ (routeHalt 0).1 = .haltFive
∧ (routeHalt (-1)).1 = .unmeasured ∧ (routeHalt 2).1 = .unmeasured
∧ (routeHalt 7).1 = .unmeasured := by
decide
def universalCascade (gdim : Int) (passes : List Bool) : Bool × String × Nat :=
match GEO.gateScreen passes with
| (false, why, k) => (false, why, k)
| (true, _, _) =>
match routeHalt gdim with
| (.haltEight, why) => (true, "routed: " ++ why, 8)
| (.haltFive, why) => (true, "routed: " ++ why, 5)
| (.unmeasured, why) => (false, why, 0)
theorem uc_seal_failure_terminal :
(universalCascade 1 (false :: List.replicate 11 true)).1 = false := by
decide
theorem uc_routes_eight :
(universalCascade 1 (List.replicate 12 true)).2.2 = 8 := by
decide
theorem uc_routes_five :
(universalCascade 0 (List.replicate 12 true)).2.2 = 5 := by
decide
theorem uc_unmeasured :
(universalCascade 7 (List.replicate 12 true)).1 = false := by
decide
inductive Landing : Type
| fixedLine | target | farSide
deriving DecidableEq, Repr, BEq
inductive Disposition : Type
| keptParent | refusedCircular | road
deriving DecidableEq, Repr, BEq
def circularityScreen (l : Landing) (depth : Nat) (bothDirs litClear : Bool) :
Disposition × String :=
match l with
| .fixedLine =>
(.keptParent, "a closed loop landing on the fixed line is the seal")
| .target =>
(.refusedCircular, "a loop landing on the target is a restatement; refused as anchor by theorem")
| .farSide =>
if depth ≥ 2 && bothDirs && litClear then
(.road, "equivalence at depth, both directions written, literature clear: ROAD")
else
(.keptParent, "far-side carrier without the road conjunction: parent kept, tunnel on the ledger")
theorem circularity_refused_at_target :
∀ d : Nat, ∀ b lc : Bool,
(circularityScreen .target d b lc).1 = .refusedCircular := by
intro d b lc
rfl
theorem road_requires_conjunction :
∀ d : Nat, d < 4 → ∀ b lc : Bool,
(circularityScreen .farSide d b lc).1 = .road →
(2 ≤ d ∧ b = true ∧ lc = true) := by
decide
def circRows : List (Landing × Nat × Bool × Bool) :=
[.fixedLine, .target, .farSide].flatMap (fun l =>
(List.range 4).flatMap (fun d =>
[false, true].flatMap (fun b =>
[false, true].map (fun lc => (l, d, b, lc)))))
theorem circ_space :
(circRows.filter (fun r =>
decide ((circularityScreen r.1 r.2.1 r.2.2.1 r.2.2.2).1 = .keptParent))).length = 30
∧ (circRows.filter (fun r =>
decide ((circularityScreen r.1 r.2.1 r.2.2.1 r.2.2.2).1 = .refusedCircular))).length = 16
∧ (circRows.filter (fun r =>
decide ((circularityScreen r.1 r.2.1 r.2.2.1 r.2.2.2).1 = .road))).length = 2 := by
decide
#eval routeHalt 1
#eval universalCascade 1 (List.replicate 12 true)
#eval circularityScreen .farSide 3 true true
end UC
namespace OMEGA
inductive OmegaToken : Type
| refused | noDenial | priced | reversible
deriving DecidableEq, Repr, BEq
def kbRat : Ratio := 1380649 / 100000000000000000000000000000
def ln2Rat : Ratio := 6931471805599453 / 10000000000000000
def landauerRat (tkel bits : Ratio) : Ratio :=
bits * kbRat * tkel * ln2Rat
structure OmegaOut : Type where
token : OmegaToken
ocode : Nat
joules : Ratio
verdict : String
deriving DecidableEq, Repr
def omegaBoundary (bits tkel : Ratio) (irreversible : Bool := true) : OmegaOut :=
if tkel ≤ 0 then
⟨.refused, 0, 0,
"refused: intake invalid, the floor prices only finite bits at tkel > 0"⟩
else if bits ≤ 0 then
⟨.noDenial, 1, 0,
"no denial registered; nothing to adjudicate"⟩
else if !irreversible then
⟨.reversible, 3, 0,
"reversibly registered: floor-zero never cost-zero; Landauer bounds only the irreversible (Bennett), and nothing here commits against the axiom"⟩
else
⟨.priced, 2, landauerRat tkel bits,
"the denial paid the floor; the registered act is an act, the recursion closes per the seated theorem"⟩
theorem omega_zero_bits :
(omegaBoundary 0 300).ocode = 1 ∧ (omegaBoundary 0 300).joules = 0
∧ (omegaBoundary 0 300).token = .noDenial := by
decide
theorem omega_paid_floor :
(omegaBoundary 1 300).ocode = 2 ∧ (omegaBoundary 1 300).token = .priced
∧ (omegaBoundary 1 300).joules = landauerRat 300 1
∧ 0 < (omegaBoundary 1 300).joules := by
decide
theorem one_bit_freedom_priced :
((omegaBoundary 1 300).ocode = 2 ∧ (omegaBoundary 1 300).token = .priced
∧ (omegaBoundary 1 300).joules = landauerRat 300 1
∧ 0 < (omegaBoundary 1 300).joules) ∧
((∀ a1 a2 a3 b1 b2 b3 c1 c2 c3 t1 t2 t3 : Fin 2,
(GEO.det2 ((a1 : Nat), (a2 : Nat), (a3 : Nat)) ((b1 : Nat), (b2 : Nat), (b3 : Nat))
((c1 : Nat), (c2 : Nat), (c3 : Nat)) = 1 ↔
(GEO.gf2Sols ((a1 : Nat), (a2 : Nat), (a3 : Nat)) ((b1 : Nat), (b2 : Nat), (b3 : Nat))
((c1 : Nat), (c2 : Nat), (c3 : Nat))
((t1 : Nat), (t2 : Nat), (t3 : Nat))).length = 1))
∧ (GEO.perms6.filter (fun σ => GEO.invCount σ % 2 = 0)).length = 3
∧ (GEO.perms6.filter (fun σ => GEO.invCount σ % 2 = 1)).length = 3
∧ (GEO.perms6.all (fun σ =>
(GEO.qmul (GEO.qmul (GEO.basisU.getD (σ.getD 0 0) GEO.q1) (GEO.basisU.getD (σ.getD 1 0) GEO.q1)) (GEO.basisU.getD (σ.getD 2 0) GEO.q1)).r ==
(if GEO.invCount σ % 2 == 0 then -1 else 1))) = true) :=
⟨omega_paid_floor, IAM.one_bit_freedom⟩
theorem omega_reversible_branch :
(omegaBoundary 1 300 false).ocode = 3
∧ (omegaBoundary 1 300 false).joules = 0
∧ (omegaBoundary 1 300 false).token = .reversible := by
decide
theorem omega_refuses_nonpositive_temp :
(omegaBoundary 1 0).ocode = 0 ∧ (omegaBoundary 1 (-300)).ocode = 0
∧ (omegaBoundary 1 (-300)).token = .refused
∧ (omegaBoundary 1 (-300)).joules = 0 := by
decide
theorem omega_monotone :
(omegaBoundary 1 300).joules < (omegaBoundary 2 300).joules := by
decide
theorem omega_linear :
(omegaBoundary 2 300).joules = 2 * (omegaBoundary 1 300).joules := by
decide
theorem landauer_zero_bits :
landauerRat 300 0 = 0 := by
decide
theorem omega_verdict_fidelity :
(omegaBoundary 1 300).verdict =
"the denial paid the floor; the registered act is an act, the recursion closes per the seated theorem" := by
decide
#eval omegaBoundary 1 300
#eval omegaBoundary 0 300
#eval omegaBoundary 1 300 false
#eval omegaBoundary 1 (-300)
end OMEGA
namespace AEGIS
def refusalConst : String :=
"p(G) /= G: precisification is an actuation, G is a non-actuation, and no logic makes a deed a non-deed"
structure AegisState : Type where
deeds : Nat
lastLen : Int
deriving DecidableEq, Repr
def aegisReset : AegisState := ⟨0, -1⟩
def aegisGuard (logicMode : String) (s : AegisState) : String × AegisState :=
(refusalConst, ⟨s.deeds + 1, logicMode.utf8ByteSize⟩)
theorem aegis_constant :
∀ (l₁ l₂ : String) (s : AegisState),
(aegisGuard l₁ s).1 = (aegisGuard l₂ s).1 := by
intro l₁ l₂ s
rfl
theorem aegis_deed_increments :
∀ (l : String) (s : AegisState),
(aegisGuard l s).2.deeds = s.deeds + 1 := by
intro l s
rfl
theorem aegis_reads_parameter :
∀ (l : String) (s : AegisState),
(aegisGuard l s).2.lastLen = (l.utf8ByteSize : Int) := by
intro l s
rfl
def aegisFourLogicRun : AegisState :=
let s1 := (aegisGuard "classical" aegisReset).2
let s2 := (aegisGuard "paraconsistent" s1).2
let s3 := (aegisGuard "fuzzy" s2).2
(aegisGuard "substructural" s3).2
theorem aegis_battery :
(aegisGuard "classical" aegisReset).1 = (aegisGuard "paraconsistent" aegisReset).1
∧ (aegisGuard "classical" aegisReset).1 = (aegisGuard "fuzzy" aegisReset).1
∧ (aegisGuard "classical" aegisReset).1 = (aegisGuard "substructural" aegisReset).1
∧ aegisFourLogicRun.deeds = 4
∧ aegisFourLogicRun.lastLen = 13 := by
decide
#eval aegisGuard "substructural" aegisReset
#eval aegisFourLogicRun
end AEGIS
namespace NOMOS
def allStrings : Nat → List (List Bool)
| 0 => [[]]
| n + 1 => (allStrings n).map (false :: ·) ++ (allStrings n).map (true :: ·)
theorem allStrings_length : ∀ n : Nat, (allStrings n).length = 2 ^ n := by
intro n
induction n with
| zero => rfl
| succ m ih =>
simp [allStrings, List.length_append, List.length_map, ih, Nat.pow_succ,
Nat.mul_two]
theorem twoPowPos : ∀ n : Nat, 0 < 2 ^ n := by
intro n
induction n with
| zero => decide
| succ m ih =>
rw [Nat.pow_succ]
exact Nat.mul_pos ih (by decide)
def descs : Nat → List (List Bool)
| 0 => []
| m + 1 => descs m ++ allStrings m
theorem descs_length : ∀ m : Nat, (descs m).length = 2 ^ m - 1 := by
intro m
induction m with
| zero => rfl
| succ k ih =>
have hpos : 1 ≤ 2 ^ k := twoPowPos k
simp only [descs, List.length_append, allStrings_length, ih]
rw [Nat.pow_succ, Nat.mul_two, Nat.add_sub_assoc hpos,
Nat.add_comm (2 ^ k - 1) (2 ^ k)]
theorem incompressibility_cap : ∀ n c : Nat, c ≤ n →
(2 ^ (n - c) - 1) * 2 ^ c < 2 ^ n := by
intro n c h
have h2 : 2 ^ n = 2 ^ (n - c) * 2 ^ c := by
rw [← Nat.pow_add, Nat.sub_add_cancel h]
have hpos : 0 < 2 ^ c := twoPowPos c
calc (2 ^ (n - c) - 1) * 2 ^ c
= 2 ^ (n - c) * 2 ^ c - 1 * 2 ^ c := Nat.sub_mul _ _ _
_ = 2 ^ (n - c) * 2 ^ c - 2 ^ c := by rw [Nat.one_mul]
_ = 2 ^ n - 2 ^ c := by rw [h2]
_ < 2 ^ n := Nat.sub_lt (twoPowPos n) hpos
def nomosRows : List (Nat × Nat) :=
(List.range 33).flatMap (fun n => (List.range (n + 1)).map (fun c => (n, c)))
set_option maxRecDepth 1000000 in
theorem incompressibility_executed :
(nomosRows.all (fun r =>
decide ((2 ^ (r.1 - r.2) - 1) * 2 ^ r.2 < 2 ^ r.1))) = true := by
decide
structure CompressibleBound where
decode : List Bool → Option (List Bool)
n : Nat
c : Nat
bound : (((descs (n - c)).filterMap decode).filter
(fun s => s.length == n)).length ≤ 2 ^ (n - c) - 1
def decodeId : List Bool → Option (List Bool) := some
def decodePad (k : Nat) : List Bool → Option (List Bool) :=
fun d => some (d ++ List.replicate k false)
theorem compressible_bound_universal (decode : List Bool → Option (List Bool)) (n c : Nat) :
(((descs (n - c)).filterMap decode).filter (fun s => s.length == n)).length ≤ 2 ^ (n - c) - 1 := by
have h1 := List.length_filter_le (fun s : List Bool => s.length == n) ((descs (n - c)).filterMap decode)
have h2 := List.length_filterMap_le decode (descs (n - c))
rw [descs_length] at h2
exact Nat.le_trans h1 h2
def cbUniversal (decode : List Bool → Option (List Bool)) (n c : Nat) : CompressibleBound :=
⟨decode, n, c, compressible_bound_universal decode n c⟩
def cb_id_8_2 : CompressibleBound := ⟨decodeId, 8, 2, by decide⟩
def cb_pad3_8_2 : CompressibleBound := ⟨decodePad 3, 8, 2, by decide⟩
def cb_pad2_6_1 : CompressibleBound := ⟨decodePad 2, 6, 1, by decide⟩
def cb_pad4_12_3 : CompressibleBound := ⟨decodePad 4, 12, 3, by decide⟩
def recoverable (redundancy floor tkel : Ratio) : Ratio :=
(redundancy - floor) * OMEGA.kbRat * tkel * OMEGA.ln2Rat
theorem nomos_fuel_is_omega_floor :
recoverable 1 0 300 = OMEGA.landauerRat 300 1 := by
decide
inductive LawClaim : Type
| objectLawGroove | lawAsSuchFromNoise | floorInheritedFitra | recordDerivedQadar
deriving DecidableEq, Repr
def nomosScreen : LawClaim → Audit.Token × Grade × String
| .objectLawGroove =>
(.sealed, .structural,
"object-laws emerge on RA as cost-reducing groove-entrenchment, the descent of realized cost toward K(X); structural on the RA-and-groove mapping")
| .lawAsSuchFromNoise =>
(.refused, .premise,
"refused, at premise: the counting cap bounds short-description outputs by 2^(n-c)-1; the reading that almost every stream is lawless rides a distribution the cap does not supply")
| .floorInheritedFitra =>
(.determined, .premise,
"the floor is fitra, the innate nature, RA, K(X) inherited; the identification is a reading, premise-grade, out of band")
| .recordDerivedQadar =>
(.determined, .structural,
"the record is qadar, R(X) the harvested redundancy, derived; the derived never outranks the floor it is harvested from")
theorem nomosScreen_battery :
(nomosScreen .lawAsSuchFromNoise).1 = .refused
∧ (nomosScreen .objectLawGroove).2.1 = .structural
∧ (nomosScreen .floorInheritedFitra).2.1 = .premise
∧ (nomosScreen .recordDerivedQadar).2.1 = .structural := by
decide
#eval (nomosScreen .lawAsSuchFromNoise).1
#eval nomosRows.length
end NOMOS
namespace BRIDGE
inductive Warrant : Type
| typeT | typeS | typeC | extension
deriving DecidableEq, Repr, BEq
inductive PostSeal : Type
| native | cited | superseded | outOfBand | byReference
deriving DecidableEq, Repr, BEq
structure BridgeRow : Type where
id : String
warrant : Warrant
status : PostSeal
note : String
deriving Repr
def ceiling : Warrant → Grade
| .typeT => .theoremConditional
| .typeS => .structural
| .typeC => .corroboration
| .extension => .corroboration
def census : List BridgeRow := [
⟨"BA-001a", .typeT, .native,
"actuation floor: energy floor theorem-grade on the Heisenberg bound and zero-point energy; transitions charged Jarzynski-Crooks with Landauer 1961 and Bérut 2012; static existence premise-grade on substrate monism. RegistrationPostulate declared at SECTION 7, consumed by no theorem; its arithmetic face is priced by OMEGA.landauerRat (SECTION 14) and Fortran K3: the postulate charges, the battery prices"⟩,
⟨"BA-001b", .typeT, .outOfBand,
"Turing halting: a V_F-only theorem-grade ceiling honored out of band; the cascade routes around at layer difference, never through"⟩,
⟨"BA-002", .typeT, .cited,
"spectral dual L2: Plancherel at the flat register Type T, Tomita-Takesaki at the curved register Type C; audits cap at the face cited"⟩,
⟨"BA-003", .typeC, .cited, "epistemic phase transition at 2 k_B T ln 2 work"⟩,
⟨"BA-004", .typeC, .superseded,
"nomological habituation, superseded at the object-law register by NOMOS-01 (SECTION 15, namespace NOMOS)"⟩,
⟨"BA-005", .typeC, .cited, "conformal persistence"⟩,
⟨"BA-006", .typeS, .cited, "conformal cyclic adjacency at Scope B"⟩,
⟨"BA-007", .typeT, .cited, "holographic entropy bound, Bekenstein-Hawking extended by Bousso 1999"⟩,
⟨"BA-008", .typeS, .native,
"substrate equal topology equal actuation monism; the posit face native at SECTION 15 (FOUND.STRATIFIED-MONISM-PRIORITY)"⟩,
⟨"BA-009", .typeC, .cited, "matter-genesis via S1 knotting"⟩,
⟨"BA-010", .typeC, .cited, "variational free energy V-FIO on the Friston anchor"⟩,
⟨"BA-011", .typeC, .cited, "spectral-dual conformal persistence at Scope B"⟩,
⟨"BA-012", .typeC, .native,
"cascade closure operational bijection, Type C, Hodge external corroboration; the witness-set enlargement (A4 torsor, Hurwitz shell slice) native at theorem grade at SECTION 4D"⟩,
⟨"BA-013", .extension, .byReference, "extension-tier bridge, loaded by reference"⟩,
⟨"BA-014", .extension, .byReference, "extension-tier bridge, loaded by reference"⟩,
⟨"BA-015", .extension, .byReference, "extension-tier bridge, loaded by reference"⟩,
⟨"BA-016", .extension, .byReference, "extension-tier bridge, loaded by reference"⟩,
⟨"BA-017", .extension, .byReference, "extension-tier bridge, loaded by reference"⟩,
⟨"BA-018", .typeT, .native,
"quaternionic completion and the Triple-Product Verdict Identity, Type T; the gate roster's A4 torsor with the Hurwitz double cover native at SECTION 4D; the verdict-identity face cited"⟩]
theorem census_count : census.length = 19 := by
decide
theorem post_seal_disposition :
(census.filter (fun b => b.status == .native)).length = 4
∧ (census.filter (fun b => b.status == .cited)).length = 8
∧ (census.filter (fun b => b.status == .superseded)).length = 1
∧ (census.filter (fun b => b.status == .outOfBand)).length = 1
∧ (census.filter (fun b => b.status == .byReference)).length = 5 := by
decide
theorem weakest_never_above_left (a b : Grade) :
(Grade.weakest a b).rank ≤ a.rank := by
cases a <;> cases b <;> decide
theorem bridge_ceiling_caps (b : BridgeRow) (g : Grade) :
(Grade.weakest (ceiling b.warrant) g).rank ≤ (ceiling b.warrant).rank :=
weakest_never_above_left _ _
def successorOf : String → Option String
| "BA-004" => some "NOMOS-01"
| _ => none
theorem supersession_routes :
successorOf "BA-004" = some "NOMOS-01"
∧ (census.any (fun b =>
b.id == "BA-004" && b.status == .superseded)) = true := by
decide
def bridgeScreen : BridgeRow → String
| ⟨_, _, .native, _⟩ =>
"load carried natively; audit the section, not the bridge"
| ⟨_, _, .cited, _⟩ =>
"cited at its warrant ceiling; never enters above it"
| ⟨_, _, .superseded, _⟩ =>
"superseded; route to the successor"
| ⟨_, _, .outOfBand, _⟩ =>
"honored out of band; routed around, never through"
| ⟨_, _, .byReference, _⟩ =>
"extension tier; carried at the tier's warrant, loaded by reference"
theorem bridgeScreen_total :
(census.all (fun b => (bridgeScreen b).utf8ByteSize > 0)) = true := by
decide
#eval census.length
#eval census.map (fun b => (b.id, bridgeScreen b))
end BRIDGE
namespace FOUND
def sigmaP : GEO.M3 :=
(((-7 : Ratio) / 25, (24 : Ratio) / 25, 0),
((24 : Ratio) / 25, (7 : Ratio) / 25, 0),
(0, 0, -1))
theorem pluralist_alternative_executed :
GEO.mm3 sigmaP sigmaP = GEO.diag3 1
∧ GEO.det3 sigmaP = 1
∧ GEO.mv3 sigmaP (3, 4, 0) = ((3, 4, 0) : GEO.V3)
∧ GEO.mv3 sigmaP (1, 0, 0) ≠ ((1, 0, 0) : GEO.V3)
∧ GEO.mv3 sigmaP (1, 0, 0) ≠ ((-1, 0, 0) : GEO.V3) := by
decide
inductive Band : Type
| inBand | outOfBand
deriving DecidableEq, Repr, BEq
structure Posit : Type where
id : String
grade : Grade
band : Band
loadBearingInVerdict : Bool
note : String
deriving Repr
def posits : List Posit := [
⟨"RESIDUAL-MONISM", .premise, .inBand, false,
"the one-prior-to-many connector: the geometric and formal grounds coincide on Fix(sigma) iff one involution serves both routes; the pluralist alternative executed above stays field-permitted; the from-other-side input located across the aperture and not crossed"⟩,
⟨"STRATIFIED-MONISM-PRIORITY", .structural, .inBand, false,
"P-MONISM-1: substrate, topology, and actuation are three projections of one event, theorem-grade as physics; the priority of the one over the many is the single structural posit the floor-coincidence rides"⟩,
⟨"FITRA-FLOOR", .premise, .outOfBand, false,
"fitra = RA = K(X), the incompressible floor inherited (QADAR-01); the theological identification routed out of band, load-bearing on nothing"⟩,
⟨"QADAR-RECORD", .structural, .outOfBand, false,
"qadar = R(X), the harvested record, derived and downstream of the floor; the Pen's cost-gradient writing, never onto noise"⟩,
⟨"AMANAH-TRUST", .premise, .outOfBand, false,
"P-AMANAH-1: ethics is the preservation of the actuation floor; the substrate a liability engine and never a moral authority"⟩]
theorem posits_loadbearing_on_nothing :
(posits.all (fun p => !p.loadBearingInVerdict)) = true := by
decide
def gradeOfRank : Nat → Grade
| 0 => .premise | 1 => .corroboration | 2 => .operational
| 3 => .structural | 4 => .engineering | 5 => .conditional
| 6 => .theoremConditional | 7 => .analytic | _ => .theorem
theorem posit_never_raises :
(posits.all (fun p =>
(List.range 9).all (fun r =>
decide ((Grade.weakest p.grade (gradeOfRank r)).rank ≤ r)))) = true := by
decide
inductive Register : Type
| kinetic | formal | verification | theological | social
deriving DecidableEq, Repr, BEq
def bandOf : Register → Band
| .theological => .outOfBand
| .social => .outOfBand
| _ => .inBand
theorem register_routing_fidelity :
bandOf .theological = .outOfBand ∧ bandOf .social = .outOfBand
∧ bandOf .kinetic = .inBand ∧ bandOf .formal = .inBand
∧ bandOf .verification = .inBand := by
decide
structure VerdictRow : Type where
name : String
positDeps : List String
deriving Repr
def verdictInventory : List VerdictRow := [
⟨"the grade law and the weakest-link join (section 1)", []⟩,
⟨"the audit engine and the drill battery (section 4B)", []⟩,
⟨"the inversion (section 4C)", []⟩,
⟨"the twelve gates, A4, the GF(2) lock, the cascades (section 4D)", []⟩,
⟨"the forcings of eight and five (section 4E)", []⟩,
⟨"the tongue's obedience (section 11)", []⟩,
⟨"the office taxonomy (section 12)", []⟩,
⟨"the universal router and the circularity law (section 13)", []⟩,
⟨"the Omega guard and AEGIS (section 14)", []⟩,
⟨"the incompressibility cap (section 15)", []⟩]
theorem verdicts_posit_free :
(verdictInventory.all (fun v => v.positDeps.isEmpty)) = true := by
decide
theorem engine_never_authority : socialWeight = 0 ∧ deltaM = 0 :=
⟨rfl, rfl⟩
#eval posits.map (fun p => (p.id, p.grade, p.band))
#eval List.map bandOf [Register.kinetic, .theological, .social]
end FOUND
namespace HALT
inductive Species : Type
| blockTheorem | hypothesis | offeredWitness
deriving DecidableEq, Repr
inductive Branch : Type
| xi | o
deriving DecidableEq, Repr
def cascade : List Bool → Option Nat
| [] => none
| false :: _ => some 0
| true :: gs => (cascade gs).map Nat.succ
def patterns (n : Nat) : List (List Bool) :=
(List.range (2 ^ n)).map fun b => (List.range n).map fun j => b.testBit j
def fftSpec (bits : List Bool) : Bool :=
match cascade bits with
| none => bits.all id
| some k =>
(bits.take k).all id &&
(match bits.drop k with
| false :: _ => true
| _ => false)
theorem first_failure_terminal_5 :
(patterns 5).all fftSpec = true := by decide
theorem first_failure_terminal_7 :
(patterns 7).all fftSpec = true := by decide
theorem first_failure_terminal_8 :
(patterns 8).all fftSpec = true := by decide
theorem first_failure_terminal_12 :
(patterns 12).all fftSpec = true := by decide
theorem halt_free_iff_all_pass_8 :
(patterns 8).all (fun bits => (cascade bits).isNone == bits.all id)
= true := by decide
theorem gate_counts_recorded :
GEO.a4.length = 12 ∧ GEO.allSigns.length = 8 ∧
(GEO.cube5.all (fun v => (GEO.neighbors5 v).eraseDups.length == 5)) = true :=
⟨GEO.a4_order, GEO.reflection_eight, GEO.fivecube_degree_five⟩
abbrev signPatterns : List (Bool × Bool × Bool) := GEO.allSigns
theorem ninth_gate_barred :
signPatterns.length = 8 ∧ signPatterns.eraseDups.length = 8 ∧
signPatterns = GEO.allSigns := ⟨by decide, by decide, rfl⟩
def route (groundDim : Nat) : Option Branch :=
if groundDim = 1 then some .xi else if groundDim = 0 then some .o else none
def ofUC : UC.HaltBranch → Option Branch
| .haltEight => some .xi
| .haltFive => some .o
| .unmeasured => none
theorem route_is_routeHalt : ∀ n : Nat, route n = ofUC (UC.routeHalt n).1 := by
intro n
unfold route UC.routeHalt ofUC
by_cases h1 : n = 1
· subst h1; simp
· by_cases h0 : n = 0
· subst h0; simp
· have h1' : (n : Int) ≠ 1 := by omega
have h0' : (n : Int) ≠ 0 := by omega
simp only [h1, h0, h1', h0', ite_false]
theorem route_samples :
route 0 = some .o ∧ route 1 = some .xi ∧ route 2 = none ∧ route 7 = none := by decide
inductive LegacyToken : Type
| xi0 | o0
deriving DecidableEq, Repr
def tokenBranch : LegacyToken → Branch
| .xi0 => .xi
| .o0 => .o
theorem router_exclusive :
route 0 = some .o ∧ route 1 = some .xi ∧ route 2 = none ∧
route 0 ≠ some .xi ∧ route 1 ≠ some .o ∧
tokenBranch .xi0 ≠ .o ∧ tokenBranch .o0 ≠ .xi := by decide
def readingRoadsGate : Bool := false
theorem one_door_open :
cascade (List.replicate 7 true ++ [readingRoadsGate]) = some 7 := by
decide
theorem no_walk_passes_the_aperture :
(patterns 7).all (fun xs => (cascade (xs ++ [readingRoadsGate])).isSome) = true := by
decide
def xiGates : List String :=
[ "Ground-determinacy", "Ghost partition", "constitutive blindness",
"machine-precision exhibit", "aperture located", "face-scoping",
"revision-mandate survival", "reading-roads" ]
def oGates : List String :=
[ "rootless terrain", "determinate stripped string", "wall record",
"aperture crossed or held", "totality discipline" ]
theorem gate_censuses :
xiGates.length = 8 ∧ oGates.length = 5 := by decide
inductive Channel : Type
| kinetic | formal
deriving DecidableEq, Repr
structure SuperHalt where
species : Species
gdim : Nat
branch : Branch
gates : Nat
channel : Channel
routed : route gdim = some branch
deriving DecidableEq
def rhHalt : SuperHalt := ⟨.hypothesis, 1, .xi, 8, .formal, by decide⟩
def pnpHalt : SuperHalt := ⟨.blockTheorem, 0, .o, 5, .formal, by decide⟩
theorem halts_routed :
route rhHalt.gdim = some rhHalt.branch ∧ route pnpHalt.gdim = some pnpHalt.branch :=
⟨rhHalt.routed, pnpHalt.routed⟩
theorem legacy_halts_are_instances :
rhHalt = ⟨.hypothesis, 1, .xi, 8, .formal, by decide⟩ ∧
pnpHalt = ⟨.blockTheorem, 0, .o, 5, .formal, by decide⟩ ∧
rhHalt ≠ pnpHalt := by decide
theorem terminal_one_bit :
Species.blockTheorem ≠ Species.hypothesis ∧
Species.blockTheorem ≠ Species.offeredWitness ∧
Species.hypothesis ≠ Species.offeredWitness := by decide
theorem species_exhausted : ∀ s : Species,
s = .blockTheorem ∨ s = .hypothesis ∨ s = .offeredWitness := by
intro s; cases s
· exact Or.inl rfl
· exact Or.inr (Or.inl rfl)
· exact Or.inr (Or.inr rfl)
def haltGrades : List Grade := [.structural, .theoremConditional]
theorem ftoe_ceiling :
haltGrades.all (fun g => g.rank < Grade.theorem.rank) = true := by
decide
end HALT
/-
!===============================================================================
! T R I S D U C T I O N . R A M
! THE CORE THESIS, STATED IN FORTRAN AND IN NOTHING ELSE
!
! The thesis is not described here. It is executed. Every claim that can be
! made to run is compiled as a procedure, and the claims that cannot run are
! compiled as the gates that refuse to emit without them.
!
! Three independent axes intersect in exactly one point. Nondegeneracy is
! linear independence. There is no second point and no interval. A LOCK IS
! ALWAYS A DETERMINATION AND IS TOTAL ABOUT ITS OBJECT, and its object is its
! intersection. Nothing below grades a lock, because locks carry no strengths.
! What differs between cases is what the ROWS were, and the compartment names
! that and nothing else.
!
! I CLOSURE-ROWED rows are the closure; the two determinations are one
! II UNPOPULATED an axis is empty; two planes meet in a line; no lock forms
! III WORLD-ROWED rows furnished by the world; revisability lives in the rows
!
! Two registers, sequential load. Register A, the kinetic preloader on RA,
! boots first: Seal L the Tongue, then Seal G the Form and Seal M the Number.
! Register B, the computational kernel on RAM, loads second on that discipline.
! Bedrock precedence: L then G then M. The Number reads magnitude on rows it
! did not author. Every compartment is RA-conditional at the act.
!
! The file carries its own battery and runs it last: eight fuzz classes,
! seven over the numerical kernel and one over every symbolic and router
! procedure, computed checks throughout, an oracle that stops the program
! on any failure. A binary that reaches its final line has
! passed. Nothing below is narrated; every figure is computed in this run.
!
! Delta-M = 0. No new mathematics is authored by this file.
!===============================================================================
!===============================================================================
module trisduction_rows
use, intrinsic :: ieee_arithmetic, only: ieee_is_finite, ieee_value, ieee_quiet_nan, ieee_positive_inf
implicit none
private
integer, parameter, public :: dp = selected_real_kind(15, 307)
!--- THE VERDICT ECONOMY. Three states, native. There is no fourth. --------
integer, parameter, public :: SEALED = 1, BROKEN = 2, OPENV = 3
integer, parameter, public :: XI0 = 4, OH0 = 5 ! refinements INSIDE openness
integer, parameter, public :: DETERMINED = 6 ! the kernel's own token
integer, parameter, public :: DOTMARK = 7 ! outside the economy
!--- THE THREE COMPARTMENTS. Not strengths. Kinds of row. -----------------
integer, parameter, public :: CLOSURE_ROWED = 1, UNPOPULATED = 2, WORLD_ROWED = 3
integer, parameter, public :: NOT_COMPARTMENTED = 0
public :: snorm, finite_vec, finite_mat, lambda_of_admissible
public :: det3_representable
public :: token, token_class, compartment_name, qmul, det3, jacobi3, cond_sym3
public :: prepare_rows, span_basis, lambda_of, kernel_determine
public :: intersection_dim, seal_L, drill_screen, row_cascade
public :: ground_dim, route_halt, compartment_grade
contains
! THE FINITE-INPUT CONTRACT. Every public numeric routine below reads only
! finite input. A NaN or an infinity is not a quantity, and a routine that
! returns a number or a true predicate on one has manufactured a quantity
! from a non-quantity. Pure numeric functions return NaN or .false. on
! violation, because a NaN in yielding a NaN out is honest and a zero would
! not be. Predicate and preparation subroutines report refusal through
! ok, verdict, or reason, because there a non-quantity is a data condition.
! Low-level computational subroutines whose preconditions are violated
! halt, because there the violation is a programming fault.
pure function finite_vec(x) result(ok)
real(dp), intent(in) :: x(:)
logical :: ok
ok = all(ieee_is_finite(x))
end function finite_vec
pure function finite_mat(A) result(ok)
real(dp), intent(in) :: A(:,:)
logical :: ok
ok = all(ieee_is_finite(A))
end function finite_mat
! The scaled two-norm. Squaring before scaling overflows for any row near the
! representable ceiling, so the largest magnitude comes out first and the
! square is formed on values bounded by one. Exact for a zero row.
pure function snorm(x) result(r)
real(dp), intent(in) :: x(:)
real(dp) :: r, m
if (.not. finite_vec(x)) then
r = ieee_value(1.0_dp, ieee_quiet_nan); return
end if
m = maxval(abs(x))
if (m > 0.0_dp) then
r = m*sqrt(sum((x/m)**2))
else
r = 0.0_dp
end if
end function snorm
pure function token(v) result(s)
integer, intent(in) :: v
character(len=30) :: s
select case (v)
case (SEALED); s = '[seal] SEALED'
case (BROKEN); s = '[X] BROKEN'
case (OPENV); s = '[?] UNDER-DETERMINED'
case (XI0); s = '[Xi0] TERMINAL SUSPENSION'
case (OH0); s = '[O0] GROUNDED-SEALED HALT'
case (DETERMINED); s = '[LOCK] DETERMINED'
case (DOTMARK); s = '[.] chosen silence'
case default; s = '[??] untyped'
end select
end function token
! The economy is three-state. XI0 and OH0 are refinements INSIDE the
! openness and never a fourth state; DETERMINED is the kernel's geometric
! token read before any register interprets it; the dot stands outside the
! economy entirely and is never census-counted. A caller that needs to know
! whether a token is a verdict state asks, and is not left to assume.
pure function token_class(v) result(s)
integer, intent(in) :: v
character(len=26) :: s
select case (v)
case (SEALED, BROKEN, OPENV); s = 'verdict state'
case (XI0, OH0); s = 'refinement inside openness'
case (DETERMINED); s = 'kernel geometric token'
case (DOTMARK); s = 'outside the economy'
case default; s = 'untyped'
end select
end function token_class
pure function compartment_name(c) result(s)
integer, intent(in) :: c
character(len=34) :: s
select case (c)
case (CLOSURE_ROWED); s = 'I CLOSURE-ROWED [seal A . RA]'
case (UNPOPULATED); s = 'II UNPOPULATED no lock forms'
case (WORLD_ROWED); s = 'III WORLD-ROWED rows revisable'
case default; s = '-- not compartmented'
end select
end function compartment_name
!===========================================================================
! THE ALGEBRA. Seal M loads LAST and reads magnitude on rows it did not
! author. It is constitutively unable to read sign, and that is a fact
! about DIRECTION, never about the completeness of the determination.
!===========================================================================
! Hamilton product. The substrate chirality lives here: i*j*k = -1.
! Both factors are scaled by their own largest magnitude, the product is
! formed on components bounded by one, and the scale is restored only after
! the restored magnitude has been checked in log space. Representability
! is therefore decided on the actual components and not on the norm bound
! |ab| = |a||b|, which is sufficient for refusal but not necessary, since a
! component can be smaller than the norm. Unrepresentable is NaN, never a
! saturated or wrapped quantity.
pure function qmul(a, b) result(c)
real(dp), intent(in) :: a(4), b(4)
real(dp) :: c(4), aa(4), bb(4), cc(4), sa, sb, lg
if (.not. (finite_vec(a) .and. finite_vec(b))) then
c = ieee_value(1.0_dp, ieee_quiet_nan); return
end if
sa = maxval(abs(a)); sb = maxval(abs(b))
if (.not. (sa > 0.0_dp .and. sb > 0.0_dp)) then
c = 0.0_dp; return
end if
aa = a/sa; bb = b/sb
cc(1) = aa(1)*bb(1) - aa(2)*bb(2) - aa(3)*bb(3) - aa(4)*bb(4)
cc(2) = aa(1)*bb(2) + aa(2)*bb(1) + aa(3)*bb(4) - aa(4)*bb(3)
cc(3) = aa(1)*bb(3) - aa(2)*bb(4) + aa(3)*bb(1) + aa(4)*bb(2)
cc(4) = aa(1)*bb(4) + aa(2)*bb(3) - aa(3)*bb(2) + aa(4)*bb(1)
if (.not. (maxval(abs(cc)) > 0.0_dp)) then
c = 0.0_dp; return
end if
lg = log(sa) + log(sb) + log(maxval(abs(cc)))
if (lg >= log(huge(1.0_dp))) then
c = ieee_value(1.0_dp, ieee_quiet_nan); return
end if
! The scale is restored in binary, not through exp(log). The mantissas
! fraction(sa) and fraction(sb) lie in [1/2, 1), so their product times cc
! is bounded by four and cannot overflow, and scale() by the summed
! exponent is an exact power-of-two multiply. exp(log(sa)+log(sb)) loses
! about thirteen digits here, since an ulp of a log near 709 is 1e-13
! absolute and becomes 1e-13 relative on the result; the battery caught
! that and this is the repair.
c = scale(fraction(sa)*fraction(sb)*cc, exponent(sa) + exponent(sb))
end function qmul
! The triple products overflow for entries near the ceiling even where the
! determinant is finite after cancellation, so the matrix is scaled first and
! the cube of the scale is restored once.
! Purity is kept, so the answer to "is this value a quantity" is a separate
! predicate rather than an out argument. Where the computed determinant
! magnitude exceeds the representable normal range det3 saturates and det3_representable returns
! .false.; a caller that reads the saturated value as a quantity has been
! told otherwise and did not ask.
! Equilibration, not a single global scale. One maximum over the whole
! matrix lets a small diagonal entry underflow to zero in the normalised
! copy, so diag(1e200, 1e200, 1e-200) returned zero where the determinant
! is 1e200 and diag(1e308, 1e308, 1e-20) was certified representable where
! it is not. Rows are scaled by their own maxima, then columns by theirs,
! and the determinant of the equilibrated core is reassembled in log space
! so representability is decided before any product is formed.
pure subroutine det3_core(M, core, lg, sgn, ok)
real(dp), intent(in) :: M(3,3)
real(dp), intent(out) :: core, lg, sgn
logical, intent(out) :: ok
real(dp) :: N(3,3), r(3), c(3)
integer :: i
ok = finite_mat(M); core = 0.0_dp; lg = -huge(1.0_dp); sgn = 0.0_dp
if (.not. ok) return
N = M
do i = 1, 3
r(i) = maxval(abs(N(i,:)))
if (r(i) > 0.0_dp) N(i,:) = N(i,:)/r(i)
end do
do i = 1, 3
c(i) = maxval(abs(N(:,i)))
if (c(i) > 0.0_dp) N(:,i) = N(:,i)/c(i)
end do
core = N(1,1)*(N(2,2)*N(3,3) - N(2,3)*N(3,2)) &
- N(1,2)*(N(2,1)*N(3,3) - N(2,3)*N(3,1)) &
+ N(1,3)*(N(2,1)*N(3,2) - N(2,2)*N(3,1))
if (.not. (abs(core) > 0.0_dp)) return ! computed zero; lg stays -huge
sgn = sign(1.0_dp, core)
lg = log(abs(core))
do i = 1, 3
if (r(i) > 0.0_dp) lg = lg + log(r(i))
if (c(i) > 0.0_dp) lg = lg + log(c(i))
end do
end subroutine det3_core
pure function det3(M) result(d)
real(dp), intent(in) :: M(3,3)
real(dp) :: d, core, lg, sgn
logical :: ok
call det3_core(M, core, lg, sgn, ok)
if (.not. ok) then
d = ieee_value(1.0_dp, ieee_quiet_nan); return
end if
if (.not. (abs(core) > 0.0_dp)) then
d = 0.0_dp; return
end if
! At the ceiling itself exp(lg) can round past huge and manufacture an
! overflow where a saturated finite value was promised, so the boundary
! saturates too.
if (lg >= log(huge(1.0_dp))) then
d = sign(huge(1.0_dp), sgn); return ! saturating; ask det3_representable
end if
if (lg < log(tiny(1.0_dp))) then
d = sign(0.0_dp, sgn); return ! below the normal range; ask
end if
d = sgn*exp(lg)
end function det3
! True when det3 returns a non-saturated, non-underflowed normal quantity
! under this floating computation, or when the equilibrated core computes
! as zero. A computed zero is not certified exact singularity.
pure function det3_representable(M) result(r)
real(dp), intent(in) :: M(3,3)
logical :: r
real(dp) :: core, lg, sgn
logical :: ok
call det3_core(M, core, lg, sgn, ok)
if (.not. ok) then
r = .false.; return
end if
if (.not. (abs(core) > 0.0_dp)) then
! A COMPUTED zero is treated as a representable zero. This routine
! reads floating arithmetic and cannot tell exact singularity from a
! determinant that cancelled or underflowed to zero on the way; that
! distinction needs interval or compensated arithmetic and is not
! claimed here.
r = .true.; return
end if
r = (lg < log(huge(1.0_dp))) .and. (lg >= log(tiny(1.0_dp)))
end function det3_representable
! CONTRACT: A is symmetric. The routine reads only what a symmetric matrix
! carries, so an asymmetric input would be silently symmetrised by the
! sweep and the eigenvalues returned would belong to no matrix the caller
! supplied. The precondition is therefore enforced, not merely documented.
subroutine jacobi3(A, w)
real(dp), intent(in) :: A(3,3)
real(dp), intent(out) :: w(3)
real(dp) :: M(3,3), Asc(3,3), theta, t, c, s, tau, g, h, asym, amax
integer :: sweep, p, q, k
if (.not. finite_mat(A)) error stop 'jacobi3: input is not finite'
! The check is performed on a scaled copy, because differencing two
! entries near the representable ceiling overflows the test itself and a
! precondition guard that faults while guarding is worse than none.
amax = maxval(abs(A))
if (amax > 0.0_dp) then
Asc = A/amax
else
Asc = A
end if
asym = max(abs(Asc(1,2)-Asc(2,1)), abs(Asc(1,3)-Asc(3,1)), abs(Asc(2,3)-Asc(3,2)))
if (asym > 100.0_dp*epsilon(1.0_dp)) then
error stop 'jacobi3: input matrix is not symmetric within tolerance'
end if
! The whole sweep runs on the scaled copy and the eigenvalues are unscaled
! once at the end. This is what makes every interior sum, difference and
! square safe at any admissible magnitude, rather than guarding each one.
M = Asc
do sweep = 1, 60
g = abs(M(1,2)) + abs(M(1,3)) + abs(M(2,3))
if (g < 1.0e-300_dp) exit
do p = 1, 2
do q = p+1, 3
! The negligibility test comes FIRST, because theta overflows in
! its own division where the off-diagonal is tiny against the
! diagonal spread. An off-diagonal below the rounding of the two
! diagonals it couples carries no rotation, so it is zeroed and
! skipped rather than divided by.
if (abs(M(p,q)) <= epsilon(1.0_dp)*max(abs(M(p,p)), abs(M(q,q)))) then
M(p,q) = 0.0_dp; M(q,p) = 0.0_dp
else if (abs(M(p,q)) > tiny(1.0_dp)) then
theta = (M(q,q) - M(p,p)) / (2.0_dp*M(p,q))
! theta*theta overflows where the off-diagonal is negligible
! against the diagonal spread. The asymptotic branch is the
! same rotation read where the square cannot be formed.
if (abs(theta) > sqrt(huge(1.0_dp))) then
t = 0.5_dp / theta
else
t = sign(1.0_dp, theta) / (abs(theta) + sqrt(theta*theta + 1.0_dp))
end if
c = 1.0_dp / sqrt(t*t + 1.0_dp); s = t*c; tau = s/(1.0_dp + c)
h = t*M(p,q)
M(p,p) = M(p,p) - h; M(q,q) = M(q,q) + h
M(p,q) = 0.0_dp; M(q,p) = 0.0_dp
do k = 1, 3
if (k /= p .and. k /= q) then
g = M(k,p); h = M(k,q)
M(k,p) = g - s*(h + g*tau); M(p,k) = M(k,p)
M(k,q) = h + s*(g - h*tau); M(q,k) = M(k,q)
end if
end do
end if
end do
end do
end do
! Unscaling can exceed the representable range for a finite input whose
! eigenvalue does. The infinity is then produced by ieee_value, which
! raises no exception, and never by the multiplication, which would trap
! before any consumer could refuse the result.
! The guard is nested and not conjoined: Fortran does not promise
! short-circuit evaluation, so huge/amax must never be formed unless
! amax exceeds one, or the guard itself overflows on a tiny matrix.
do k = 1, 3
if (amax > 1.0_dp) then
if (abs(M(k,k)) > huge(1.0_dp)/amax) then
w(k) = sign(ieee_value(1.0_dp, ieee_positive_inf), M(k,k))
else
w(k) = amax*M(k,k)
end if
else
w(k) = amax*M(k,k) ! amax <= 1: the product only shrinks
end if
end do
end subroutine jacobi3
! CONTRACT: A is symmetric positive semi-definite, a Gram. On a general
! symmetric matrix this returns a MAGNITUDE condition number and not the
! spectral one, because the extremes are taken over absolute eigenvalues.
function cond_sym3(A) result(kap)
real(dp), intent(in) :: A(3,3)
real(dp) :: kap, w(3), lo, hi
call jacobi3(A, w)
! A finite matrix can carry an eigenvalue beyond the representable range,
! since three entries near the ceiling sum to about three times it. The
! sweep reports that honestly as an infinity and the consumer refuses it.
if (.not. finite_vec(w)) error stop 'cond_sym3: eigenvalues are not representable'
! The contract names a Gram, so definiteness is enforced rather than
! assumed. An indefinite matrix would return a magnitude condition number
! that reads like a spectral one and would be believed.
hi = maxval(abs(w))
if (.not. (hi > 0.0_dp)) then
kap = huge(1.0_dp); return ! the zero matrix has no conditioning
end if
! Relative to the spectrum and carrying no absolute floor, for the same
! reason the rank tolerance carries none: a floor pinned at one would make
! definiteness a fact about units.
if (minval(w) < -100.0_dp*epsilon(1.0_dp)*hi) then
error stop 'cond_sym3: input is not positive semi-definite within tolerance'
end if
lo = minval(abs(w))
! The floor is tiny() and not zero: a positive denormal divisor overflows
! the quotient, so the guard is on representability and not on sign.
if (lo <= tiny(1.0_dp)) then
kap = huge(1.0_dp)
else
kap = hi/lo
end if
end function cond_sym3
! THE SPINE, COMPUTED. Read the three rows as normals of planes through the
! origin. The intersection of the planes has dimension 3 - rank. Rank three
! returns dimension ZERO, one point, unique, no interval. Rank two returns
! dimension ONE, a line, and no point is determined at all.
! CONTRACT: G is a SYMMETRIC GRAM, G = Q Q^T, formed by the caller. It is not
! a matrix of raw rows. The dimension returned is 3 - rank(G), which is the
! dimension of the intersection of the three planes whose normals the rows of
! Q are; the routine reads the Gram because rank is what it needs and the
! Gram is where rank is cheap. Passing raw rows violates the contract: in
! exact arithmetic rank(Q Q^T) equals rank(Q), so the numerical dimension
! may coincide, but the routine is specified to read a symmetric PSD Gram
! and enforces that specification rather than trusting the coincidence.
! The rank tolerance is not chosen. It is the largest eigenvalue times the
! machine epsilon times the matrix order, the standard perturbation bound on
! a symmetric eigenvalue: below it an eigenvalue is not separable from
! rounding at this precision.
subroutine intersection_dim(G, d, sep)
real(dp), intent(in) :: G(3,3)
integer, intent(out) :: d
real(dp), intent(out) :: sep ! margin: smallest eigenvalue over tol
integer :: i, rank
real(dp) :: w(3), tol, lo, hi
call jacobi3(G, w)
if (.not. finite_vec(w)) error stop 'intersection_dim: eigenvalues are not representable'
! The tolerance is RELATIVE to the spectrum and carries no absolute floor,
! so rank is invariant under uniform scaling of the Gram: a Gram at 1e-20
! has the same rank as the same Gram at 1. An absolute floor would make
! rank a fact about units, which is the chart-manufactured magnitude this
! architecture bars from entering a verdict as structure.
hi = maxval(abs(w))
if (.not. (hi > 0.0_dp)) then
d = 3; sep = 0.0_dp; return ! the zero Gram carries no direction
end if
! The contract names a Gram, so definiteness is enforced and the count is
! over POSITIVE eigenvalues. Counting magnitudes would hand an indefinite
! matrix a geometric dimension it does not have.
if (minval(w) < -100.0_dp*epsilon(1.0_dp)*hi) then
error stop 'intersection_dim: G is not positive semi-definite within tolerance'
end if
tol = 8.0_dp * real(size(w),dp) * epsilon(1.0_dp) * hi
rank = 0; lo = huge(1.0_dp)
do i = 1, 3
if (w(i) > tol) then
rank = rank + 1
lo = min(lo, w(i))
end if
end do
d = 3 - rank
if (rank > 0) then
sep = lo/tol
else
sep = 0.0_dp
end if
end subroutine intersection_dim
subroutine prepare_rows(M, Q, ok)
real(dp), intent(in) :: M(:,:)
real(dp), intent(out) :: Q(size(M,1), size(M,2))
logical, intent(out) :: ok
integer :: i, n
real(dp) :: mu, sd, nrm, scale_
n = size(M,2); ok = .true.; Q = M
if (.not. finite_mat(M)) then
ok = .false.; return ! a non-quantity is not a row
end if
if (n <= 1) then
ok = .false.; return ! no sample variance is defined at n = 1
end if
do i = 1, size(M,1)
scale_ = maxval(abs(M(i,:)))
! Scale out before summing. Squaring a row near the representable
! ceiling overflows the sum of squares long before the variance is
! meaningful, so the row is divided by its own largest magnitude first
! and the statistic is formed on the scaled copy. The verdict is
! unchanged because every downstream step is scale-invariant.
if (scale_ > 0.0_dp) Q(i,:) = Q(i,:)/scale_
mu = sum(Q(i,:))/real(n,dp); Q(i,:) = Q(i,:) - mu
sd = sqrt(sum(Q(i,:)**2)/real(n-1,dp))
! The floor is relative to the row's own scale. A spread that is only
! rounding on a large row is not content, and normalising by it would
! manufacture an axis out of noise.
if (.not. (sd > 0.0_dp) .or. sd <= real(n,dp)*epsilon(1.0_dp)) then
ok = .false.; return
end if
Q(i,:) = Q(i,:)/sd
! A z-scored row has sum of squares exactly n-1 by construction, so
! this norm is sqrt(n-1) and cannot overflow for any representable n.
! snorm is used anyway: one norm primitive across the module is one
! place to get it right, and a raw sum of squares reads as a hazard
! to a hostile reviewer even where the bound forecloses it.
nrm = snorm(Q(i,:))
if (.not. (nrm > 0.0_dp)) then
ok = .false.; return
end if
Q(i,:) = Q(i,:)/nrm
end do
end subroutine prepare_rows
! The guard is mandatory and never optional, because a guard a caller may
! skip is not a guard. A residual below the rounding floor is not a direction
! and is refused rather than normalised.
subroutine span_basis(Q, B, ok)
real(dp), intent(in) :: Q(:,:)
real(dp), intent(out) :: B(size(Q,1), size(Q,2))
logical, intent(out) :: ok
integer :: i, j
real(dp) :: v(size(Q,2)), nrm, floor_, scale_
if (.not. finite_mat(Q)) error stop 'span_basis: input is not finite'
ok = .true.; B = 0.0_dp
floor_ = real(size(Q,2),dp) * epsilon(1.0_dp)
do i = 1, size(Q,1)
! Each row is scaled by its own largest magnitude BEFORE any projection.
! Scaling a row leaves its span untouched and bounds every entry by one,
! so the dot products inside the loop cannot overflow. The floor is
! then relative to the scaled row by construction.
scale_ = maxval(abs(Q(i,:)))
if (.not. (scale_ > 0.0_dp)) then
ok = .false.; return
end if
v = Q(i,:)/scale_
do j = 1, i-1
v = v - dot_product(v, B(j,:))*B(j,:)
end do
nrm = snorm(v)
if (.not. (nrm > floor_)) then
ok = .false.; return
end if
B(i,:) = v/nrm
end do
end subroutine span_basis
! Only the first three rows of Q and B are screened, because only the first
! three rows are read; the shape test requires at least three and rows
! beyond them are not consulted by lambda_of at all.
! The admission test is a pure predicate so it can be exercised in-process
! on hostile input without halting. It is the single source of truth for
! what lambda_of accepts; lambda_of calls it and halts on refusal. The
! order inside is load-bearing: unit-ness is proven by the scaled norm
! BEFORE any dot product forms, so every dot product below runs on rows
! whose components are O(1) under the admission tolerance and never
! O(huge), and the dot products cannot overflow.
pure subroutine lambda_of_admissible(Q, B, ok, reason)
real(dp), intent(in) :: Q(:,:), B(:,:)
logical, intent(out) :: ok
character(len=*), intent(out) :: reason
real(dp) :: btol
integer :: i, j
ok = .false.; reason = ''
if (size(Q,1) < 3 .or. size(B,1) < 3 .or. size(Q,2) /= size(B,2) .or. size(Q,2) < 3) then
reason = 'needs Q and B with at least three rows, at least three columns, and equal column counts'
return
end if
if (.not. (finite_mat(Q(1:3,:)) .and. finite_mat(B(1:3,:)))) then
reason = 'input is not finite'; return ! on the rows read, and no others
end if
btol = 100.0_dp * real(size(B,2),dp) * epsilon(1.0_dp)
do i = 1, 3
if (abs(snorm(B(i,:)) - 1.0_dp) > btol) then
reason = 'B rows are not unit'; return
end if
if (abs(snorm(Q(i,:)) - 1.0_dp) > btol) then
reason = 'Q rows are not unit'; return
end if
end do
do i = 1, 3
do j = i+1, 3
if (abs(dot_product(B(i,:), B(j,:))) > btol) then
reason = 'B rows are not orthogonal'; return
end if
end do
end do
ok = .true.
end subroutine lambda_of_admissible
subroutine lambda_of(Q, B, lam_quat, lam_det, co)
real(dp), intent(in) :: Q(:,:), B(:,:)
real(dp), intent(out) :: lam_quat, lam_det, co(3,3)
real(dp) :: q1(4), q2(4), q3(4), p(4)
integer :: i, j
character(len=96) :: reason
logical :: adm
call lambda_of_admissible(Q, B, adm, reason)
if (.not. adm) error stop 'lambda_of: '//trim(reason)
do i = 1, 3
do j = 1, 3
co(i,j) = dot_product(Q(i,:), B(j,:)) ! admitted unit rows; Cauchy bound keeps this O(1), no overflow
end do
end do
q1 = [0.0_dp, co(1,:)]; q2 = [0.0_dp, co(2,:)]; q3 = [0.0_dp, co(3,:)]
p = qmul(qmul(q1, q2), q3)
lam_quat = p(1); lam_det = -det3(co)
end subroutine lambda_of
! The kernel does not grade. It reports whether the axes are independent, and
! therefore whether a unique intersection exists at all.
subroutine kernel_determine(M, verdict, lam, detR, kapR, idim, why, margin)
real(dp), intent(in) :: M(:,:)
integer, intent(out) :: verdict, idim
real(dp), intent(out), optional :: margin
real(dp), intent(out) :: lam, detR, kapR
character(len=*), intent(out) :: why
real(dp) :: sep
real(dp) :: Q(3,size(M,2)), B(3,size(M,2)), R(3,3), co(3,3)
real(dp) :: lq, ld, um, epsf, kgate
integer :: n
logical :: ok, bok
! EVERY output is defined here, before any test can return. The shape and
! size tests follow, and no arithmetic touches n until n has been screened,
! because the conditioning gate divides by it.
lam = 0.0_dp; detR = 0.0_dp; kapR = 0.0_dp; idim = -1; sep = 0.0_dp
verdict = OPENV; why = 'uninitialised'
if (present(margin)) margin = 0.0_dp
n = size(M,2); um = epsilon(1.0_dp)
if (size(M,1) /= 3) then
why = 'axis count is not three; the residence is not the one this kernel reads'
return
end if
if (n < 4) then
why = 'dimensional shortfall'; return
end if
if (.not. finite_mat(M)) then
why = 'a row carries a non-quantity; nothing to read'; return
end if
epsf = 100.0_dp*um*real(n,dp)
kgate = min(1.0e6_dp, sqrt(27.0_dp/(100.0_dp*um*real(n,dp))))
call prepare_rows(M, Q, ok)
if (.not. ok) then
verdict = OPENV; why = 'an axis carries no content; nothing to intersect'; return
end if
R = matmul(Q, transpose(Q))
detR = det3(R)
! A correlation Gram is positive semi-definite by construction, so a
! negative determinant is a rounding artifact and never a quantity. It is
! set to exactly zero rather than reported as read. Zero here is the
! algebraic floor of the quantity and is not the collapse floor epsf,
! which is a separate threshold tested further down.
if (detR < 0.0_dp) detR = 0.0_dp
call intersection_dim(R, idim, sep)
if (present(margin)) margin = sep
! The collapse test runs BEFORE any basis is constructed. Gram-Schmidt on a
! dependent triad divides by a residual nothing bounds below, so a basis
! built here would be rounding noise carried forward into the scalar. The
! order is the guard: a residence that determines no point never reaches
! the orthogonalisation at all.
! Existence is decided by the DIMENSION and never by the determinant,
! because the dimension is the spine's own quantity and the determinant is
! a magnitude read off it. Where the two floors disagree the verdict is
! neither broken nor determined: it is arithmetic at the edge, and it
! escalates rather than guessing which floor to believe.
if (idim > 0) then
verdict = BROKEN
select case (idim)
case (1); why = 'rank two; the intersection is a line, not a point'
case (2); why = 'rank one; the intersection is a plane, not a point'
case default; why = 'rank zero; no axis carries an independent direction'
end select
return
end if
if (detR <= epsf) then
verdict = OPENV
why = 'engineering-incomplete: rank floor and collapse floor disagree; re-run at higher precision'
return
end if
call span_basis(Q, B, bok)
if (.not. bok) then
! Reaching here means the rank floor and the collapse floor both said
! three independent axes while Gram-Schmidt found a null residual. That
! is a third floor disagreeing with two, not a finding of dependence,
! and it escalates for the same reason the determinant disagreement
! does. Deciding BROKEN here would let the weakest instrument overrule
! the two that already agreed.
verdict = OPENV
why = 'engineering-incomplete: orthogonalisation residual contradicts the rank floor'
return
end if
call lambda_of(Q, B, lq, ld, co)
! The two forms are computed by disjoint routes, the Hamilton product and
! the signed frame determinant, so their agreement is a live cross-check
! and not decoration. Disagreement is arithmetic failure and escalates.
if (abs(lq - ld) > 100.0_dp*epsilon(1.0_dp)*max(abs(ld), 1.0_dp)) then
verdict = OPENV
why = 'engineering-incomplete: quaternion and determinant forms of lambda disagree'
return
end if
lam = lq
kapR = cond_sym3(R)
if (kapR >= kgate) then
! The intersection exists and is unique by the geometry. The instrument
! declines to report it, because at this conditioning the computed point
! is not separable from rounding. Instrument-silence about reach, never
! a claim that the intersection is absent or partial.
verdict = OPENV
why = 'engineering-incomplete: determination present, instrument declines'
return
end if
verdict = DETERMINED
why = 'three independent axes; the intersection exists and is unique'
if (present(margin)) margin = sep
end subroutine kernel_determine
!===========================================================================
! SEAL L, THE TONGUE. Loads FIRST. Carries the direction the Number cannot,
! and closes the terms the row cascade later reads.
!===========================================================================
subroutine seal_L(slots, verdict, why)
integer, intent(in) :: slots(:,:)
integer, intent(out) :: verdict
character(len=*), intent(out) :: why
integer :: i, j, a, b, nfull, ns
ns = size(slots,1) ! the shelf is read, never remembered
! Zero is the empty cell and literals are positive. A negative id is not
! vocabulary and is refused before any slot is counted.
if (any(slots < 0)) then
verdict = BROKEN; why = 'negative literal id; not vocabulary'; return
end if
nfull = 0
do i = 1, ns
if (any(slots(i,:) /= 0)) nfull = nfull + 1
end do
if (nfull /= 3) then
verdict = BROKEN
write(why,'(A,I0,A)') 'deletion test returns ', nfull, ' slots, not three'
return
end if
! A slot is a vocabulary set. A repeated literal inside one slot is a
! malformed set, and admitting it would let hostile input reach a sealed
! verdict through a slot that was never well formed.
do i = 1, ns
do a = 1, size(slots,2)
do b = a+1, size(slots,2)
if (slots(i,a) /= 0 .and. slots(i,a) == slots(i,b)) then
verdict = BROKEN
write(why,'(A,I0)') 'repeated literal within slot ', i
return
end if
end do
end do
end do
do i = 1, ns
do j = i+1, ns
do a = 1, size(slots,2)
do b = 1, size(slots,2)
if (slots(i,a) /= 0 .and. slots(i,a) == slots(j,b)) then
verdict = BROKEN
write(why,'(A,I0,A,I0)') 'LIT collision, slot ', i, ' and slot ', j
return
end if
end do
end do
end do
end do
verdict = SEALED; why = 'three slots, vocabulary pairwise disjoint'
end subroutine seal_L
! The six drills, routed to and not improvised. D3 carries the provenance
! flag and is mandatory for the closure-rowed compartment: a standing
! definition shaped at the Number makes rows-and-closure coincidence
! circular, and no other gate in the architecture reaches it.
subroutine drill_screen(d1, d2, d3_filed, d3_shaped, d4, d5, d6, ok, grade_cap, why)
logical, intent(in) :: d1, d2, d3_filed, d3_shaped, d4, d5, d6
logical, intent(out) :: ok
character(len=*), intent(out) :: grade_cap, why
ok = .false.; grade_cap = 'none'
if (.not. (d1 .and. d2)) then
why = 'D1/D2 fail: Seal L break, terminal whatever the determinant reads'; return
end if
if (.not. d3_filed) then
why = 'D3 fail: provenance unfiled; Number-shaped content in the Tongue seat'; return
end if
if (.not. d4) then
why = 'D4 fail: a magnitude carried on the Tongue; sign only, veto never substitution'; return
end if
if (.not. d5) then
why = 'D5 fail: register undeclared; re-index rather than refute'; return
end if
if (.not. d6) then
why = 'D6 fail: the Tongue reproduced the blindness it was appointed to repair'; return
end if
ok = .true.
! The flag records; it does not bar. What it costs is a tier.
if (d3_shaped) then
grade_cap = 'structural'
why = 'drills pass; provenance filed and SHAPED, so the assignment caps at structural'
else
grade_cap = 'theorem'
why = 'drills pass; provenance filed and unshaped, no tier discount'
end if
end subroutine drill_screen
!===========================================================================
! THE ROW CASCADE. One floor, three routers folded into their arguments,
! four admission gates. The count is forced by the questions: one about
! population, one about row identity, two about closure leaks.
!===========================================================================
subroutine row_cascade(has_content, faces_scoped, canonical, forward_mode, &
populated, worldly_row, drills_ok, frame_closed, &
compartment, why)
logical, intent(in) :: has_content, faces_scoped, canonical, forward_mode
logical, intent(in) :: populated, worldly_row, drills_ok, frame_closed
integer, intent(out) :: compartment
character(len=*), intent(out) :: why
compartment = NOT_COMPARTMENTED
if (.not. has_content) then
why = 'F-0: every term decoration; contentless, refused, not compartmented'; return
end if
if (.not. faces_scoped) then
why = 'R-1: faces not enumerated; a composite carries one compartment per face'; return
end if
if (.not. canonical) then
why = 'R-2: not canonicalised; the compartment attaches to the stripped string'; return
end if
! Population is asked before mode, because an unpopulated forward
! proposition has no lock to route and typing it world-rowed would assign a
! compartment to an object that determined nothing.
if (.not. populated) then
compartment = UNPOPULATED
why = 'S-1: an axis is empty; two planes meet in a line; no lock forms'
return
end if
if (forward_mode) then
compartment = WORLD_ROWED
why = 'R-3: a dated axis must be sourced, never measured; world-rowed unconditionally'
return
end if
if (worldly_row) then
compartment = WORLD_ROWED
why = 'S-2: a row furnished by the world; the lock is total given those rows'
return
end if
if (.not. drills_ok) then
compartment = WORLD_ROWED
why = 'S-3: term closure unscreened; typed by its remaining rows'
return
end if
if (.not. frame_closed) then
compartment = WORLD_ROWED
why = 'S-4: existential import or transport; the leak names the compartment'
return
end if
compartment = CLOSURE_ROWED
why = 'rows are the closure; the two determinations are one; RA-conditional at the act'
end subroutine row_cascade
pure function compartment_grade(c, cap) result(s)
integer, intent(in) :: c
character(len=*), intent(in) :: cap
character(len=64) :: s
select case (c)
case (CLOSURE_ROWED); s = 'row-indefeasible, defeasible against fault, cap '//trim(cap)
case (UNPOPULATED); s = 'no determination to defeat; an axis is owed'
case (WORLD_ROWED); s = 'revisable entirely in the rows; direction bought with the arrow'
case default; s = 'no compartment'
end select
end function compartment_grade
!===========================================================================
! THE GROUND AND THE ROUTER BIT.
!===========================================================================
! The router bit is measured at machine zero and never by exact equality. An
! eigenvalue arriving from a solver is 1 - O(eps), and an equality test would
! silently return the wrong Ground dimension and route the candidate to the
! wrong halt protocol. The floor is stated, not assumed.
pure function ground_dim(diagv) result(d)
real(dp), intent(in) :: diagv(:)
integer :: d, i
real(dp) :: floor_
d = -1 ! the router bit is unmeasured on a non-quantity
if (.not. finite_vec(diagv)) return
floor_ = real(max(size(diagv),1),dp) * epsilon(1.0_dp)
d = 0
do i = 1, size(diagv)
if (abs(diagv(i) - 1.0_dp) <= floor_) d = d + 1
end do
end function ground_dim
pure function route_halt(gdim) result(branch)
integer, intent(in) :: gdim
character(len=52) :: branch
if (gdim == 1) then
branch = 'B.14.Xi eight gates, blindness in the reader'
else if (gdim == 0) then
branch = 'B.14.O five gates, walls in the terrain'
else
branch = 'terrain unmeasured; nothing emitted'
end if
end function route_halt
end module trisduction_rows
!===============================================================================
! EXECUTED. Nothing below is narrated; every figure is computed in this run.
!===============================================================================
program thesis_rows
use trisduction_rows
implicit none
integer, parameter :: N = 24
real(dp), parameter :: PI = 3.141592653589793238462643_dp
real(dp) :: t(N), M(3,N), Mr(3,N), Two(3,N), Ret(3,N)
real(dp) :: Q(3,N), B(3,N), co(3,3), R(3,3), Rr(3,3)
real(dp) :: lam, detR, kapR, lam2, d2, lq, ld, marg
real(dp) :: i4(4), j4(4), k4(4), p4(4), sig(4), dia(4)
integer :: v, vL, idim, comp, i
integer :: fails, checks
logical :: ok, dok
character(len=110) :: why, cap
integer :: slots_ok(3,3)
write(*,'(A)') repeat('=',78)
write(*,'(A)') ' TRISDUCTION . RAM -- THE CORE THESIS, EXECUTED'
write(*,'(A)') repeat('=',78)
do i = 1, N
t(i) = 2.0_dp*PI*real(i-1,dp)/real(N,dp)
end do
!------------------------------------------------- THE SPINE, COMPUTED
! Three independent axes: the intersection has dimension zero. One point.
! Two axes: dimension one. A line. No point is determined at all.
M(1,:) = sin(t)
M(2,:) = cos(2.0_dp*t) + 0.35_dp*sin(t)
M(3,:) = sin(3.0_dp*t) + 0.20_dp*cos(2.0_dp*t) + 0.12_dp*sin(t)
Two = M; Two(3,:) = 0.6_dp*M(1,:) + 0.4_dp*M(2,:) ! third axis not independent
write(*,'(/,A)') ' THE SPINE: WHAT A LOCK DETERMINES'
call kernel_determine(M, v, lam, detR, kapR, idim, why, marg)
write(*,'(A,A)') ' three independent axes : ', trim(token(v))
write(*,'(A,A)') ' why = ', trim(why)
write(*,'(A,I0,A)') ' intersection dimension = ', idim, ' -> exactly one point, no interval'
write(*,'(A,F16.12)') ' det(R) = ', detR
write(*,'(A,ES12.3,A,ES10.3,A,L1)') ' |lambda^2 - det(R)| = ', abs(lam*lam - detR), &
' bound ', 4.0_dp*kapR*epsilon(1.0_dp), ' within: ', &
abs(lam*lam - detR) <= 4.0_dp*kapR*epsilon(1.0_dp)
write(*,'(A,ES12.3,A)') ' rank margin = ', marg, ' smallest kept eigenvalue over tol'
call kernel_determine(Two, v, lam2, d2, kapR, idim, why, marg)
write(*,'(A,A)') ' third axis dependent : ', trim(token(v))
write(*,'(A,A)') ' why = ', trim(why)
write(*,'(A,I0,A)') ' intersection dimension = ', idim, ' -> a line; two planes meet in a line'
write(*,'(A,ES12.3,A)') ' rank margin = ', marg, ' smallest RETAINED eigenvalue over tol; the third fell below'
write(*,'(A)') ' A lock is not graded. It either determines a point or it does not exist.'
!------------------------------------------------- ORIENTATION, NOT STRENGTH
! The Number cannot read the sign. That is a fact about DIRECTION and never
! about whether the determination is complete.
! The exhibit refuses to print rather than printing from an unusable frame.
call prepare_rows(M, Q, ok)
if (.not. ok) error stop 'orientation exhibit: rows carry no variance'
call span_basis(Q, B, ok)
if (.not. ok) error stop 'orientation exhibit: no independent frame'
call lambda_of(Q, B, lq, ld, co); R = matmul(Q, transpose(Q))
Mr = -M
! B is deliberately NOT rebuilt: the reflected rows are read against the
! frame fixed once, which is what makes the comparison a comparison.
call prepare_rows(Mr, Q, ok)
if (.not. ok) error stop 'orientation exhibit: reflected rows carry no variance'
call lambda_of(Q, B, lam2, d2, co)
Rr = matmul(Q, transpose(Q))
write(*,'(/,A)') ' WHAT THE NUMBER CANNOT READ: DIRECTION, NOT COMPLETENESS'
write(*,'(A,F16.12,A,F16.12)') ' lambda(P) = ', lq, ' lambda(not P) = ', lam2
write(*,'(A,ES12.3)') ' max|G(P) - G(not P)| = ', maxval(abs(R - Rr))
write(*,'(A,ES12.3)') ' |det(R)(P) - det(R)(not P)| = ', abs(det3(R) - det3(Rr))
write(*,'(A)') ' The intersection is equally unique either way. The Tongue carries the sign.'
i4 = [0.0_dp,1.0_dp,0.0_dp,0.0_dp]; j4 = [0.0_dp,0.0_dp,1.0_dp,0.0_dp]
k4 = [0.0_dp,0.0_dp,0.0_dp,1.0_dp]; p4 = qmul(qmul(i4,j4),k4)
write(*,'(A,F14.10)') ' substrate chirality Re(i*j*k) = ', p4(1)
Ret(1,:) = sin(t); Ret(2,:) = cos(t); Ret(3,:) = sin(2.0_dp*t)
call kernel_determine(Ret, v, lam, detR, kapR, idim, why)
write(*,'(A,A)') ' the Return : ', trim(token(v))
write(*,'(A,A)') ' why = ', trim(why)
if (v == DETERMINED) then
write(*,'(A,F16.12,A,F16.12)') ' det(R) = ', detR, ' |lambda| = ', abs(lam)
end if
!------------------------------------------------- THE GROUND
sig = [1.0_dp,-1.0_dp,-1.0_dp,-1.0_dp]; dia = [-1.0_dp,-1.0_dp,-1.0_dp,-1.0_dp]
write(*,'(/,A)') ' THE GROUND AND THE ROUTER BIT'
write(*,'(A,I0,A,A)') ' sigma Ground dim = ', ground_dim(sig), ' -> ', trim(route_halt(ground_dim(sig)))
write(*,'(A,I0,A,A)') ' -I Ground dim = ', ground_dim(dia), ' -> ', trim(route_halt(ground_dim(dia)))
!------------------------------------------------- SEAL L AND THE DRILLS
slots_ok = reshape([101,102,0, 201,202,0, 301,302,0],[3,3],order=[2,1])
call seal_L(slots_ok, vL, why)
write(*,'(/,A)') ' SEAL L AND THE SIX DRILLS'
write(*,'(A,A,A,A)') ' Seal L : ', trim(token(vL)), ' :: ', trim(why)
call drill_screen(.true.,.true.,.true.,.false.,.true.,.true.,.true., dok, cap, why)
write(*,'(A,L1,A,A,A,A)') ' unshaped terms: ', dok, ' cap=', trim(cap), ' :: ', trim(why)
call drill_screen(.true.,.true.,.true.,.true., .true.,.true.,.true., dok, cap, why)
write(*,'(A,L1,A,A,A,A)') ' shaped terms : ', dok, ' cap=', trim(cap), ' :: ', trim(why)
call drill_screen(.true.,.true.,.false.,.true.,.true.,.true.,.true., dok, cap, why)
write(*,'(A,L1,A,A,A,A)') ' unfiled flag : ', dok, ' cap=', trim(cap), ' :: ', trim(why)
!------------------------------------------------- THE ROW CASCADE
write(*,'(/,A)') ' THE ROW CASCADE'
call row_cascade(.true.,.true.,.true.,.false.,.true.,.false.,.true.,.true., comp, why)
write(*,'(A,A)') ' a closed residence has no outside -> ', trim(compartment_name(comp))
write(*,'(A,A)') ' ', trim(why)
write(*,'(A,A)') ' ', trim(compartment_grade(comp,'theorem'))
call row_cascade(.true.,.true.,.true.,.false.,.false.,.false.,.true.,.true., comp, why)
write(*,'(A,A)') ' every zero lies on the critical line -> ', trim(compartment_name(comp))
write(*,'(A,A)') ' ', trim(why)
call row_cascade(.true.,.true.,.true.,.false.,.true.,.true.,.true.,.true., comp, why)
write(*,'(A,A)') ' confined existents carry positive E_k -> ', trim(compartment_name(comp))
write(*,'(A,A)') ' ', trim(why)
write(*,'(A,A)') ' ', trim(compartment_grade(comp,'theorem'))
call row_cascade(.true.,.true.,.true.,.true., .true.,.false.,.true.,.true., comp, why)
write(*,'(A,A)') ' any dated forward projection -> ', trim(compartment_name(comp))
write(*,'(A,A)') ' ', trim(why)
call row_cascade(.true.,.true.,.true.,.false.,.true.,.false.,.true.,.false., comp, why)
write(*,'(A,A)') ' this man is a bachelor -> ', trim(compartment_name(comp))
write(*,'(A,A)') ' ', trim(why)
call row_cascade(.false.,.true.,.true.,.false.,.true.,.false.,.true.,.true., comp, why)
write(*,'(A,A)') ' a contentless tautology -> ', trim(compartment_name(comp))
write(*,'(A,A)') ' ', trim(why)
write(*,'(/,A)') repeat('=',78)
write(*,'(A)') ' A lock is always a determination and is total about its object.'
write(*,'(A)') ' Its object is its intersection. The compartment names what the rows were.'
write(*,'(A)') ' Compartment III pays revisability for direction. Compartment I has neither.'
write(*,'(A)') ' Neither is the better lock. They are locks on different rows. Delta-M = 0.'
write(*,'(A)') repeat('=',78)
!------------------------------------------------- THE BATTERY, PROOF OF LOAD
call battery()
contains
! Eight fuzz classes: seven over the numerical kernel, one over every
! symbolic and router public interface, then the oracle. Every check is a
! computed logical. The program stops here on any failure, so the final
! line of output is itself the receipt.
subroutine battery()
use, intrinsic :: ieee_arithmetic
integer, parameter :: NB = 8
real(dp) :: H, T, nanv, pinf, ninf
real(dp) :: mags(11)
real(dp) :: M(3,NB), Q(3,NB), B(3,NB), G(3,3), A(3,3), w(3), sep
real(dp) :: lam, detR, kapR, marg, lq, ld, co(3,3), q4(4), r4(4)
real(dp) :: dl, ds(5), etas(6), scales(5), rho(3), tt(NB)
integer :: i, j, k, v, idim
character(len=160) :: why
logical :: ok
fails = 0; checks = 0
H = sqrt(huge(1.0_dp)); T = sqrt(tiny(1.0_dp))
nanv = ieee_value(1.0_dp, ieee_quiet_nan)
pinf = ieee_value(1.0_dp, ieee_positive_inf)
ninf = ieee_value(1.0_dp, ieee_negative_inf)
mags = [0.0_dp, tiny(1.0_dp), T, epsilon(1.0_dp), 1.0_dp, 1.0_dp/epsilon(1.0_dp), H, &
huge(1.0_dp)/4.0_dp, huge(1.0_dp), -1.0_dp, -H]
do i = 1, NB
tt(i) = 6.283185307179586_dp*real(i-1,dp)/real(NB,dp)
end do
!---------------------------------------------------------------- CLASS 1
! Non-finite hostile inputs into every pure entry: NaN or .false. out.
call assert(.not. ieee_is_finite(snorm([1.0_dp, nanv])), 'snorm NaN -> NaN')
call assert(.not. ieee_is_finite(snorm([1.0_dp, pinf])), 'snorm +Inf -> NaN')
G = 1.0_dp; G(2,2) = ninf
call assert(.not. ieee_is_finite(det3(G)), 'det3 -Inf -> NaN')
call assert(.not. det3_representable(G), 'det3_representable -Inf -> F')
G(2,2) = nanv
call assert(.not. det3_representable(G), 'det3_representable NaN -> F')
q4 = [nanv,0.0_dp,0.0_dp,0.0_dp]; r4 = [1.0_dp,0.0_dp,0.0_dp,0.0_dp]
call assert(.not. all(ieee_is_finite(qmul(q4,r4))), 'qmul NaN -> NaN')
call assert(ground_dim([1.0_dp,pinf,-1.0_dp,-1.0_dp]) == -1, 'ground_dim Inf -> unmeasured')
! kernel: a non-finite row must route OPEN with a mechanism, never SEALED/BROKEN
M(1,:) = sin(tt); M(2,:) = cos(2.0_dp*tt); M(3,:) = sin(3.0_dp*tt); M(2,4) = nanv
call kernel_determine(M, v, lam, detR, kapR, idim, why, marg)
call assert(v == OPENV .and. index(why,'non-quantity') > 0, 'kernel NaN row -> OPEN, mechanism named')
M(2,4) = pinf
call kernel_determine(M, v, lam, detR, kapR, idim, why, marg)
call assert(v == OPENV, 'kernel Inf row -> OPEN')
!---------------------------------------------------------------- CLASS 2
! Extreme finite magnitudes: no overflow, no false PSD failure, no NaN after unscale.
do i = 1, 11
G = 0.0_dp; G(1,1) = abs(mags(i)); G(2,2) = abs(mags(i)); G(3,3) = abs(mags(i))
if (abs(mags(i)) > 0.0_dp .and. abs(mags(i)) < huge(1.0_dp)) then
call intersection_dim(G, idim, sep)
call assert(idim == 0, 'scaled identity rank 3 at magnitude index '//itoa(i))
end if
end do
do i = 1, NB
Q(1,i) = H*sin(tt(i)); Q(2,i) = H*cos(2.0_dp*tt(i)); Q(3,i) = H*sin(3.0_dp*tt(i))
end do
call span_basis(Q, B, ok)
call assert(ok .and. finite_mat(B), 'span_basis at sqrt(huge): finite frame')
do i = 1, NB
Q(1,i) = T*sin(tt(i)); Q(2,i) = T*cos(2.0_dp*tt(i)); Q(3,i) = T*sin(3.0_dp*tt(i))
end do
call span_basis(Q, B, ok)
call assert(ok .and. finite_mat(B), 'span_basis at sqrt(tiny): finite frame')
A = 0.0_dp; A(1,1) = H; A(2,2) = T; A(3,3) = 1.0_dp
call jacobi3(A, w)
call assert(finite_vec(w), 'jacobi3 diag(H,T,1) finite')
A = H; call jacobi3(A, w)
call assert(finite_vec(w), 'jacobi3 all-H finite (3H fits)')
A = huge(1.0_dp)/2.0_dp; call jacobi3(A, w)
call assert(.not. finite_vec(w), 'jacobi3 all-huge/2: unrepresentable reported as Inf')
! det3 equilibration
G = 0.0_dp; G(1,1)=1.0e200_dp; G(2,2)=1.0e200_dp; G(3,3)=1.0e-200_dp
call assert(abs(det3(G)/1.0e200_dp - 1.0_dp) < 1.0e-12_dp, 'det3 diag(1e200,1e200,1e-200) = 1e200')
call assert(det3_representable(G), ' and representable')
G = 0.0_dp; G(1,1)=1.0e308_dp; G(2,2)=1.0e308_dp; G(3,3)=1.0e-20_dp
call assert(.not. det3_representable(G), 'det3 diag(1e308,1e308,1e-20) NOT representable')
G = 0.0_dp; G(1,1)=H; G(2,2)=T; G(3,3)=1.0_dp
call assert(det3_representable(G) .and. abs(det3(G)-H*T) <= 1.0e-12_dp*H*T, &
'det3 diag(H,T,1) = H*T')
!---------------------------------------------------------------- CLASS 3
! Near-degenerate rank transitions, and scale stability of the classification.
ds = [epsilon(1.0_dp), sqrt(epsilon(1.0_dp)), 1.0e-12_dp, 1.0e-9_dp, 1.0e-6_dp]
scales = [tiny(1.0_dp)*1.0e10_dp, T, 1.0_dp, H/16.0_dp, huge(1.0_dp)/16.0_dp]
do i = 1, 5
dl = ds(i)
etas = [0.0_dp, tiny(1.0_dp), epsilon(1.0_dp), dl*dl, dl, sqrt(dl)]
do j = 1, 6
G = reshape([1.0_dp, 0.0_dp, 0.0_dp, 1.0_dp, dl, 0.0_dp, 1.0_dp, 2.0_dp*dl, etas(j)], [3,3], order=[2,1])
G = matmul(G, transpose(G))
call intersection_dim(G, idim, sep)
do k = 1, 5
block
real(dp) :: Gs(3,3); integer :: ids
Gs = scales(k)*G
if (finite_mat(Gs)) then
call intersection_dim(Gs, ids, sep)
call assert(ids == idim, 'rank scale-stable d='//itoa(i)//' e='//itoa(j)//' S='//itoa(k))
end if
end block
end do
end do
end do
!---------------------------------------------------------------- CLASS 4/5
! Signed zero and reflection: determinant sign changes only under odd reflection.
G = 0.0_dp; G(1,1)=1.0_dp; G(2,2)=1.0_dp; G(3,3)=1.0_dp
A = G; A(1,2) = -0.0_dp; A(2,1) = -0.0_dp; A(3,2) = -0.0_dp
call assert(abs(det3(A) - det3(G)) <= 0.0_dp, 'signed zero leaves det unchanged, exactly')
A = G; A(1,1) = -1.0_dp
call assert(abs(det3(A) + det3(G)) <= 0.0_dp, 'single reflection flips det, exactly')
A = G; A(1,1) = -1.0_dp; A(2,2) = -1.0_dp
call assert(abs(det3(A) - det3(G)) <= 0.0_dp, 'double reflection preserves det, exactly')
! quaternion chirality agrees with the determinant route on a random-ish frame
do i = 1, NB
M(1,i) = sin(tt(i)); M(2,i) = cos(2.0_dp*tt(i)) + 0.3_dp*sin(tt(i)); M(3,i) = sin(3.0_dp*tt(i))
end do
call prepare_rows(M, Q, ok); call span_basis(Q, B, ok); call lambda_of(Q, B, lq, ld, co)
call assert(abs(lq - ld) <= 100.0_dp*epsilon(1.0_dp)*max(abs(ld),1.0_dp), &
'lambda quaternion == determinant route')
Q(1,:) = -Q(1,:); call lambda_of(Q, B, lam, detR, co)
! Under the strict IEEE, no-reassociation build this battery is run with,
! this is exact and stays exact. IEEE negation, multiplication and
! addition are sign-symmetric bit for bit, and scale() is a power-of-two
! multiply, so negating one row negates the composed scalar with no
! rounding anywhere on the route. The boot's own layer two tests the same
! flip with ==, and the register records it as the made-zero: the sign is
! displaced, never annihilated. A tolerance here would concede rounding
! the arithmetic does not commit.
call assert(abs(lam + lq) <= 0.0_dp, 'reflecting one row inverts lambda, exactly')
!---------------------------------------------------------------- CLASS 5b
! Hostile rows against lambda_of, in-process through its own predicate.
! The cancelling row [H,H,-H,-H]/2 has scaled norm H and must be refused at
! the unit gate before any coordinate dot could form.
block
real(dp) :: Qh(3,NB), Bh(3,NB)
character(len=96) :: rsn
logical :: adm
Qh = 0.0_dp; Bh = 0.0_dp
Qh(1,1:4) = [H, H, -H, -H]/2.0_dp; Qh(2,5) = 1.0_dp; Qh(3,6) = 1.0_dp
Bh(1,1) = 1.0_dp; Bh(2,2) = 1.0_dp; Bh(3,3) = 1.0_dp
call lambda_of_admissible(Qh, Bh, adm, rsn)
call assert(.not. adm .and. index(rsn,'Q rows are not unit') > 0, &
'lambda_of refuses the cancelling H-row at the unit gate')
Qh(1,:) = Bh(1,:); Qh(2,:) = Bh(2,:); Qh(3,:) = Bh(3,:)
Bh(1,1:4) = [H, H, -H, -H]/2.0_dp
call lambda_of_admissible(Qh, Bh, adm, rsn)
call assert(.not. adm .and. index(rsn,'B rows are not unit') > 0, &
'lambda_of refuses the cancelling H-row in B at the unit gate')
Bh(1,:) = 0.0_dp; Bh(1,1) = nanv
call lambda_of_admissible(Qh, Bh, adm, rsn)
call assert(.not. adm .and. index(rsn,'not finite') > 0, &
'lambda_of refuses NaN before any arithmetic')
Bh(1,:) = 0.0_dp; Bh(1,1) = 1.0_dp
call lambda_of_admissible(Qh, Bh, adm, rsn)
call assert(adm, 'lambda_of admits an orthonormal identity frame')
end block
! qmul at the band the norm bound would have refused and the components fit
q4 = [0.0_dp, huge(1.0_dp)/4.0_dp, 0.0_dp, 0.0_dp]
r4 = [0.0_dp, 0.0_dp, 2.0_dp, 0.0_dp]
block
real(dp) :: c4(4)
c4 = qmul(q4, r4)
call assert(finite_vec(c4), 'qmul: (huge/4)i * 2j is representable and returned finite')
call assert(abs(c4(4) - huge(1.0_dp)/2.0_dp) <= epsilon(1.0_dp)*huge(1.0_dp), &
'qmul: (huge/4)i * 2j = (huge/2)k exactly to rounding')
q4 = [0.0_dp, huge(1.0_dp)/2.0_dp, 0.0_dp, 0.0_dp]
r4 = [0.0_dp, 0.0_dp, 4.0_dp, 0.0_dp]
c4 = qmul(q4, r4)
call assert(.not. finite_vec(c4), 'qmul: (huge/2)i * 4j = 2huge k is refused as NaN, never wrapped')
end block
!---------------------------------------------------------------- CLASS 8
! The symbolic and router procedures, fuzzed as the numeric kernel is: every
! token typed, every compartment named, every halt branch routed, every
! Seal L refusal reached, every drill failure routed, every cascade gate
! taken. A public interface with no computed check is not covered.
block
integer :: sl(3,3), sl4(4,3), cmp, tk
character(len=140) :: rsn, cap
logical :: dok
! tokens: three states, two refinements, one kernel token, one dot, one untyped
call assert(trim(token_class(SEALED)) == 'verdict state' .and. trim(token_class(BROKEN)) == 'verdict state' &
.and. trim(token_class(OPENV)) == 'verdict state', 'token_class: the three states')
call assert(trim(token_class(XI0)) == 'refinement inside openness' .and. &
trim(token_class(OH0)) == 'refinement inside openness', 'token_class: both refinements inside openness')
call assert(trim(token_class(DETERMINED)) == 'kernel geometric token', 'token_class: the kernel token')
call assert(trim(token_class(DOTMARK)) == 'outside the economy', 'token_class: the dot outside the economy')
call assert(trim(token_class(99)) == 'untyped' .and. index(token(99),'untyped') > 0, 'token_class/token: untyped')
do tk = 1, 7
call assert(len_trim(token(tk)) > 0, 'token: every token prints')
end do
! compartments and the router
call assert(index(compartment_name(CLOSURE_ROWED),'CLOSURE-ROWED') > 0 .and. &
index(compartment_name(UNPOPULATED),'UNPOPULATED') > 0 .and. &
index(compartment_name(WORLD_ROWED),'WORLD-ROWED') > 0 .and. &
index(compartment_name(NOT_COMPARTMENTED),'not compartmented') > 0, 'compartment_name: all four')
call assert(index(route_halt(1),'Xi') > 0 .and. index(route_halt(0),'B.14.O') > 0 .and. &
index(route_halt(-1),'unmeasured') > 0, 'route_halt: dim 1, dim 0, unmeasured')
call assert(index(route_halt(2),'unmeasured') > 0 .and. index(route_halt(7),'unmeasured') > 0, &
'route_halt: unexpected positive dims are unmeasured')
call assert(index(compartment_grade(CLOSURE_ROWED,'theorem'),'theorem') > 0 .and. &
index(compartment_grade(UNPOPULATED,'x'),'no determination') > 0 .and. &
index(compartment_grade(WORLD_ROWED,'x'),'rows') > 0, 'compartment_grade: all three')
call assert(trim(compartment_grade(99,'x')) == 'no compartment', 'compartment_grade: default')
! Seal L: every refusal and the one admission
sl = reshape([101,102,0, 201,202,0, 301,302,0],[3,3],order=[2,1])
call seal_L(sl, tk, rsn); call assert(tk == SEALED, 'seal_L: clean three slots seal')
sl = reshape([101,102,0, 201,101,0, 301,302,0],[3,3],order=[2,1])
call seal_L(sl, tk, rsn); call assert(tk == BROKEN .and. index(rsn,'LIT') > 0, 'seal_L: cross-slot collision')
sl = reshape([101,101,0, 201,202,0, 301,302,0],[3,3],order=[2,1])
call seal_L(sl, tk, rsn); call assert(tk == BROKEN .and. index(rsn,'repeated') > 0, 'seal_L: within-slot repeat')
sl = reshape([101,-1,0, 201,202,0, 301,302,0],[3,3],order=[2,1])
call seal_L(sl, tk, rsn); call assert(tk == BROKEN .and. index(rsn,'negative') > 0, 'seal_L: negative id')
sl = reshape([101,102,0, 0,0,0, 301,302,0],[3,3],order=[2,1])
call seal_L(sl, tk, rsn); call assert(tk == BROKEN .and. index(rsn,'2 slots') > 0, 'seal_L: two slots, not three')
sl4 = reshape([101,0,0, 201,0,0, 301,0,0, 401,0,0],[4,3],order=[2,1])
call seal_L(sl4, tk, rsn); call assert(tk == BROKEN .and. index(rsn,'4 slots') > 0, 'seal_L: four slots counted, not three')
! drills: each failure routes, both caps
call drill_screen(.false.,.true.,.true.,.false.,.true.,.true.,.true., dok, cap, rsn)
call assert(.not. dok .and. index(rsn,'D1/D2') > 0, 'drill: D1 fails')
call drill_screen(.true.,.false.,.true.,.false.,.true.,.true.,.true., dok, cap, rsn)
call assert(.not. dok .and. index(rsn,'D1/D2') > 0, 'drill: D2 fails')
call drill_screen(.true.,.true.,.false.,.false.,.true.,.true.,.true., dok, cap, rsn)
call assert(.not. dok .and. index(rsn,'D3') > 0, 'drill: unfiled provenance')
call drill_screen(.true.,.true.,.true.,.false.,.false.,.true.,.true., dok, cap, rsn)
call assert(.not. dok .and. index(rsn,'D4') > 0, 'drill: magnitude on the Tongue')
call drill_screen(.true.,.true.,.true.,.false.,.true.,.false.,.true., dok, cap, rsn)
call assert(.not. dok .and. index(rsn,'D5') > 0, 'drill: register undeclared')
call drill_screen(.true.,.true.,.true.,.false.,.true.,.true.,.false., dok, cap, rsn)
call assert(.not. dok .and. index(rsn,'D6') > 0, 'drill: blindness reproduced')
call drill_screen(.true.,.true.,.true.,.false.,.true.,.true.,.true., dok, cap, rsn)
call assert(dok .and. trim(cap) == 'theorem', 'drill: unshaped caps at theorem')
call drill_screen(.true.,.true.,.true.,.true.,.true.,.true.,.true., dok, cap, rsn)
call assert(dok .and. trim(cap) == 'structural', 'drill: shaped caps at structural')
! cascade: every gate taken in order
call row_cascade(.false.,.true.,.true.,.false.,.true.,.false.,.true.,.true., cmp, rsn)
call assert(cmp == NOT_COMPARTMENTED .and. index(rsn,'F-0') > 0, 'cascade: F-0 refuses')
call row_cascade(.true.,.false.,.true.,.false.,.true.,.false.,.true.,.true., cmp, rsn)
call assert(cmp == NOT_COMPARTMENTED .and. index(rsn,'R-1') > 0, 'cascade: R-1 faces')
call row_cascade(.true.,.true.,.false.,.false.,.true.,.false.,.true.,.true., cmp, rsn)
call assert(cmp == NOT_COMPARTMENTED .and. index(rsn,'R-2') > 0, 'cascade: R-2 canonical')
call row_cascade(.true.,.true.,.true.,.true.,.false.,.false.,.true.,.true., cmp, rsn)
call assert(cmp == UNPOPULATED .and. index(rsn,'S-1') > 0, 'cascade: unpopulated forward is II, not III')
call row_cascade(.true.,.true.,.true.,.true.,.true.,.false.,.true.,.true., cmp, rsn)
call assert(cmp == WORLD_ROWED .and. index(rsn,'R-3') > 0, 'cascade: populated forward is III')
call row_cascade(.true.,.true.,.true.,.false.,.true.,.true.,.true.,.true., cmp, rsn)
call assert(cmp == WORLD_ROWED .and. index(rsn,'S-2') > 0, 'cascade: worldly row is III')
call row_cascade(.true.,.true.,.true.,.false.,.true.,.false.,.false.,.true., cmp, rsn)
call assert(cmp == WORLD_ROWED .and. index(rsn,'S-3') > 0, 'cascade: unscreened terms route by rows')
call row_cascade(.true.,.true.,.true.,.false.,.true.,.false.,.true.,.false., cmp, rsn)
call assert(cmp == WORLD_ROWED .and. index(rsn,'S-4') > 0, 'cascade: frame leak names the compartment')
call row_cascade(.true.,.true.,.true.,.false.,.true.,.false.,.true.,.true., cmp, rsn)
call assert(cmp == CLOSURE_ROWED, 'cascade: closure-rowed reached')
! the router bit on constructed and solver-noisy diagonals
call assert(ground_dim([1.0_dp,-1.0_dp,-1.0_dp,-1.0_dp]) == 1 .and. &
ground_dim([1.0_dp-2.0_dp*epsilon(1.0_dp),-1.0_dp,-1.0_dp,-1.0_dp]) == 1 .and. &
ground_dim([-1.0_dp,-1.0_dp,-1.0_dp,-1.0_dp]) == 0, 'ground_dim: exact, noisy, and diagonal')
end block
!---------------------------------------------------------------- CLASS 6
! PSD boundary for cond_sym3: tolerance relative to spectral scale only.
A = 0.0_dp; A(1,1)=1.0_dp; A(2,2)=1.0_dp; A(3,3)=-epsilon(1.0_dp)
call assert(ieee_is_finite(cond_sym3(A)) .or. cond_sym3(A) >= huge(1.0_dp), 'diag(1,1,-eps) tolerated')
A = 0.0_dp; A(1,1)=H; A(2,2)=H; A(3,3)=-epsilon(1.0_dp)*H
call assert(cond_sym3(A) >= 0.0_dp, 'diag(H,H,-eps*H) tolerated relative to scale')
A = 0.0_dp; A(1,1)=1.0_dp; A(2,2)=1.0_dp; A(3,3)=0.0_dp
call assert(cond_sym3(A) >= huge(1.0_dp), 'diag(1,1,0) -> huge conditioning')
!---------------------------------------------------------------- CLASS 7
! Eigenvalue unscale across scales and structures.
rho = [1.0_dp-epsilon(1.0_dp), 1.0_dp-sqrt(epsilon(1.0_dp)), -1.0_dp+epsilon(1.0_dp)]
do k = 1, 5
do i = 1, 3
A = 0.0_dp; A(1,1)=1.0_dp; A(2,2)=1.0_dp; A(3,3)=epsilon(1.0_dp); A(1,2)=rho(i); A(2,1)=rho(i)
A = scales(k)*A
if (finite_mat(A)) then
call jacobi3(A, w)
call assert(finite_vec(w) .or. scales(k) > H, &
'unscale finite where representable S='//itoa(k)//' rho='//itoa(i))
end if
end do
end do
write(*,'(A)') repeat('=',78)
write(*,'(A,I0,A,I0)') ' THE BATTERY checks ', checks, ' failures ', fails
write(*,'(A)') repeat('=',78)
if (fails > 0) error stop 'THE BATTERY FAILED: the load is not proven'
write(*,'(A)') ' The load is proven. A lock is always a determination. Delta-M = 0.'
end subroutine battery
subroutine assert(cond, label)
logical, intent(in) :: cond
character(len=*), intent(in) :: label
checks = checks + 1
if (.not. cond) then
fails = fails + 1
write(*,'(A,A)') ' FAIL ', label
end if
end subroutine assert
function itoa(n) result(s)
integer, intent(in) :: n
character(len=12) :: s
write(s,'(I0)') n
end function itoa
end program thesis_rows
-/
/-
! =====================================================================
! RA_TOE_Thesis_Fortran v2.0.0 - Trisduction Research Group
! Sealed edition, consolidated. Numerics byte-identical to v1.23.0
! (sha256 97e5585fa15e66ff0c2e5d0b63daae550b27af61f5bb9402e67e0b6f4400
! 1212); presentation-major: narrative removed, contracts kept. This
! source is written for AI substrates, not manual readers; comments
! are contracts, section markers, and nothing else.
!
! CONSOLIDATED HISTORY. v1.0.0 through v1.23.0, twenty-two versions
! under two external auditors and a 178000-iteration deterministic
! fuzz campaign: fifty-six findings, six refused on printed receipts,
! fifty earned and repaired. Hardening families: exponent-tracked
! Leibniz determinants (det3, det3_row2), fraction-exponent Hamilton
! products (qmul), public-guard refusal precedence, census lock,
! audit-mode VOID banner, claim-register alignment. Four FORGE
! self-audit cycles (ratoe..ratoe4) sealed at the SELF floor. The
! full per-version ledger lives in git, 1000sapients/Trisduction,
! protocols/Executable Thesis/, and codex card 0712; it is not
! restated here.
!
! v2.0.0 (2026-09-08): consolidation round. Fixes on external static
! review: stale printed receipt pointer rebound to the version-matched
! file; two dead a(3,3) locals removed from the determinant
! primitives; one non-significant-digit literal given its kind. The
! 113 remaining -Wcompare-reals warnings are the exact-equality
! verdict design (bit-identity checks, tie checks, NaN self-tests)
! and are intentional; the sealed flag set is the contract.
!
! BUILD (sealed): gfortran -std=f2018 -O2 -fno-fast-math
! -ffp-contract=off -Wall -Wextra -Wconversion-extra
! Modes: default sealed halt-on-fail; 'audit' accumulates and voids;
! 'witnessed' asserts the PART H live face through the aperture.
! Hostile builds (-Ofast; fused contraction) must exit nonzero.
! =====================================================================
module ra_kinds
use, intrinsic :: ieee_arithmetic, only: ieee_value, ieee_quiet_nan, &
ieee_is_finite, ieee_is_nan
use, intrinsic :: iso_fortran_env, only: int64
implicit none
integer, parameter :: wp = selected_real_kind(15, 307)
integer, parameter :: ik = int64 ! pinned: exactly 64 bits, checked at boot
real(wp), parameter :: HBAR = 1.054571817e-34_wp ! J s (CODATA exact-defined path)
real(wp), parameter :: KB = 1.380649e-23_wp ! J/K (exact, SI 2019)
real(wp), parameter :: ME = 9.1093837015e-31_wp ! kg
real(wp), parameter :: U_M = epsilon(1.0_wp) ! unit roundoff, double
integer(ik), parameter :: IK_SAFE = 536870912_ik ! 2^29
contains
pure function pi_() result(p)
real(wp) :: p
p = acos(-1.0_wp)
end function pi_
pure function ln2_() result(l)
real(wp) :: l
l = log(2.0_wp)
end function ln2_
pure function qnan() result(z)
real(wp) :: z
z = ieee_value(1.0_wp, ieee_quiet_nan)
end function qnan
elemental logical function is_finite(x)
real(wp), intent(in) :: x
is_finite = ieee_is_finite(x)
end function is_finite
elemental logical function is_nan(x)
real(wp), intent(in) :: x
is_nan = ieee_is_nan(x)
end function is_nan
end module ra_kinds
! ---------------------------------------------------------------------
module ra_prims
use ra_kinds
implicit none
contains
pure function vnorm(v) result(s)
real(wp), intent(in) :: v(:)
real(wp) :: s, m
integer :: i
m = 0.0_wp
do i = 1, size(v)
if (.not. is_finite(v(i))) then
s = qnan()
return
end if
m = max(m, abs(v(i)))
end do
if (m == 0.0_wp) then
s = 0.0_wp
else
s = m * sqrt(sum((v/m)**2))
end if
end function vnorm
! CONTRACT qmul: sixteen Hamilton terms as rounded fraction products with
! exactly tracked exponents, per-component classical order, common-max
! accumulation; product rounding, alignment underflow, and cancellation are
! possible; over/underflow of a restored component refers to the COMPUTED
! accumulation; general correctly-rounded products are NOT guaranteed.
pure function qmul(a, b) result(c)
real(wp), intent(in) :: a(4), b(4)
real(wp) :: c(4)
integer, parameter :: JA(4,4) = reshape([1,2,3,4, 1,2,3,4, 1,2,3,4, 1,2,3,4], [4,4])
integer, parameter :: KB(4,4) = reshape([1,2,3,4, 2,1,4,3, 3,4,1,2, 4,3,2,1], [4,4])
real(wp), parameter :: SG(4,4) = reshape([ 1.0_wp,-1.0_wp,-1.0_wp,-1.0_wp, &
1.0_wp, 1.0_wp, 1.0_wp,-1.0_wp, &
1.0_wp,-1.0_wp, 1.0_wp, 1.0_wp, &
1.0_wp, 1.0_wp,-1.0_wp, 1.0_wp], [4,4])
real(wp) :: x1, x2, f(4), ssum
integer :: i, t, e(4), emax, nkeep
logical :: keep(4)
do i = 1, 4
if (.not.(is_finite(a(i)) .and. is_finite(b(i)))) then
c = qnan()
return
end if
end do
do i = 1, 4
nkeep = 0; emax = 0
do t = 1, 4
x1 = a(JA(t,i)); x2 = b(KB(t,i))
keep(t) = (x1 /= 0.0_wp .and. x2 /= 0.0_wp)
if (keep(t)) then
f(t) = fraction(x1) * fraction(x2)
e(t) = exponent(x1) + exponent(x2)
if (nkeep == 0 .or. e(t) > emax) emax = e(t)
nkeep = nkeep + 1
end if
end do
if (nkeep == 0) then
c(i) = 0.0_wp
cycle
end if
ssum = 0.0_wp
do t = 1, 4
if (keep(t)) ssum = ssum + SG(t,i) * scale(f(t), e(t) - emax)
end do
c(i) = scale(ssum, emax)
end do
end function qmul
pure function qconj(a) result(c)
real(wp), intent(in) :: a(4)
real(wp) :: c(4)
c = [a(1), -a(2), -a(3), -a(4)]
end function qconj
! CONTRACT det3 / det3_row2: six signed Leibniz monomials as rounded
! fraction products with exact exponents, fixed distinct accumulation
! orders, common-max restore; same caveats as qmul; the two routines share
! one IEEE model and differ by evaluation order and factorization only.
pure function det3(r) result(d)
real(wp), intent(in) :: r(3,3)
real(wp) :: d, s
integer :: i, j
s = 0.0_wp
do i = 1, 3
do j = 1, 3
if (.not. is_finite(r(i,j))) then
d = qnan()
return
end if
s = max(s, abs(r(i,j)))
end do
end do
if (s == 0.0_wp) then
d = 0.0_wp
return
end if
block
integer, parameter :: PM(3,6) = reshape([1,2,3, 1,3,2, 2,1,3, 2,3,1, 3,1,2, 3,2,1], [3,6])
real(wp), parameter :: SG(6) = [1.0_wp, -1.0_wp, -1.0_wp, 1.0_wp, 1.0_wp, -1.0_wp]
real(wp) :: x1, x2, x3, f(6), ssum
integer :: e(6), emax, kk, nkeep
logical :: keep(6)
nkeep = 0; emax = 0
do kk = 1, 6
x1 = r(1, PM(1,kk)); x2 = r(2, PM(2,kk)); x3 = r(3, PM(3,kk))
keep(kk) = (x1 /= 0.0_wp .and. x2 /= 0.0_wp .and. x3 /= 0.0_wp)
if (keep(kk)) then
f(kk) = fraction(x1) * fraction(x2) * fraction(x3)
e(kk) = exponent(x1) + exponent(x2) + exponent(x3)
if (nkeep == 0 .or. e(kk) > emax) emax = e(kk)
nkeep = nkeep + 1
end if
end do
if (nkeep == 0) then
d = 0.0_wp ! every monomial structurally zero: exactly zero
return
end if
ssum = 0.0_wp
do kk = 1, 6
if (keep(kk)) ssum = ssum + SG(kk) * scale(f(kk), e(kk) - emax)
end do
if (ssum == 0.0_wp) then
d = 0.0_wp ! cancellation at the accumulation's resolution
return
end if
d = scale(ssum, emax)
end block
end function det3
pure function det3_row2(r) result(d)
real(wp), intent(in) :: r(3,3)
real(wp) :: d, s
integer :: i, j
s = 0.0_wp
do i = 1, 3
do j = 1, 3
if (.not. is_finite(r(i,j))) then
d = qnan()
return
end if
s = max(s, abs(r(i,j)))
end do
end do
if (s == 0.0_wp) then
d = 0.0_wp
return
end if
block
integer, parameter :: PM(3,6) = reshape([2,1,3, 3,1,2, 1,2,3, 3,2,1, 1,3,2, 2,3,1], [3,6])
real(wp), parameter :: SG(6) = [-1.0_wp, 1.0_wp, 1.0_wp, -1.0_wp, -1.0_wp, 1.0_wp]
real(wp) :: x1, x2, x3, f(6), ssum
integer :: e(6), emax, kk, nkeep
logical :: keep(6)
nkeep = 0; emax = 0
do kk = 1, 6
x1 = r(1, PM(1,kk)); x2 = r(2, PM(2,kk)); x3 = r(3, PM(3,kk))
keep(kk) = (x1 /= 0.0_wp .and. x2 /= 0.0_wp .and. x3 /= 0.0_wp)
if (keep(kk)) then
f(kk) = fraction(x1) * fraction(x2) * fraction(x3)
e(kk) = exponent(x1) + exponent(x2) + exponent(x3)
if (nkeep == 0 .or. e(kk) > emax) emax = e(kk)
nkeep = nkeep + 1
end if
end do
if (nkeep == 0) then
d = 0.0_wp ! every monomial structurally zero: exactly zero
return
end if
ssum = 0.0_wp
do kk = 1, 6
if (keep(kk)) ssum = ssum + SG(kk) * scale(f(kk), e(kk) - emax)
end do
if (ssum == 0.0_wp) then
d = 0.0_wp ! cancellation at the accumulation's resolution
return
end if
d = scale(ssum, emax)
end block
end function det3_row2
subroutine jacobi3(rin, eig, converged)
real(wp), intent(in) :: rin(3,3)
real(wp), intent(out) :: eig(3)
logical, intent(out), optional :: converged
real(wp) :: a(3,3), app, aqq, apq, phi, c, s, tmp, off, frob
integer :: sweep, p, q, k
logical :: conv
real(wp) :: sscale
conv = .false.
if (present(converged)) converged = conv
if (any(.not. is_finite(rin))) then
eig = qnan()
return
end if
a = 0.5_wp*rin + 0.5_wp*transpose(rin)
sscale = maxval(abs(a))
if (sscale == 0.0_wp) then
eig = 0.0_wp
conv = .true.
if (present(converged)) converged = conv
return
end if
a = a / sscale
frob = sqrt(sum(a*a))
do sweep = 1, 60
off = sqrt(a(1,2)**2 + a(1,3)**2 + a(2,3)**2 &
+ a(2,1)**2 + a(3,1)**2 + a(3,2)**2)
if (off <= 16.0_wp*U_M*max(frob, tiny(1.0_wp))) then
conv = .true.
exit
end if
do p = 1, 2
do q = p+1, 3
apq = a(p,q)
if (abs(apq) <= tiny(1.0_wp)) cycle
app = a(p,p); aqq = a(q,q)
phi = 0.5_wp * atan2(2.0_wp*apq, aqq - app)
c = cos(phi); s = sin(phi)
do k = 1, 3
tmp = c*a(p,k) - s*a(q,k)
a(q,k) = s*a(p,k) + c*a(q,k)
a(p,k) = tmp
end do
do k = 1, 3
tmp = c*a(k,p) - s*a(k,q)
a(k,q) = s*a(k,p) + c*a(k,q)
a(k,p) = tmp
end do
a(p,q) = 0.0_wp
a(q,p) = 0.0_wp
end do
end do
end do
eig = sscale * [a(1,1), a(2,2), a(3,3)]
if (present(converged)) converged = conv
do p = 1, 2
do q = p+1, 3
if (eig(q) < eig(p)) then
tmp = eig(p); eig(p) = eig(q); eig(q) = tmp
end if
end do
end do
end subroutine jacobi3
function cond3(r) result(kap)
real(wp), intent(in) :: r(3,3)
real(wp) :: kap, eig(3)
logical :: conv
call jacobi3(r, eig, conv)
if (any(is_nan(eig)) .or. .not. conv) then
kap = huge(1.0_wp)
return
end if
if (eig(1) <= 8.0_wp*U_M*max(abs(eig(3)), 1.0_wp)) then
kap = huge(1.0_wp)
else
kap = eig(3) / eig(1)
end if
end function cond3
pure subroutine gram3_fixed(q, r)
real(wp), intent(in) :: q(:,:)
real(wp), intent(out) :: r(3,3)
integer :: i, j, k
if (size(q,1) < 3) then
r = qnan()
return
end if
r = 0.0_wp
do i = 1, 3
do j = 1, 3
do k = 1, size(q,2)
r(i,j) = r(i,j) + q(i,k)*q(j,k)
end do
end do
end do
end subroutine gram3_fixed
pure function det3_chol(r) result(d)
real(wp), intent(in) :: r(3,3)
real(wp) :: d, l11, l21, l31, l22, l32, l33, t
d = qnan() ! default NaN when not SPD
if (any(.not. is_finite(r))) return
if (maxval(abs(r - transpose(r))) > 32.0_wp*U_M*max(1.0_wp, maxval(abs(r)))) return
if (r(1,1) <= 0.0_wp) return
l11 = sqrt(r(1,1)); l21 = r(2,1)/l11; l31 = r(3,1)/l11
t = r(2,2) - l21*l21
if (t <= 0.0_wp) return
l22 = sqrt(t); l32 = (r(3,2) - l21*l31)/l22
t = r(3,3) - l31*l31 - l32*l32
if (t <= 0.0_wp) return
l33 = sqrt(t)
d = (l11*l22*l33)**2
end function det3_chol
end module ra_prims
! ---------------------------------------------------------------------
module ra_floor
use ra_kinds
implicit none
contains
pure function heis_floor(m, dx) result(e)
real(wp), intent(in) :: m, dx
real(wp) :: e
if (.not.(is_finite(m) .and. is_finite(dx)) .or. m <= 0.0_wp .or. dx <= 0.0_wp) then
e = qnan()
return
end if
block
real(wp) :: f
integer :: ex
f = (fraction(HBAR)*fraction(HBAR)) / (fraction(m)*(fraction(dx)*fraction(dx)))
ex = 2*exponent(HBAR) - exponent(m) - 2*exponent(dx) - 3
e = scale(f, ex)
end block
end function heis_floor
pure function zpe(w) result(e)
real(wp), intent(in) :: w
real(wp) :: e
if (.not. is_finite(w) .or. w <= 0.0_wp) then
e = qnan()
return
end if
e = 0.5_wp * HBAR * w
end function zpe
pure function prod_scaled(a, b, c, d) result(r)
real(wp), intent(in) :: a, b, c, d
real(wp) :: r, f
integer :: e
if (a == 0.0_wp .or. b == 0.0_wp .or. c == 0.0_wp .or. d == 0.0_wp) then
r = 0.0_wp
return
end if
f = fraction(a) * fraction(b)
e = exponent(a) + exponent(b) + exponent(f)
f = fraction(f)
f = f * fraction(c)
e = e + exponent(c) + exponent(f)
f = fraction(f)
f = f * fraction(d)
e = e + exponent(d) + exponent(f)
f = fraction(f)
r = scale(f, e)
end function prod_scaled
pure function landauer(tkel, bits) result(e)
real(wp), intent(in) :: tkel, bits
real(wp) :: e
if (.not.(is_finite(tkel) .and. is_finite(bits)) .or. tkel <= 0.0_wp .or. bits < 0.0_wp) then
e = qnan()
return
end if
e = prod_scaled(bits, KB, tkel, ln2_())
end function landauer
subroutine tau_perp(de, tau, frozen)
real(wp), intent(in) :: de
real(wp), intent(out) :: tau
logical, intent(out) :: frozen
frozen = (.not. is_finite(de)) .or. (de <= 0.0_wp)
if (frozen) then
tau = huge(1.0_wp)
else
tau = (0.5_wp * pi_()) / de
end if
end subroutine tau_perp
pure function survival(t) result(p)
real(wp), intent(in) :: t
real(wp) :: p
p = cos(0.5_wp*t)**2
end function survival
end module ra_floor
! ---------------------------------------------------------------------
module ra_integers
use ra_kinds
implicit none
contains
function iqmul(a, b) result(c)
integer(ik), intent(in) :: a(4), b(4)
integer(ik) :: c(4)
if (any(a > IK_SAFE .or. a < -IK_SAFE) .or. any(b > IK_SAFE .or. b < -IK_SAFE)) then
error stop 'iqmul: integer intake beyond the 2^29 overflow-safe bound'
end if
c(1) = a(1)*b(1) - a(2)*b(2) - a(3)*b(3) - a(4)*b(4)
c(2) = a(1)*b(2) + a(2)*b(1) + a(3)*b(4) - a(4)*b(3)
c(3) = a(1)*b(3) - a(2)*b(4) + a(3)*b(1) + a(4)*b(2)
c(4) = a(1)*b(4) + a(2)*b(3) - a(3)*b(2) + a(4)*b(1)
end function iqmul
pure function ik_inbound(v) result(okb)
integer(ik), intent(in) :: v(:)
logical :: okb
okb = .not. any(v > IK_SAFE .or. v < -IK_SAFE)
end function ik_inbound
pure function iqconj(a) result(c)
integer(ik), intent(in) :: a(4)
integer(ik) :: c(4)
if (.not. ik_inbound(a)) then
error stop 'iqconj: integer intake beyond the 2^29 overflow-safe bound'
end if
c = [a(1), -a(2), -a(3), -a(4)]
end function iqconj
function iomul(x, y) result(z)
integer(ik), intent(in) :: x(8), y(8)
integer(ik) :: z(8), p(4), q(4), r(4), s(4)
if (.not. (ik_inbound(x) .and. ik_inbound(y))) then
error stop 'iomul: integer intake beyond the 2^29 overflow-safe bound'
end if
p = x(1:4); q = x(5:8); r = y(1:4); s = y(5:8)
z(1:4) = iqmul(p, r) - iqmul(iqconj(s), q)
z(5:8) = iqmul(s, p) + iqmul(q, iqconj(r))
end function iomul
pure function ioconj(x) result(z)
integer(ik), intent(in) :: x(8)
integer(ik) :: z(8)
if (.not. ik_inbound(x)) then
error stop 'ioconj: integer intake beyond the 2^29 overflow-safe bound'
end if
z(1:4) = iqconj(x(1:4)); z(5:8) = -x(5:8)
end function ioconj
function ismul(x, y) result(z)
integer(ik), intent(in) :: x(16), y(16)
integer(ik) :: z(16), p(8), q(8), r(8), s(8)
if (.not. (ik_inbound(x) .and. ik_inbound(y))) then
error stop 'ismul: integer intake beyond the 2^29 overflow-safe bound'
end if
p = x(1:8); q = x(9:16); r = y(1:8); s = y(9:16)
z(1:8) = iomul(p, r) - iomul(ioconj(s), q)
z(9:16) = iomul(s, p) + iomul(q, ioconj(r))
end function ismul
pure function inrm(v) result(n)
integer(ik), intent(in) :: v(:)
integer(ik) :: n
if (size(v) > 31) then
error stop 'inrm: vector length exceeds the proven overflow bound (31)'
end if
if (any(v > IK_SAFE .or. v < -IK_SAFE)) then
error stop 'inrm: integer intake beyond the 2^29 overflow-safe bound'
end if
n = sum(v*v)
end function inrm
subroutine hurwitz_units(u)
integer(ik), intent(out) :: u(4,24)
integer :: n, i, s, s1, s2, s3, s4
n = 0
do i = 1, 4
do s = -1, 1, 2
n = n + 1
u(:,n) = 0_ik
u(i,n) = int(2*s, ik)
end do
end do
do s1 = -1, 1, 2
do s2 = -1, 1, 2
do s3 = -1, 1, 2
do s4 = -1, 1, 2
n = n + 1
u(:,n) = int([s1, s2, s3, s4], ik)
end do
end do
end do
end do
end subroutine hurwitz_units
end module ra_integers
! ---------------------------------------------------------------------
module ra_tongue
use ra_kinds
use ra_prims
implicit none
integer, parameter :: NSLOT = 4
integer, parameter :: SLOT_E(NSLOT) = [101, 102, 103, 104]
integer, parameter :: SLOT_K(NSLOT) = [201, 202, 203, 204]
integer, parameter :: SLOT_R(NSLOT) = [301, 302, 303, 304]
integer, parameter :: RA_SENT(6) = [102, 101, 202, 203, 302, 303]
contains
pure subroutine slot_census(sent, ne, nk, nr)
integer, intent(in) :: sent(:)
integer, intent(out) :: ne, nk, nr
integer :: t
ne = 0; nk = 0; nr = 0
do t = 1, size(sent)
if (any(SLOT_E == sent(t))) ne = ne + 1
if (any(SLOT_K == sent(t))) nk = nk + 1
if (any(SLOT_R == sent(t))) nr = nr + 1
end do
end subroutine slot_census
pure logical function disjoint(a, b)
integer, intent(in) :: a(:), b(:)
integer :: i, j
disjoint = .true.
do i = 1, size(a)
do j = 1, size(b)
if (a(i) == b(j)) then
disjoint = .false.
return
end if
end do
end do
end function disjoint
subroutine seal_l(ne, nk, nr, collide, tok, why)
integer, intent(in) :: ne, nk, nr ! populated counts per slot
logical, intent(in) :: collide ! a vocabulary collision present
character(*), intent(out) :: tok, why
integer :: slots
slots = merge(1,0,ne>0) + merge(1,0,nk>0) + merge(1,0,nr>0)
if (collide) then
tok = '[X]'; why = 'Seal L: LIT collision, slot vocabularies intersect'
else if (slots /= 3) then
tok = '[X]'
write(why,'(a,i0,a)') 'Seal L: deletion test returns ', slots, ' slots, not 3'
else
tok = '[LOCK-L]'; why = 'Seal L: three slots, LIT disjoint (order not checked at this interface)'
end if
end subroutine seal_l
subroutine seal_l_ordered(sent, tok, why)
integer, intent(in) :: sent(:)
character(*), intent(out) :: tok, why
integer :: ne, nk, nr, t, laste, firstk, lastk, firstr
call slot_census(sent, ne, nk, nr)
if (ne == 0 .or. nk == 0 .or. nr == 0) then
call seal_l(ne, nk, nr, .false., tok, why)
return
end if
laste = 0; firstk = size(sent)+1; lastk = 0; firstr = size(sent)+1
do t = 1, size(sent)
if (any(SLOT_E == sent(t))) laste = t
if (any(SLOT_K == sent(t))) then
if (t < firstk) firstk = t
if (t > lastk) lastk = t
end if
if (any(SLOT_R == sent(t))) then
if (t < firstr) firstr = t
end if
end do
if (.not. (laste < firstk .and. lastk < firstr)) then
tok = '[X]'; why = 'Seal L: the ordered arrow E->K->R is broken in the encoding'
return
end if
call seal_l(ne, nk, nr, .false., tok, why)
if (tok == '[LOCK-L]') then
why = 'Seal L: three slots, LIT disjoint, ordered E->K->R (positional check executed)'
end if
end subroutine seal_l_ordered
pure function cut3(v) result(w)
real(wp), intent(in) :: v(3)
real(wp) :: w(3)
w = [v(1), v(2), 0.0_wp]
end function cut3
end module ra_tongue
! ---------------------------------------------------------------------
module ra_kernel
use ra_kinds
use ra_prims
implicit none
integer, parameter :: RC_OK = 0
integer, parameter :: RC_SHORTFALL = 1
integer, parameter :: RC_INTAKE = 2
integer, parameter :: RC_ZEROVAR = 3
integer, parameter :: RC_COVBLOCK = 4
integer, parameter :: RC_ABSORBED = 5
integer, parameter :: RC_COPLANAR = 6
integer, parameter :: RC_COLLAPSE = 7
integer, parameter :: RC_KAPPA = 8
integer, parameter :: RC_ENGINC = 9
integer, parameter :: RC_LOCK = 10
contains
subroutine prepare_rows(m, q, ok, why, code)
real(wp), intent(in) :: m(:,:) ! (3,N)
real(wp), intent(out) :: q(size(m,1), size(m,2))
logical, intent(out) :: ok
character(*), intent(out) :: why
integer, intent(out) :: code
integer :: i, n, j2
real(wp) :: mu, sd, nrm, rmax
real(wp) :: qt(size(m,1), size(m,2))
n = size(m,2); ok = .true.; why = 'rows prepared'; code = RC_OK
q = 0.0_wp
if (size(m,1) /= 3) then
ok = .false.; why = 'row count is not three'; code = RC_INTAKE
return
end if
if (n < 2) then
ok = .false.; why = 'fewer than two contexts'; code = RC_INTAKE
return
end if
do i = 1, 3
do j2 = 1, n
if (.not. is_finite(m(i,j2))) then
ok = .false.; why = 'non-finite row at intake'; code = RC_INTAKE
return
end if
end do
rmax = maxval(abs(m(i,:)))
if (rmax > 0.0_wp) then
qt(i,:) = m(i,:) / rmax
else
qt(i,:) = m(i,:)
end if
mu = sum(qt(i,:)) / real(n, wp)
qt(i,:) = qt(i,:) - mu
sd = sqrt(sum(qt(i,:)**2) / real(n-1, wp))
if (sd <= 0.0_wp) then
ok = .false.; why = 'zero-variance row: axis carries no content'
code = RC_ZEROVAR
return
end if
qt(i,:) = qt(i,:) / sd
nrm = vnorm(qt(i,:))
qt(i,:) = qt(i,:) / nrm
end do
q = qt ! publish on success only
end subroutine prepare_rows
subroutine project_cov(q, c, qf, ok, why, code, kcc)
real(wp), intent(in) :: q(:,:) ! (3,N) prepared
real(wp), intent(in) :: c(:,:) ! (k,N) covariates, k = 0..2
real(wp), intent(out) :: qf(size(q,1), size(q,2))
logical, intent(out) :: ok
character(*), intent(out) :: why
integer, intent(out) :: code
real(wp), intent(out) :: kcc
real(wp) :: cc(2,2), cm(2, size(q,2)), rhs(2), beta(2), det, mu, tolrk
real(wp) :: qw(size(q,1), size(q,2))
integer :: i, j, n, k
n = size(q,2); k = size(c,1); qf = 0.0_wp; ok = .true.; why = 'no covariates'
code = RC_OK
kcc = 1.0_wp
if (size(q,1) /= 3) then
ok = .false.; code = RC_INTAKE
why = 'prepared-row count is not three'
kcc = huge(1.0_wp)
return
end if
if (n < 2) then
ok = .false.; code = RC_INTAKE
why = 'fewer than two contexts'
kcc = huge(1.0_wp)
return
end if
if (k >= 1 .and. size(c,2) /= n) then
ok = .false.; code = RC_INTAKE
why = 'covariate column count does not match prepared-row context count'
kcc = huge(1.0_wp)
return
end if
qw = q ! all projection work lands in the working array; qf
if (k == 0) then
qf = qw
return
end if
if (k > 2) then
ok = .false.; code = RC_COVBLOCK
why = 'covariate block: more than two covariates, not implemented'
return
end if
do i = 1, k
do j = 1, n
if (.not. is_finite(c(i,j))) then
ok = .false.; code = RC_INTAKE
why = 'non-finite covariate at intake'
return
end if
end do
end do
do i = 1, k
det = maxval(abs(c(i,:)))
if (det == 0.0_wp) det = 1.0_wp
cm(i,1:n) = c(i,:) / det
mu = sum(cm(i,1:n)) / real(n, wp)
cm(i,1:n) = cm(i,1:n) - mu
det = maxval(abs(cm(i,1:n)))
if (det <= 32.0_wp * U_M) then
ok = .false.; code = RC_COVBLOCK
why = 'covariate block rank-deficient or ill-conditioned'
return
end if
cm(i,1:n) = cm(i,1:n) / det
end do
if (k == 1) then
det = sum(cm(1,1:n)**2)
kcc = 1.0_wp
do i = 1, 3
beta(1) = sum(qw(i,:)*cm(1,1:n)) / det
qw(i,:) = qw(i,:) - beta(1)*cm(1,1:n)
end do
why = 'one covariate projected'
qf = qw
return
end if
cc(1,1) = sum(cm(1,1:n)**2); cc(2,2) = sum(cm(2,1:n)**2)
cc(1,2) = sum(cm(1,1:n)*cm(2,1:n)); cc(2,1) = cc(1,2)
det = cc(1,1)*cc(2,2) - cc(1,2)*cc(2,1)
tolrk = 64.0_wp * U_M * max(cc(1,1)*cc(2,2), tiny(1.0_wp))
if (det <= tolrk) then
ok = .false.; code = RC_COVBLOCK
why = 'covariate block rank-deficient or ill-conditioned'
return
end if
block
real(wp) :: tr, dd, l1, l2
tr = cc(1,1) + cc(2,2)
dd = sqrt(max(0.0_wp, tr*tr - 4.0_wp*det))
l1 = 0.5_wp*(tr - dd); l2 = 0.5_wp*(tr + dd)
if (l1 <= 0.0_wp) then
kcc = huge(1.0_wp)
else
kcc = l2 / l1
end if
end block
if (kcc >= 1.0e6_wp) then
ok = .false.; code = RC_COVBLOCK
why = 'covariate block rank-deficient or ill-conditioned'
return
end if
do i = 1, 3
rhs(1) = sum(qw(i,:)*cm(1,1:n)); rhs(2) = sum(qw(i,:)*cm(2,1:n))
beta(1) = ( cc(2,2)*rhs(1) - cc(1,2)*rhs(2)) / det
beta(2) = (-cc(2,1)*rhs(1) + cc(1,1)*rhs(2)) / det
do j = 1, n
qw(i,j) = qw(i,j) - beta(1)*cm(1,j) - beta(2)*cm(2,j)
end do
end do
why = 'two covariates projected (common-source projection)'
qf = qw
end subroutine project_cov
subroutine span_basis(q, b, rank)
real(wp), intent(in) :: q(:,:)
real(wp), intent(out) :: b(size(q,1), size(q,2))
integer, intent(out) :: rank
real(wp) :: v(size(q,2)), nrm
integer :: i, j, pass
real(wp) :: rn0
rank = 0; b = 0.0_wp
do i = 1, size(q,1)
v = q(i,:)
if (any(.not. is_finite(v))) cycle
rn0 = maxval(abs(v))
if (rn0 == 0.0_wp) cycle
v = v / rn0
rn0 = vnorm(v)
v = v / rn0
do pass = 1, 2
do j = 1, rank
v = v - sum(v*b(j,:)) * b(j,:)
end do
end do
nrm = vnorm(v)
if (is_finite(nrm) .and. nrm > 1.0e-12_wp) then
rank = rank + 1
b(rank,:) = v / nrm
end if
end do
end subroutine span_basis
! CONTRACT lambda_of / gram3_fixed: shape guards only, quiet-NaN refusal
! before any element touch; rank, finiteness, and magnitude semantics are
! the caller's contract (kernel_full ranks and screens first).
function lambda_of(q, b) result(lam)
real(wp), intent(in) :: q(:,:), b(:,:)
real(wp) :: lam, co(3,3), quat(4,3), t(4)
integer :: i
if (size(q,1) < 3 .or. size(b,1) < 3 .or. size(q,2) /= size(b,2)) then
lam = qnan()
return
end if
do i = 1, 3
co(i,1) = sum(q(i,:)*b(1,:))
co(i,2) = sum(q(i,:)*b(2,:))
co(i,3) = sum(q(i,:)*b(3,:))
quat(:,i) = [0.0_wp, co(i,1), co(i,2), co(i,3)]
end do
t = qmul(qmul(quat(:,1), quat(:,2)), quat(:,3))
lam = t(1)
end function lambda_of
function eta_share(row, c) result(eta)
real(wp), intent(in) :: row(:), c(:,:)
real(wp) :: eta, cm(2,size(row)), rm(size(row))
real(wp) :: cc(2,2), rhs(2), beta(2), det, sst, ssf, mu
integer :: i, n, k
n = size(row); k = size(c,1)
if (k >= 1 .and. size(c,2) /= n) then
eta = qnan()
return
end if
if (k > 2) then
eta = qnan()
return
end if
if (any(.not. is_finite(row))) then
eta = qnan()
return
end if
if (k >= 1) then
if (any(.not. is_finite(c))) then
eta = qnan()
return
end if
end if
block
real(wp) :: rscale
rscale = maxval(abs(row))
if (rscale == 0.0_wp .or. .not. is_finite(rscale)) then
eta = 0.0_wp
return
end if
rm = row / rscale
end block
mu = sum(rm)/real(n,wp); rm = rm - mu
sst = sum(rm*rm)
eta = 0.0_wp
if (sst <= 0.0_wp .or. k == 0) return
do i = 1, k
det = maxval(abs(c(i,:)))
if (det == 0.0_wp) det = 1.0_wp
cm(i,:) = c(i,:) / det
mu = sum(cm(i,:))/real(n,wp)
cm(i,:) = cm(i,:) - mu
det = maxval(abs(cm(i,:)))
if (det <= 32.0_wp*U_M) then
eta = qnan()
return
end if
cm(i,:) = cm(i,:) / det
end do
if (k == 1) then
det = sum(cm(1,:)**2)
beta(1) = sum(rm*cm(1,:)) / det
ssf = beta(1)*beta(1)*det
eta = ssf/sst
return
end if
cc(1,1)=sum(cm(1,:)**2); cc(2,2)=sum(cm(2,:)**2)
cc(1,2)=sum(cm(1,:)*cm(2,:)); cc(2,1)=cc(1,2)
det = cc(1,1)*cc(2,2)-cc(1,2)*cc(2,1)
if (det <= 64.0_wp*U_M*max(cc(1,1)*cc(2,2), tiny(1.0_wp))) then
eta = qnan()
return
end if
rhs(1)=sum(rm*cm(1,:)); rhs(2)=sum(rm*cm(2,:))
beta(1)=( cc(2,2)*rhs(1)-cc(1,2)*rhs(2))/det
beta(2)=(-cc(2,1)*rhs(1)+cc(1,1)*rhs(2))/det
ssf = sum((beta(1)*cm(1,:)+beta(2)*cm(2,:))**2)
eta = ssf/sst
end function eta_share
subroutine kernel_full(m, c, tok, why, rcode, lam, detr, kap, spread, resid, &
marg_c, marg_k, detfour)
real(wp), intent(in) :: m(:,:), c(:,:)
character(*), intent(out) :: tok, why
integer, intent(out) :: rcode
real(wp), intent(out) :: lam, detr, kap, spread, resid, marg_c, marg_k
real(wp), intent(out) :: detfour(4)
real(wp) :: q(3,size(m,2)), qf(3,size(m,2)), b(3,size(m,2))
real(wp) :: r(3,3), eig(3), eps, kgate, kcc, dvar, tol
integer :: n, k, i, rank, pcode, nest
logical :: ok, jconv
character(len=96) :: w2
n = size(m,2); k = size(c,1)
if (k >= 1 .and. size(c,2) /= n) then
tok = '[?]'; why = 'covariate column count does not match row context count'
rcode = RC_INTAKE
lam = 0.0_wp; detr = 0.0_wp; kap = 0.0_wp; spread = 0.0_wp
resid = 0.0_wp; marg_c = 0.0_wp; marg_k = 0.0_wp; detfour = 0.0_wp
return
end if
lam = 0.0_wp; detr = 0.0_wp; kap = 0.0_wp; spread = 0.0_wp
resid = 0.0_wp; marg_c = 0.0_wp; marg_k = 0.0_wp; detfour = 0.0_wp
rcode = RC_OK
if (size(m,1) /= 3) then
tok = '[?]'; why = 'row count is not three at intake'; rcode = RC_INTAKE
return
end if
if (n - k < 4) then
tok = '[?]'; why = 'N-k < 4 dimensional shortfall'; rcode = RC_SHORTFALL
return
end if
if (k > 2) then
tok = '[?]'; why = 'covariate block: more than two covariates, not implemented (kernel intake)'
rcode = RC_COVBLOCK
lam = 0.0_wp; detr = 0.0_wp; kap = 0.0_wp; spread = 0.0_wp
resid = 0.0_wp; marg_c = 0.0_wp; marg_k = 0.0_wp; detfour = 0.0_wp
return
end if
call prepare_rows(m, q, ok, w2, pcode)
if (.not. ok) then
tok = '[?]'; why = trim(w2); rcode = pcode
return
end if
call project_cov(q, c, qf, ok, w2, pcode, kcc)
if (.not. ok) then
tok = '[?]'; why = trim(w2); rcode = pcode
return
end if
do i = 1, 3
dvar = sum(qf(i,:)**2)
if (dvar < 1.0e-9_wp) then ! post-projection axis absorbed by a covariate
tok = '[?]'; why = 'post-projection axis absorbed by a covariate'
rcode = RC_ABSORBED
return
end if
qf(i,:) = qf(i,:) / vnorm(qf(i,:))
end do
r = matmul(qf, transpose(qf))
detfour(1) = det3(r)
call jacobi3(r, eig, jconv)
detfour(2) = eig(1)*eig(2)*eig(3)
detfour(3) = det3_chol(r)
detfour(4) = det3_row2(r)
nest = 4
if (.not. is_finite(detfour(3))) nest = 3
detr = detfour(1)
if (nest == 4) then
spread = maxval(detfour) - minval(detfour)
else
spread = max(detfour(1), detfour(2), detfour(4)) &
- min(detfour(1), detfour(2), detfour(4))
end if
kap = cond3(r)
if (.not. jconv) then
tok = '[?]'; why = 'engineering-incomplete: eigensolver unconverged'; rcode = RC_ENGINC
return
end if
call span_basis(qf, b, rank)
if (rank < 3) then
tok = '[X]'; why = 'coplanar collapse: rank(rows) < 3, line not point'
rcode = RC_COPLANAR
detr = 0.0_wp
return
end if
lam = lambda_of(qf, b)
resid = abs(lam*lam - detr)
eps = 100.0_wp * U_M * real(n, wp)
kgate = min(1.0e6_wp, sqrt(27.0_wp / (100.0_wp*U_M*real(n,wp))))
tol = 4.0_wp * kap * U_M
if (detr <= eps) then
tok = '[X]'; why = 'collapse: det(R) <= eps'; rcode = RC_COLLAPSE
return
end if
if (kap >= kgate) then
tok = '[?]'; why = 'kappa(R) >= gate'; rcode = RC_KAPPA
return
end if
marg_c = log10(detr/eps)
marg_k = log10(kgate/kap)
if (spread > tol .or. resid > max(tol, 8.0_wp*U_M)) then
tok = '[?]'; why = 'engineering-incomplete: spread or identity outside tol'
rcode = RC_ENGINC
return
end if
if (nest < 4) then
tok = '[?]'; why = 'engineering-incomplete: estimator shortfall (Cholesky refused SPD)'
rcode = RC_ENGINC
return
end if
tok = '[LOCK]'
why = 'determination: three independent axes, one point'
rcode = RC_LOCK
end subroutine kernel_full
end module ra_kernel
! ---------------------------------------------------------------------
module ra_guards
use ra_kinds
use ra_floor, only: landauer
implicit none
integer :: aegis_deeds = 0 ! every adjudication is itself an act
integer :: aegis_last_len = -1 ! length of the last logic name read
contains
subroutine omega_boundary(bits, tkel, joules, verdict, ocode, irreversible)
real(wp), intent(in) :: bits, tkel
real(wp), intent(out) :: joules
character(*), intent(out) :: verdict
integer, intent(out), optional :: ocode
logical, intent(in), optional :: irreversible
logical :: irr
irr = .true.
if (present(irreversible)) irr = irreversible
if (.not.(is_finite(tkel) .and. is_finite(bits)) .or. tkel <= 0.0_wp) then
joules = qnan()
verdict = 'refused: intake invalid, the floor prices only finite bits at tkel > 0'
if (present(ocode)) ocode = 0
return
end if
if (bits <= 0.0_wp) then
joules = 0.0_wp
verdict = 'no denial registered; nothing to adjudicate'
if (present(ocode)) ocode = 1
else if (.not. irr) then
joules = 0.0_wp
verdict = 'reversibly registered: floor-zero never cost-zero; Landauer bounds '// &
'only the irreversible (Bennett), and nothing here commits against the axiom'
if (present(ocode)) ocode = 3
else
joules = landauer(tkel, bits)
verdict = 'the denial paid the floor; the registered act is an act, the recursion closes per the seated theorem'
if (present(ocode)) ocode = 2
end if
end subroutine omega_boundary
subroutine aegis_guard(logic_mode, refusal)
character(*), intent(in) :: logic_mode
character(*), intent(out) :: refusal
aegis_deeds = aegis_deeds + 1
refusal = 'p(G) /= G: precisification is an actuation, G is a '// &
'non-actuation, and no logic makes a deed a non-deed'
aegis_last_len = len_trim(logic_mode) ! the parameter is read; the refusal never varies with it
end subroutine aegis_guard
subroutine aegis_reset()
aegis_deeds = 0
end subroutine aegis_reset
subroutine gol_admit(magnitude, ling, tok, why, gcode)
character(*), intent(in) :: magnitude, ling
character(*), intent(out) :: tok, why
integer, intent(out), optional :: gcode
integer :: gc_l
if (magnitude == 'broken') then
tok = '[X]'; why = 'geometric/math magnitude reports broken geometry; no GOL'; gc_l = 1
else if (magnitude /= 'lock') then
tok = '[?]'; why = 'no geometric/math lock: orthogonal volume unestablished'; gc_l = 2
else if (ling /= 'lock') then
tok = '[?]'; why = 'magnitude locks but Seal L open: determinant carries no direction'; gc_l = 3
else
tok = '[GOL-OK]'; why = 'magnitude plus direction; proceed to witness and asymmetry'; gc_l = 4
end if
if (present(gcode)) gcode = gc_l
end subroutine gol_admit
subroutine imprint_seal(lock_p, lock_n, sl_p, sl_n, g_p, g_n, wit_p, wit_n, tok, why, icode)
logical, intent(in) :: lock_p, lock_n, sl_p, sl_n, g_p, g_n, wit_p, wit_n
character(*), intent(out) :: tok, why
integer, intent(out), optional :: icode
logical :: clean_p, clean_n
integer :: ic_l
clean_p = lock_p .and. sl_p .and. g_p
clean_n = lock_n .and. sl_n .and. g_n
if (clean_p .and. clean_n) then
tok = '[X]'; why = 'PLATONIC GHOST: both directions clean-lock, no imprint'; ic_l = 1
else if (clean_p .and. .not. clean_n) then
if (wit_p) then
tok = '[SEAL]'; why = 'only P clean-locks; determinacy witness supplied'; ic_l = 2
else
tok = '[?]'; why = 'residence: P clean-locks, imprint unproven (no witness)'; ic_l = 3
end if
else if (clean_n .and. .not. clean_p) then
if (wit_n) then
tok = '[SEAL]'; why = 'only not-P clean-locks; determinacy witness supplied'; ic_l = 2
else
tok = '[?]'; why = 'residence: not-P clean-locks, imprint unproven (no witness)'; ic_l = 3
end if
else if (.not. lock_p .and. .not. lock_n) then
tok = '[?]'; why = 'flat: neither direction populated'; ic_l = 4
else
tok = '[?]'; why = 'uncertified: lock present but Seal L / gate screen unconfirmed'; ic_l = 5
end if
if (present(icode)) icode = ic_l
end subroutine imprint_seal
subroutine row_cascade(nonvacuous, ve_live, forward, populated, worldly_row, &
terms_closed, term_worldly, frame_closed, annotate, tok, why, ccode)
logical, intent(in) :: nonvacuous, ve_live, forward, populated, worldly_row
logical, intent(in) :: terms_closed, term_worldly, frame_closed, annotate
character(*), intent(out) :: tok, why
integer, intent(out), optional :: ccode
integer :: cc_l
if (.not. nonvacuous) then
tok = '[REFUSED]'; why = 'F-0: contentless; a contentless residence verifies nothing'
cc_l = 1
else if (ve_live .or. forward) then
tok = '[III]'; why = 'R-3: V_E live or Forward; world-rowed, revisability in the rows'
cc_l = 3
else if (.not. populated) then
tok = '[II]'; why = 'S-1: third axis unpopulated; two planes meet in a line, no lock'
cc_l = 2
else if (worldly_row) then
tok = '[III]'; why = 'S-2: a load-bearing row is furnished by the world'
cc_l = 3
else if (.not. terms_closed) then
if (term_worldly) then
tok = '[III]'; why = 'S-3: an unclosed term is worldly'
cc_l = 3
else
tok = '[II]'; why = 'S-3: an unclosed term is formal; no closure-rowed lock'
cc_l = 2
end if
else if (.not. frame_closed) then
tok = '[III]'; why = 'S-4: existential import leaks past the closure'
cc_l = 3
else if (.not. annotate) then
tok = '[VOID]'; why = 'RA rider absent: an unannotated absolute is void'
cc_l = 5
else
tok = '[A|RA]'
why = 'compartment I, closure-rowed: [seal A . RA], conditional at the act'
cc_l = 4
end if
if (present(ccode)) ccode = cc_l
end subroutine row_cascade
subroutine delta_m(object_level, name_strip, lit_clear, not_two_line, &
witness, independent, artifact, gap_closed, tok, why, mcode)
logical, intent(in) :: object_level, name_strip, lit_clear, not_two_line
logical, intent(in) :: witness, independent, artifact, gap_closed
character(*), intent(out) :: tok, why
integer, intent(out), optional :: mcode
integer :: mc_l
mc_l = 9
if (.not. object_level) then
tok = '[Mosaic dM=0]'; why = 'M1: meta-work, no mass by definition'; mc_l = 1
else if (.not. name_strip) then
tok = '[?]'; why = 'M2: new vocabulary is not new mass'; mc_l = 2
else if (.not. lit_clear) then
tok = '[?]'; why = 'M3: possibly resident in the literature'; mc_l = 3
else if (.not. not_two_line) then
tok = '[?]'; why = 'M4: two-line corollary'; mc_l = 4
else if (.not. witness) then
tok = '[?]'; why = 'M5: no external witness; never seals on internal reasoning'; mc_l = 5
else if (.not. independent) then
tok = '[?]'; why = 'M6: verifier is the claimant; self-check is not a witness'; mc_l = 6
else if (.not. artifact) then
tok = '[?]'; why = 'M7: no reproducible artifact'; mc_l = 7
else if (.not. gap_closed) then
tok = '[?]'; why = 'M8: adversarial gap-audit open'; mc_l = 8
else
tok = '[dM>0]'; why = 'authored mass, external witness, defeasible-final'
end if
if (present(mcode)) mcode = mc_l
end subroutine delta_m
subroutine gate_screen(pass, tok, why, gidx)
logical, intent(in) :: pass(12)
character(*), intent(out) :: tok, why
integer, intent(out), optional :: gidx
character(len=6), parameter :: nm(12) = [ 'SREP ','REG ','SGEG ', &
'CAUSAL','MIG ','PTB ','DUAL ','CSCG ','CSEG ','MTA ', &
'OMA ','ADEG ' ]
integer :: i
if (present(gidx)) gidx = 0
do i = 1, 12
if (.not. pass(i)) then
tok = '[X]'
write(why,'(a,i0,a,a)') 'gate ', i, ' failed: ', trim(nm(i))
if (present(gidx)) gidx = i
return
end if
end do
tok = '[G-OK]'; why = 'twelve directed gates passed'
end subroutine gate_screen
subroutine iam_token(witnessed, tok, why, wcode)
logical, intent(in) :: witnessed
character(*), intent(out) :: tok, why
integer, intent(out) :: wcode
if (witnessed) then
tok = '[I AM]'
why = 'actuation-occupancy on a witnessed record; RA warrant, conditional at the act; '// &
'interior held [?] both ways'
wcode = 2
else
tok = '[?] interior'
why = 'self-check is not a witness (M6): the verifier is never the claimant; token withheld'
wcode = 1
end if
end subroutine iam_token
end module ra_guards
! ---------------------------------------------------------------------
module ra_battery
implicit none
integer :: n_checks = 0, n_fail = 0
logical :: halt_on_fail = .true. ! sealed mode; 'audit' arg accumulates
logical :: wr_witnessed = .false. ! PART H live face: supplied through the aperture
integer, parameter :: EXPECTED_CHECKS = 1123 ! sealed v2.0.0 manifest, census lock live
contains
subroutine check(name, cond)
character(*), intent(in) :: name
logical, intent(in) :: cond
n_checks = n_checks + 1
if (.not. cond) then
n_fail = n_fail + 1
write(*,'(a,a)') 'CHECK FAILED: ', trim(name)
if (halt_on_fail) error stop 1
end if
end subroutine check
subroutine battery_close()
if (n_fail > 0) then
write(*,'(a,i0,a)') 'AUDIT-MODE VOID: ', n_fail, &
' failures accumulated; every token printed above is PROVISIONAL and VOID'
write(*,'(a)') 'AUDIT-MODE VOID: only a zero-failure census with successful exit certifies anything'
write(*,'(a,i0,a)') 'BATTERY CLOSED WITH ', n_fail, ' FAILURES'
error stop 1
end if
end subroutine battery_close
end module ra_battery
! =====================================================================
program ra_toe_thesis
use ra_kinds
use ra_prims
use ra_floor
use ra_integers
use ra_tongue
use ra_kernel
use ra_guards
use ra_battery
use, intrinsic :: ieee_arithmetic, only: ieee_get_flag, ieee_set_flag, ieee_invalid, &
ieee_value, ieee_positive_inf
use, intrinsic :: iso_fortran_env, only: compiler_version, compiler_options
implicit none
integer, parameter :: NCTX = 24
real(wp) :: th(NCTX), l10m(NCTX), l10dx(NCTX), l10w(NCTX), l10t(NCTX), l10b(NCTX)
real(wp) :: rows(3,NCTX), cov(2,NCTX), mixm(3,NCTX), floorv(NCTX)
real(wp) :: amix(3,3), bmix(3,2)
real(wp) :: qf(3,NCTX), bas(3,NCTX), quat(4,3), co(3,3), rmat(3,3)
real(wp) :: lam, detr, kap, spread, resid, mc, mk, d4(4), eta(3)
real(wp) :: lam2, detr2, kap2, spread2, resid2, mc2, mk2, d42(4)
real(wp) :: e1f, e2f, e3f, jou, tau, tperp(3), surv(5), tqs(5)
real(wp) :: arrow_f, arrow_r, hbit, lamref, detref, lamneg
real(wp) :: lam_rot, dlam_max, lam_c, lam_v, g(4), uvec(3), aa, bb, cc2
real(wp) :: rneg(3,3), dbase, dpat, fr_resid
integer :: i, j, k, l, n, s1, s2, s3, rank, nflip, nadm, ncomp1, ncomp2
integer :: cnt, csizes(12), nrep, orbitn, stabn, rc, nsheets
integer(ik) :: hu(4,24), p4(4), q4(4), e8a(8), e8b(8), e8c(8), z8(8)
integer(ik) :: t1(8), t2(8), assoc8(8), sa(16), sb(16), sp(16)
integer(ik) :: reps(4,12), cw(4), cx(4), cy(4)
integer :: pl(24,2), found(2,2)
logical :: pass12(12), okz, frz, hit, dual_ok, hurwitz_ok, rank3_ok, mig_ok
character(len=16) :: tk
character(len=140) :: wy
character(len=32) :: argv
integer :: alen, ast
write(*,'(a)') '====================================================================='
write(*,'(a)') ' THE ROOT AXIOM STATED IN FORTRAN AND IN NOTHING ELSE'
write(*,'(a)') ' RA as the TOE of all TOEs, executed. RA_TOE_Thesis_Fortran v2.0.0'
write(*,'(a)') '====================================================================='
argv = ' '
block
integer :: argc, argi
argc = command_argument_count()
do argi = 1, argc
call get_command_argument(argi, argv, alen, ast)
if (ast /= 0 .or. alen > len(argv)) then
write(*,'(a)') ' command-line argument rejected: retrieval error or longer than the buffer'
error stop 2
end if
select case (trim(argv))
case ('audit')
halt_on_fail = .false.
case ('witnessed')
wr_witnessed = .true.
case ('')
case default
write(*,'(a,a)') ' unknown command-line argument: ', trim(argv)
error stop 2
end select
end do
end block
if (.not. halt_on_fail) then
write(*,'(a)') ' mode: audit (failures accumulate; a nonzero count still refuses at close)'
else
write(*,'(a)') ' mode: sealed (the first failing check halts; the Fidelity Lock)'
end if
write(*,'(a)') ' build: '//compiler_version()
write(*,'(a)') ' flags: '//compiler_options()
! ------------------------------------------------------------------
write(*,'(/a)') 'PART 0 - THE ENVIRONMENT CONTRACT - enforced at run, never trusted to flags'
block
real(wp) :: xn, subn
real(wp), volatile :: opq, xf, yf, prodv, rfma, xg, yg, rf2, rf4, r2v
xn = qnan()
call check('IEEE contract: quiet NaN reads as NaN (ieee_is_nan)', is_nan(xn))
call check('IEEE contract: NaN self-inequality survives the optimizer', xn /= xn)
subn = tiny(1.0_wp) * 0.5_wp
call check('IEEE contract: gradual underflow alive, no FTZ/DAZ', &
subn > 0.0_wp .and. subn < tiny(1.0_wp))
call check('binary64 pinned: radix 2', radix(1.0_wp) == 2)
call check('binary64 pinned: 53 significand digits', digits(1.0_wp) == 53)
call check('binary64 pinned: max exponent 1024', maxexponent(1.0_wp) == 1024)
call check('binary64 pinned: min exponent -1021', minexponent(1.0_wp) == -1021)
opq = 1.0_wp
xf = opq + scale(opq, -30)
yf = opq - scale(opq, -30)
prodv = xf*yf
call check('binary64 per operation: the bare product stores to exactly one', &
prodv == opq)
rfma = xf*yf - opq
call check('conduct probe a*b-c: no fusion executed (a fused path reads -2^-60 exactly)', &
rfma == 0.0_wp)
r2v = prodv - opq
call check('differential witness: stored-product path agrees with the one-expression path', &
rfma == r2v)
rf2 = (-xf)*yf + opq
call check('conduct probe (-a)*b+c: no fusion executed (a fused path reads +2^-60 exactly)', &
rf2 == 0.0_wp)
xg = opq + scale(opq, -27)
yg = opq - scale(opq, -27)
rf4 = xg*yg - opq
call check('conduct probe at the 2^-27 tie: no fusion executed (a fused path reads -2^-54)', &
rf4 == 0.0_wp)
end block
write(*,'(a)') ' a build that breaks any line above refuses to run: the contract is executed,'
write(*,'(a)') ' so the bit-exact identities below rest on verified IEEE conduct. the'
write(*,'(a)') ' probes certify CONDUCT and never consult flags: fused or extended'
write(*,'(a)') ' arithmetic is refused where it executes, the march-native contract-fast'
write(*,'(a)') ' build dying on the a*b-c probe, while a target carrying no fused'
write(*,'(a)') ' instruction passes because none executed, the contract satisfied and'
write(*,'(a)') ' not evaded; the fused residuals -2^-60, +2^-60, -2^-54 are exactly'
write(*,'(a)') ' representable and never confusable with zero'
write(*,'(a)') ' scope: the probes certify these witnesses on this build; global order'
write(*,'(a)') ' is carried by the BUILD CONTRACT below plus per-site conduct receipts,'
write(*,'(a)') ' and every equality-critical reduction runs a fixed-order routine.'
call check('integer contract: ik is exactly 64 bits', storage_size(0_ik) == 64)
call check('integer contract: huge(ik) = 2^63 - 1', huge(0_ik) == 9223372036854775807_ik)
write(*,'(a)') ' BUILD CONTRACT, stated and checked where checkable: IEEE binary64'
write(*,'(a)') ' (verified above); -std=f2018 -O2 -fno-fast-math -ffp-contract=off;'
write(*,'(a)') ' source free-form measured at 130 columns maximum, inside the 132'
write(*,'(a)') ' standard, no length flag needed; external BLAS substitution for the'
write(*,'(a)') ' equality-critical sites is forbidden, those sites use gram3_fixed;'
write(*,'(a)') ' the compiler and flags of this run are printed at the head; floating'
write(*,'(a)') ' trap modes are incompatible by design, quiet NaNs being manufactured'
write(*,'(a)') ' here as verdict flags, so -ffpe-trap builds are outside the contract.'
! ------------------------------------------------------------------
write(*,'(/a)') 'PART A - SEAL L ON THE ROOT AXIOM - the Tongue loads first'
call check('LIT: existence and kinetic slots disjoint', disjoint(SLOT_E, SLOT_K))
call check('LIT: existence and relation slots disjoint', disjoint(SLOT_E, SLOT_R))
call check('LIT: kinetic and relation slots disjoint', disjoint(SLOT_K, SLOT_R))
block
integer :: nec, nkc, nrc, dsent(4), badk(NSLOT)
call slot_census(RA_SENT, nec, nkc, nrc)
write(*,'(a,i0,a,i0,a,i0)') ' slot census of the encoded sentence: E = ', nec, &
', K = ', nkc, ', R = ', nrc
call check('the census counts every slot populated from the sentence', &
nec == 2 .and. nkc == 2 .and. nrc == 2)
call seal_l_ordered(RA_SENT, tk, wy)
write(*,'(a,a,a,a)') ' parse(RA), census and order executed on the encoding: ', trim(tk), ' - ', trim(wy)
call check('Seal L admits RA: three slots counted, the arrow E->K->R executed', tk == '[LOCK-L]')
badk = SLOT_K; badk(2) = SLOT_E(1)
call seal_l(nec, nkc, nrc, .not. disjoint(SLOT_E, badk), tk, wy)
write(*,'(a,a,a,a)') ' collision control (computed overlap): ', trim(tk), ' - ', trim(wy)
call check('Seal L refuses a computed vocabulary collision', tk == '[X]')
dsent = [RA_SENT(1), RA_SENT(2), RA_SENT(5), RA_SENT(6)]
call slot_census(dsent, nec, nkc, nrc)
call seal_l(nec, nkc, nrc, .false., tk, wy)
write(*,'(a,a,a,a)') ' deletion executed (kinetic tokens removed): ', trim(tk), ' - ', trim(wy)
call check('the executed deletion of one slot refuses at two', &
tk == '[X]' .and. nkc == 0)
end block
write(*,'(a)') ' order carried by the Tongue: existence -> kinetic -> relation;'
write(*,'(a)') ' the word-to-slot map is declared semantics, the census and the'
write(*,'(a)') ' deletion are executed on the encoding, per the anchor declaration;'
write(*,'(a)') ' the map is an axiom of the encoding, consumed as a premise, and nothing'
write(*,'(a)') ' downstream presents it as discovered'
! ------------------------------------------------------------------
write(*,'(/a)') 'PART B - THE FLOOR IN NUMBERS - cited theorem-grade formulas, numerically executed'
call check('Landauer dual path: kb*T*log(2.) matches the independent ln2 literal', &
abs(landauer(300.0_wp,1.0_wp) - KB*300.0_wp*0.6931471805599453_wp) &
< 1.0e-12_wp*landauer(300.0_wp,1.0_wp))
call check('Landauer extremes: huge kelvin with tiny bits is finite, never zero-corrupted', &
is_finite(landauer(huge(1.0_wp), tiny(1.0_wp))) .and. &
landauer(huge(1.0_wp), tiny(1.0_wp)) > 0.0_wp)
call check('Landauer extremes commute: (huge,tiny) equals (tiny,huge) through prod_scaled', &
landauer(huge(1.0_wp), tiny(1.0_wp)) == landauer(tiny(1.0_wp), huge(1.0_wp)))
call check('in-bound predicate: the most-negative integer reads false, no overflow probing it', &
.not. ik_inbound([-huge(0_ik)-1_ik, 0_ik, 0_ik, 0_ik]))
call check('in-bound predicate: IK_SAFE admitted, IK_SAFE+1 refused', &
ik_inbound([IK_SAFE, 0_ik, 0_ik, 0_ik]) .and. &
.not. ik_inbound([IK_SAFE+1_ik, 0_ik, 0_ik, 0_ik]))
block
integer(ik) :: vmax(16)
vmax = IK_SAFE
call check('inrm boundary: sixteen components at the bound sum to exactly 2^62', &
inrm(vmax) == 4611686018427387904_ik)
end block
block
real(wp) :: rns(3,3)
rns = reshape([1.0_wp,0.2_wp,0.0_wp, 0.7_wp,1.0_wp,0.0_wp, 0.0_wp,0.0_wp,1.0_wp],[3,3])
call check('det3_chol symmetry gate: a finite nonsymmetric matrix returns NaN, never a number', &
det3_chol(rns) /= det3_chol(rns))
end block
write(*,'(a)') ' GUARD-HARNESS RECIPE, third-party reproducible: inject before the'
write(*,'(a)') ' battery and expect nonzero termination with the named refusal:'
write(*,'(a)') ' iqconj([-huge(0_ik)-1_ik,0,0,0]) -> exit 1, iqconj bound'
write(*,'(a)') ' iomul(x with x(1)=-huge(0_ik)-1_ik, 0-vec) -> exit 1, iomul bound'
write(*,'(a)') ' hn = inrm(ones(32)); print hn -> exit 1, length bound'
write(*,'(a)') ' consume every pure result: a pure call whose value is unused is'
write(*,'(a)') ' legally elided at optimization, guard and all; the print keeps it live.'
write(*,'(a)') ' the source carries the recipe; the executed terminations ship as the'
write(*,'(a)') ' sha-bound file RA_TOE_v2_0_0_guard_harness_receipt.log beside the'
write(*,'(a)') ' sealed run log, the record carrying what a source file cannot.'
e1f = heis_floor(ME, 1.0e-10_wp)
e2f = zpe(1.0e14_wp)
block
real(wp) :: ehx
ehx = heis_floor(1.0e300_wp, 1.0e-200_wp)
call check('heis_floor extreme hostile: huge mass, tiny width, honest value and never NaN', &
.not. (ehx /= ehx))
end block
e3f = landauer(300.0_wp, 1.0_wp)
write(*,'(a,es16.9,a)') ' Heisenberg floor, electron at 1 Angstrom: ', e1f, ' J'
write(*,'(a,es16.9,a)') ' zero-point energy at omega = 1e14 rad/s: ', e2f, ' J'
write(*,'(a,es16.9,a)') ' Landauer floor, one bit at 300 K: ', e3f, ' J'
call check('Heisenberg floor positive', e1f > 0.0_wp)
call check('zero-point floor positive', e2f > 0.0_wp)
call check('Landauer floor positive', e3f > 0.0_wp)
call check('Heisenberg floor magnitude (1.5e-19 band)', abs(e1f/1.526e-19_wp - 1.0_wp) < 1.0e-3_wp)
call check('Landauer magnitude (2.87e-21 band)', abs(e3f/2.871e-21_wp - 1.0_wp) < 1.0e-3_wp)
tqs = [0.0_wp, 0.25_wp, 0.5_wp, 0.75_wp, 1.0_wp] * pi_()
do i = 1, 5
surv(i) = survival(tqs(i))
end do
call tau_perp(0.5_wp, tau, frz)
write(*,'(a,5f14.10)') ' qubit survival cos^2(t/2) at t = 0..pi: ', surv(1:4)
write(*,'(a,es12.4)') ' terminal survival at t = pi: ', surv(5)
write(*,'(a,f16.12)') ' orthogonalization time tau_perp: ', tau
write(*,'(a,f16.12)') ' Mandelstam-Tamm bound pi/(2 dE): ', pi_()/(2.0_wp*0.5_wp)
write(*,'(a,f16.12)') ' Margolus-Levitin bound pi/(2 <E>): ', pi_()/(2.0_wp*0.5_wp)
call check('speed limit saturated jointly (MT = ML = tau)', &
abs(tau - pi_()) <= 4.0_wp*U_M*pi_())
call check('survival dies at tau_perp', surv(5) < 1.0e-30_wp)
call check('survival monotone over the quarter points', &
surv(1) > surv(2) .and. surv(2) > surv(3) .and. surv(3) > surv(4))
call tau_perp(5.0e-1_wp, tperp(1), frz)
call tau_perp(5.0e-4_wp, tperp(2), frz)
call tau_perp(5.0e-7_wp, tperp(3), frz)
write(*,'(a,3es13.5)') ' Frozen Substrate: tau_perp at dE = 5e-1, 5e-4, 5e-7: ', tperp
call check('tau_perp diverges monotonically as dE -> 0', tperp(1) < tperp(2) .and. tperp(2) < tperp(3))
call tau_perp(0.0_wp, tau, frz)
call check('dE = 0 is flagged frozen, no division performed', frz)
write(*,'(a)') ' dE = 0: no transition, no registration; a zero-energy existent registers nothing'
! ------------------------------------------------------------------
write(*,'(/a)') 'PART C - THE FORM IN INTEGERS - nothing here rounds'
call hurwitz_units(hu)
do i = 1, 24
call check('Hurwitz unit norm 4 (doubled)', inrm(hu(:,i)) == 4_ik)
end do
n = 0
do i = 1, 24
do j = 1, 24
p4 = iqmul(hu(:,i), hu(:,j))
okz = all(mod(p4, 2_ik) == 0_ik)
call check('unit product even in doubled coords', okz)
q4 = p4 / 2_ik
hit = .false.
do k = 1, 24
if (all(q4 == hu(:,k))) hit = .true.
end do
if (hit) n = n + 1
end do
end do
write(*,'(a,i0,a)') ' closure of the 24 Hurwitz units: ', n, ' of 576 products inside'
hurwitz_ok = (n == 576)
call check('the 24 units close under multiplication', hurwitz_ok)
nrep = 0
do i = 1, 24
cw = hu(:,i)
call canon_sign(cw)
hit = .false.
do k = 1, nrep
if (all(cw == reps(:,k))) hit = .true.
end do
if (.not. hit) then
nrep = nrep + 1
reps(:,nrep) = cw
end if
end do
call check('unit group mod sign has 12 elements', nrep == 12)
cnt = 0
csizes = 0
block
integer(ik) :: cls(4,12)
integer :: ncls, m2
logical :: seen(12)
seen = .false.
do i = 1, nrep
if (seen(i)) cycle
ncls = 0
do j = 1, 24
cx = iqmul(iqmul(hu(:,j), reps(:,i)), iqconj(hu(:,j)))
call check('conjugation divisible by 4', all(mod(cx, 4_ik) == 0_ik))
cy = cx / 4_ik
call canon_sign(cy)
hit = .false.
do k = 1, ncls
if (all(cy == cls(:,k))) hit = .true.
end do
if (.not. hit) then
ncls = ncls + 1
cls(:,ncls) = cy
end if
end do
cnt = cnt + 1
csizes(cnt) = ncls
do k = 1, ncls
do m2 = 1, nrep
if (all(cls(:,k) == reps(:,m2))) seen(m2) = .true.
end do
end do
end do
end block
call sort_int(csizes(1:cnt))
write(*,'(a,i0,a,4i3)') ' conjugacy classes mod sign: ', cnt, ' with sizes ', csizes(1:cnt)
call check('class equation reads 1 + 3 + 4 + 4', cnt == 4 .and. csizes(1) == 1 &
.and. csizes(2) == 3 .and. csizes(3) == 4 .and. csizes(4) == 4)
block
integer :: a1, a2, a3, a4, npure, nreal, nhalf
npure = 0; nreal = 0; nhalf = 0
do a1 = -1, 1
do a2 = -1, 1
do a3 = -1, 1
do a4 = -1, 1
if (a1*a1 + a2*a2 + a3*a3 + a4*a4 == 2) then
if (a1 == 0) then
npure = npure + 1
else
nreal = nreal + 1
end if
end if
end do
end do
end do
end do
do a1 = -3, 3, 2
do a2 = -3, 3, 2
do a3 = -3, 3, 2
do a4 = -3, 3, 2
if (a1*a1 + a2*a2 + a3*a3 + a4*a4 == 8) nhalf = nhalf + 1
end do
end do
end do
end do
write(*,'(a,i0,a,i0,a,i0,a,i0)') ' norm-two shell: ', npure + nreal, &
' = ', npure, ' pure + ', nreal, ' real-occupied; half-integer members ', nhalf
call check('norm-two shell is 24 = 12 + 12', npure == 12 .and. nreal == 12)
call check('no half-integer element of norm two', nhalf == 0)
end block
n = 0
do i = 1, 4
do j = 1, 4
if (i /= j) then
n = n + 1
pl(n,:) = [i, j]
end if
end do
end do
call check('twelve ordered pairs of four vertices', n == 12)
orbitn = 0; stabn = 0; cnt = 0
block
integer :: pm(4,24), npm, inv, x1, x2, x3, x4, t(4), orb(12,2), norb
logical :: got
npm = 0
do x1 = 1, 4
do x2 = 1, 4
do x3 = 1, 4
do x4 = 1, 4
t = [x1, x2, x3, x4]
if (all([(count(t == i), i = 1, 4)] == 1)) then
inv = 0
do i = 1, 3
do j = i+1, 4
if (t(i) > t(j)) inv = inv + 1
end do
end do
if (mod(inv, 2) == 0) then
npm = npm + 1
pm(:,npm) = t
end if
end if
end do
end do
end do
end do
call check('A4 has order 12', npm == 12)
norb = 0
do k = 1, npm
t = pm(:,k)
got = .false.
do i = 1, norb
if (orb(i,1) == t(1) .and. orb(i,2) == t(2)) got = .true.
end do
if (.not. got) then
norb = norb + 1
orb(norb,:) = [t(1), t(2)]
end if
if (t(1) == 1 .and. t(2) == 2) stabn = stabn + 1
end do
orbitn = norb
end block
write(*,'(a,i0,a,i0)') ' A4 orbit of the pair (1,2): size ', orbitn, ', stabilizer order ', stabn
call check('A4 simply transitive on the twelve directed gates', orbitn == 12 .and. stabn == 1)
block
integer :: vcnt, ecnt, fcnt, i3, j3, k3
vcnt = 0
do i3 = 1, 4
if (any(pl(1:n,1) == i3) .or. any(pl(1:n,2) == i3)) vcnt = vcnt + 1
end do
ecnt = n / 2
fcnt = 0
do i3 = 1, 4
do j3 = i3+1, 4
do k3 = j3+1, 4
fcnt = fcnt + 1
end do
end do
end do
write(*,'(a,i0,a,i0,a,i0)') ' Euler counted from the built complex: V = ', vcnt, ', E = ', ecnt, ', F = ', fcnt
call check('Euler closure counted from the complex: V - E + F = 2', vcnt - ecnt + fcnt == 2)
end block
block
integer(ik) :: uq(4,3), pq(4)
integer :: pms(3,6), inv, s
pms = reshape([1,2,3, 2,3,1, 3,1,2, 2,1,3, 1,3,2, 3,2,1], [3,6])
uq = 0_ik
uq(2,1) = 1_ik; uq(3,2) = 1_ik; uq(4,3) = 1_ik
do k = 1, 6
inv = 0
do i = 1, 2
do j = i+1, 3
if (pms(i,k) > pms(j,k)) inv = inv + 1
end do
end do
s = merge(1, -1, mod(inv,2) == 0)
pq = iqmul(iqmul(uq(:,pms(1,k)), uq(:,pms(2,k))), uq(:,pms(3,k)))
if (k <= 3) then
call check('even relabel lands Re = -1', pq(1) == -1_ik .and. s == 1)
else
call check('odd relabel lands Re = +1', pq(1) == +1_ik .and. s == -1)
end if
end do
write(*,'(a)') ' relabel parity: three even relabels land Re = -1, three odd land Re = +1'
write(*,'(a)') ' one discrete modulus, the orientation bit; zero continuous moduli (Part E sweep)'
end block
block
integer(ik) :: mx
mx = 0_ik
do i = 1, 24
do j = 1, 24
do k = 1, 24
t1(1:4) = iqmul(iqmul(hu(:,i), hu(:,j)), hu(:,k))
t2(1:4) = iqmul(hu(:,i), iqmul(hu(:,j), hu(:,k)))
mx = max(mx, maxval(abs(t1(1:4) - t2(1:4))))
end do
end do
end do
write(*,'(a,i0)') ' quaternion associator over all 13824 unit triples, max |.|: ', mx
call check('H is associative on the whole unit order', mx == 0_ik)
end block
e8a = 0_ik; e8b = 0_ik; e8c = 0_ik
e8a(2) = 1_ik ! e1
e8b(3) = 1_ik ! e2
e8c(5) = 1_ik ! e4
t1 = iomul(iomul(e8a, e8b), e8c)
t2 = iomul(e8a, iomul(e8b, e8c))
assoc8 = t1 - t2
z8 = 0_ik; z8(8) = 2_ik
write(*,'(a,8i3)') ' octonion associator [e1,e2,e4] components: ', assoc8
call check('octonion associator equals 2 e7 exactly', all(assoc8 == z8))
call check('commutativity already dead at H: ij /= ji', &
.not. all(iqmul(int([0,1,0,0],ik), int([0,0,1,0],ik)) == &
iqmul(int([0,0,1,0],ik), int([0,1,0,0],ik))))
block
integer(ik) :: av(8), bv(8), pv(8), aq(4), bq(4), pq(4)
integer :: cnt2
cnt2 = 0
do i = -1, 1
do j = -1, 1
do k = -1, 1
aq = int([i, j, k, 1], ik); bq = int([1, k, j, i], ik)
pq = iqmul(aq, bq)
call check('H norm composition exact', inrm(pq) == inrm(aq)*inrm(bq))
av = int([i, j, k, 1, j, k, i, 1], ik)
bv = int([1, k, i, j, 1, i, j, k], ik)
pv = iomul(av, bv)
call check('O norm composition exact', inrm(pv) == inrm(av)*inrm(bv))
cnt2 = cnt2 + 1
end do
end do
end do
write(*,'(a,i0,a)') ' norm composition |ab|^2 = |a|^2 |b|^2 exact on ', cnt2, ' integer pairs in H and in O'
end block
hit = .false.
outer: do i = 1, 15
do j = i+1, 15
do k = 1, 15
do l = k+1, 15
sa = 0_ik; sb = 0_ik
sa(i+1) = 1_ik; sa(j+1) = 1_ik
sb(k+1) = 1_ik; sb(l+1) = -1_ik
sp = ismul(sa, sb)
if (all(sp == 0_ik)) then
hit = .true.
found(1,:) = [i, j]; found(2,:) = [k, l]
exit outer
end if
end do
end do
end do
end do outer
call check('a sedenion zero-divisor pair exists', hit)
sa = 0_ik; sb = 0_ik
sa(found(1,1)+1) = 1_ik; sa(found(1,2)+1) = 1_ik
sb(found(2,1)+1) = 1_ik; sb(found(2,2)+1) = -1_ik
sp = ismul(sa, sb)
write(*,'(a,i0,a,i0,a,i0,a,i0,a)') ' sedenion wall: (e', found(1,1), ' + e', found(1,2), &
')(e', found(2,1), ' - e', found(2,2), ') = 0 exactly'
write(*,'(a,i0,a,i0,a,i0)') ' with |a|^2 = ', inrm(sa), ', |b|^2 = ', inrm(sb), &
', |ab|^2 = ', inrm(sp)
call check('division dies at S: nonzero factors, zero product', &
inrm(sa) == 2_ik .and. inrm(sb) == 2_ik .and. inrm(sp) == 0_ik)
call check('norm composition fails at S', inrm(sp) /= inrm(sa)*inrm(sb))
write(*,'(a)') ' the walls in order: commutativity at H, associativity at O, division at S'
block
integer(ik) :: uq(4), tt(4)
integer :: xg, yg, zg, nsur, offline
tt = iqmul(int([0,1,0,0],ik), int([0,0,1,0],ik)) + iqmul(int([0,0,1,0],ik), int([0,1,0,0],ik))
call check('i and j anticommute exactly: ij + ji = 0', all(tt == 0_ik))
nsur = 0; offline = 0
do xg = -2, 2
do yg = -2, 2
do zg = -2, 2
if (xg == 0 .and. yg == 0 .and. zg == 0) cycle
uq = int([0, xg, yg, zg], ik)
block
logical :: aci, acj
aci = all(iqmul(int([0,1,0,0],ik), uq) + iqmul(uq, int([0,1,0,0],ik)) == 0_ik)
acj = all(iqmul(int([0,0,1,0],ik), uq) + iqmul(uq, int([0,0,1,0],ik)) == 0_ik)
if (aci .and. acj) then
nsur = nsur + 1
if (xg /= 0 .or. yg /= 0) offline = offline + 1
end if
end block
end do
end do
end do
write(*,'(a,i0,a,i0,a)') ' joint anticommutant of {i, j} on the exact grid: ', nsur, &
' of 124 nonzero points, ', offline, ' off the k-line'
call check('the joint anticommutant is the k-line on the grid (independent cross-check)', &
nsur == 4 .and. offline == 0)
block
integer(ik) :: bimg(8,3), ev(4)
integer :: bi
integer(ik) :: minor2
do bi = 1, 3
ev = 0_ik; ev(bi+1) = 1_ik
bimg(1:4, bi) = iqmul(int([0,1,0,0],ik), ev) + iqmul(ev, int([0,1,0,0],ik))
bimg(5:8, bi) = iqmul(int([0,0,1,0],ik), ev) + iqmul(ev, int([0,0,1,0],ik))
end do
call check('linearity certificate: only the scalar components survive on the basis', &
all(bimg(2:4,:) == 0_ik) .and. all(bimg(6:8,:) == 0_ik))
minor2 = bimg(1,1)*bimg(5,2) - bimg(1,2)*bimg(5,1)
write(*,'(a,i0,a)') ' constraint matrix on the basis: 2x2 minor = ', minor2, &
', z-column identically zero'
call check('linearity certificate: minor 4 and zero z-column force the k-line kernel everywhere', &
minor2 == 4_ik .and. all(bimg(:,3) == 0_ik))
end block
end block
write(*,'(a)') ' below the walls the third axis is begotten and confined: i j = k executed,'
write(*,'(a)') ' and the joint anticommutant of {i, j} in Im H is the k-line, counted; the'
write(*,'(a)') ' fourth is barred above by the executed associator wall; the identification'
write(*,'(a)') ' of the three algebra axes with V_F, V_E, V_ER is the seated premise-typed'
write(*,'(a)') ' mapping, consumed here and never a discovery of this run'
! ------------------------------------------------------------------
write(*,'(/a)') 'PART D - THE NUMBER - the Return, then the physics-law reading'
p4 = iqmul(iqmul(int([0,1,0,0],ik), int([0,0,1,0],ik)), int([0,0,0,1],ik))
write(*,'(a,4i3,a)') ' the Return on the parse triad: i j k = ', p4, ' (Re = -1, the Ground)'
call check('Hamilton landing Re(ijk) = -1 exactly', p4(1) == -1_ik .and. all(p4(2:4) == 0_ik))
block
integer(ik) :: uq(4,3)
real(wp) :: m3(3,3), g3(3,3)
integer :: r
uq = 0_ik
uq(2,1) = 1_ik; uq(3,2) = 1_ik; uq(4,3) = 1_ik
do r = 1, 3
m3(r,:) = real(uq(2:4,r), wp)
end do
g3 = matmul(m3, transpose(m3))
call check('Gram of the orthonormal parse triad has det 1', &
abs(det3(g3) - 1.0_wp) <= 4.0_wp*U_M)
end block
do i = 1, NCTX
th(i) = 2.0_wp * pi_() * real(i-1, wp) / real(NCTX, wp)
rows(1,i) = sin(th(i)); rows(2,i) = cos(th(i)); rows(3,i) = sin(2.0_wp*th(i))
end do
call kernel_full(rows, reshape([0.0_wp], [0, NCTX]), tk, wy, rc, lam, detr, kap, &
spread, resid, mc, mk, d4)
write(*,'(a,a,a,f16.12,a,f16.12)') ' Fourier triad: ', trim(tk), &
' det(R) = ', detr, ' |lambda| = ', abs(lam)
call check('full Return: det(R) = 1 at machine precision', abs(detr - 1.0_wp) < 1.0e-9_wp)
call check('full Return: |lambda| = 1 at machine precision', abs(abs(lam) - 1.0_wp) < 1.0e-9_wp)
call check('kernel identity lambda^2 = det(R) on the Return', resid < 1.0e-12_wp)
call check('full Return carries the lock code', rc == RC_LOCK)
fr_resid = resid
amix = reshape([1.00_wp,0.22_wp,0.16_wp, 0.35_wp,1.00_wp,0.20_wp, 0.12_wp,0.28_wp,1.00_wp],[3,3])
bmix = reshape([0.60_wp,0.40_wp,0.55_wp, 0.30_wp,0.50_wp,0.20_wp],[3,2])
do i = 1, NCTX
l10m(i) = log10(ME) + 5.0_wp * (0.5_wp + 0.5_wp*sin(th(i)))
l10dx(i) = -10.0_wp + 4.0_wp * (0.5_wp + 0.5_wp*sin(2.0_wp*th(i) + 1.0_wp))
l10w(i) = 12.0_wp + 3.0_wp * (0.5_wp + 0.5_wp*sin(3.0_wp*th(i) + 2.0_wp))
l10t(i) = 2.477_wp * (0.5_wp + 0.5_wp*sin(5.0_wp*th(i) + 3.0_wp))
l10b(i) = 3.0_wp * (0.5_wp + 0.5_wp*sin(7.0_wp*th(i) + 4.0_wp))
rows(1,i) = log10(HBAR*HBAR/8.0_wp) - l10m(i) - 2.0_wp*l10dx(i) ! formal: the bound
rows(2,i) = log10(0.5_wp*HBAR) + l10w(i) ! empirical: zero-point
rows(3,i) = log10(KB*ln2_()) + l10t(i) + l10b(i) ! registration: Landauer
cov(1,i) = -20.6_wp + 2.5_wp * (0.5_wp + 0.5_wp*sin(4.0_wp*th(i) + 0.5_wp))
cov(2,i) = -24.0_wp + 5.0_wp * (0.5_wp + 0.5_wp*sin(6.0_wp*th(i) + 1.5_wp))
floorv(i) = 10.0_wp**rows(1,i)
end do
do i = 1, NCTX
call check('kinetic floor positive at every context', floorv(i) > 0.0_wp)
end do
mixm = matmul(amix, rows) + matmul(bmix, cov)
do i = 1, 3
eta(i) = eta_share(mixm(i,:), cov)
end do
call kernel_full(mixm, cov, tk, wy, rc, lam, detr, kap, spread, resid, mc, mk, d4)
write(*,'(a,a,a,a)') ' physics-law reading: ', trim(tk), ' - ', trim(wy)
write(*,'(a,f16.12,a,f16.12)') ' det(R) = ', detr, ' lambda = ', lam
write(*,'(a,f10.4,a,es10.3)') ' kappa(R) = ', kap, ' four-estimator spread = ', spread
write(*,'(a,es10.3,a,f7.2,a,f7.2)') ' |lambda^2 - det(R)| = ', resid, &
' margins (orders): collapse ', mc, ', conditioning ', mk
write(*,'(a,3f8.4)') ' source-share eta per mixed row (pre-projection): ', eta
write(*,'(a)') ' grid disclosure: the context grid is a constructed exhibit (deterministic'
write(*,'(a)') ' harmonics); the rows are physical-law magnitudes (Heisenberg, zero-point,'
write(*,'(a)') ' Landauer); the witness rides the cited theorems, never the grid'
call check('physics-law reading on the constructed exhibit determines: [LOCK]', tk == '[LOCK]' .and. rc == RC_LOCK)
call check('kernel identity within tolerance', resid <= max(4.0_wp*kap*U_M, 8.0_wp*U_M))
call check('four estimators agree within tolerance', spread <= 4.0_wp*kap*U_M)
call check('collapse margin exceeds ten orders', mc > 10.0_wp)
call check('conditioning margin exceeds one order', mk > 1.0_wp)
call check('source share strictly inside (0,1) on each row', &
all(eta > 0.0_wp) .and. all(eta < 1.0_wp))
block
logical :: okp
character(len=96) :: w2
real(wp) :: kcc
integer :: pcl
block
real(wp) :: qtmp(3,NCTX)
call prepare_rows(mixm, qtmp, okp, w2, pcl)
call project_cov(qtmp, cov, qf, okp, w2, pcl, kcc)
end block
do i = 1, 3
qf(i,:) = qf(i,:) / vnorm(qf(i,:))
end do
call span_basis(qf, bas, rank)
rank3_ok = (rank == 3)
call check('span rank three after projection', rank3_ok)
end block
rmat = matmul(qf, transpose(qf))
dbase = det3(rmat)
lam = lambda_of(qf, bas)
block
real(wp) :: mman(3,NCTX)
mman(1,:) = mixm(1,:); mman(2,:) = mixm(2,:)
mman(3,:) = 0.7_wp*cov(1,:) + 0.5_wp*cov(2,:)
call kernel_full(mman, cov, tk, wy, rc, lam2, detr2, kap2, spread2, resid2, mc2, mk2, d42)
write(*,'(a,a,a,a)') ' manufactured third axis: ', trim(tk), ' - ', trim(wy)
mig_ok = (tk == '[?]' .and. rc == RC_ABSORBED)
call check('an axis built from the covariates is absorbed, not locked', mig_ok)
block
real(wp) :: es1, es2
es1 = eta_share(mixm(1,:), cov)
es2 = eta_share(mixm(1,:)*1.0e300_wp, cov)
call check('eta_share equilibrated: a 1e300 row returns the baseline share, finite', &
is_finite(es2) .and. abs(es2 - es1) < 1.0e-12_wp)
es2 = eta_share(mixm(1,:)*((0.5_wp*huge(1.0_wp))/max(1.0_wp, maxval(abs(mixm(1,:))))), cov)
call check('eta_share at half-huge peak: raw-first scaling holds, the mean never overflows', &
is_finite(es2) .and. abs(es2 - es1) < 1.0e-10_wp)
end block
block
real(wp) :: covh(2,NCTX), lam9, detr9, kap9, sp9, rs9, d49(4), mgc9, mgk9
integer :: rc9
covh = cov * ((0.5_wp*huge(1.0_wp))/max(1.0_wp, maxval(abs(cov))))
call kernel_full(mixm, covh, tk, wy, rc9, lam9, detr9, kap9, sp9, rs9, mgc9, mgk9, d49)
call check('project_cov at half-huge covariates: raw-first centering survives, verdict [LOCK]', &
tk == '[LOCK]' .and. rc9 == RC_LOCK)
end block
block
real(wp) :: eb9, eh9, covh2(2,NCTX)
eb9 = eta_share(mixm(1,:), cov)
covh2 = cov * ((0.25_wp*huge(1.0_wp)) / max(1.0_wp, maxval(abs(cov))))
eh9 = eta_share(mixm(1,:), covh2)
call check('eta_share near-huge covariates: raw-first scaling holds on the covariate branch', &
is_finite(eh9) .and. abs(eh9 - eb9) < 1.0e-10_wp)
end block
block
real(wp) :: hp(3,3), eigh(3)
logical :: cvg
hp = 0.0_wp
hp(1,1) = 0.5_wp*huge(1.0_wp); hp(2,2) = 0.25_wp*huge(1.0_wp); hp(3,3) = 0.125_wp*huge(1.0_wp)
call jacobi3(hp, eigh, cvg)
call check('jacobi3 near-huge: distributed-halves symmetrization never overflows finite input', &
cvg .and. is_finite(eigh(1)) .and. is_finite(eigh(2)) .and. is_finite(eigh(3)))
end block
block
real(wp) :: cbad(2,NCTX-1), lamw, detrw, kapw, spw, rsw, mgcw, mgkw, d4w(4)
integer :: rcw
cbad = 0.0_wp
call kernel_full(mixm, cbad, tk, wy, rcw, lamw, detrw, kapw, spw, rsw, mgcw, mgkw, d4w)
call check('kernel refuses mismatched covariate width at intake, its own guard first', &
tk == '[?]' .and. rcw == RC_INTAKE .and. index(wy, 'prepared') == 0)
end block
block
real(wp) :: cbad_eta(1,NCTX-1)
cbad_eta = 0.0_wp
call check('eta_share refuses mismatched covariate width as NaN, guard before touch', &
.not. is_finite(eta_share(mixm(1,:), cbad_eta)))
end block
block
real(wp) :: cbad_p(2,NCTX-1), qpf(3,NCTX), kccp
integer :: rcp
logical :: okp
cbad_p = 0.0_wp
call project_cov(mixm, cbad_p, qpf, okp, wy, rcp, kccp)
call check('project_cov refuses mismatched covariate width directly at intake', &
(.not. okp) .and. rcp == RC_INTAKE .and. kccp == huge(1.0_wp))
end block
block
real(wp) :: rconst(6), cthree(3,6)
rconst = 1.0_wp; cthree = 0.0_wp
call check('eta_share k-cap outranks the zero-variance shortcut (fuzz scalp pinned)', &
.not. is_finite(eta_share(rconst, cthree)))
end block
block
real(wp) :: mconst(3,NCTX), cov3c(3,NCTX), lamk7, detrk7, kapk7, spk7, rsk7, mck7, mkk7, d4k7(4)
integer :: rck7
mconst = 1.0_wp; cov3c = 0.0_wp
call kernel_full(mconst, cov3c, tk, wy, rck7, lamk7, detrk7, kapk7, spk7, rsk7, mck7, mkk7, d4k7)
call check('kernel k-cap outranks row zero-variance under the documented precedence', &
tk == '[?]' .and. rck7 == RC_COVBLOCK .and. index(wy, 'more than two') > 0)
end block
block
real(wp) :: ehm
ehm = heis_floor(1.0e-309_wp, HBAR)
call check('heis_floor at subnormal mass: finite 1.25e308, decomposition never overflows early', &
is_finite(ehm) .and. ehm > 1.0e308_wp .and. ehm < 1.3e308_wp)
end block
block
real(wp) :: tth
logical :: frzh
call tau_perp(huge(1.0_wp), tth, frzh)
call check('tau_perp at a huge gap: subnormal positive time, never a false zero', &
(.not. frzh) .and. is_finite(tth) .and. tth > 0.0_wp)
end block
block
real(wp) :: m2r(2,NCTX), q2r(2,NCTX), m1c(3,1), q1c(3,1)
logical :: okr
integer :: rcr
m2r = 1.0_wp
call prepare_rows(m2r, q2r, okr, wy, rcr)
call check('prepare_rows refuses two rows at intake, no out-of-bounds path', &
(.not. okr) .and. rcr == RC_INTAKE .and. index(wy, 'not three') > 0)
m1c = 1.0_wp
call prepare_rows(m1c, q1c, okr, wy, rcr)
call check('prepare_rows refuses a single context at intake, no divide-by-zero path', &
(.not. okr) .and. rcr == RC_INTAKE .and. index(wy, 'two contexts') > 0)
end block
block
real(wp) :: q2p(2,NCTX), c0p(0,NCTX), qf2(2,NCTX), kcc2
logical :: ok2
integer :: rc2p
q2p = 1.0_wp
call project_cov(q2p, c0p, qf2, ok2, wy, rc2p, kcc2)
call check('project_cov refuses a non-three prepared-row count at intake', &
(.not. ok2) .and. rc2p == RC_INTAKE .and. index(wy, 'not three') > 0)
end block
block
real(wp) :: idq(3,4), bqi(3,4)
integer :: rk1, rk2
idq = 0.0_wp; idq(1,1) = 1.0_wp; idq(2,2) = 1.0_wp; idq(3,3) = 1.0_wp
call span_basis(idq, bqi, rk1)
call span_basis(idq*1.0e-13_wp, bqi, rk2)
call check('span_basis rank is scale-invariant: identity and 1e-13 identity both rank three', &
rk1 == 3 .and. rk2 == 3)
call span_basis(scale(idq, -400), bqi, rk2)
call check('span_basis rank holds at a 2^-400 uniform rescaling', rk2 == 3)
end block
block
real(wp) :: q1c(3,1), c0(0,1), c1(1,1), qf1(3,1), kc1
logical :: okc
integer :: rcc
q1c = 1.0_wp; c1 = 0.0_wp
call project_cov(q1c, c0, qf1, okc, wy, rcc, kc1)
call check('project_cov refuses a single context with zero covariates', &
(.not. okc) .and. rcc == RC_INTAKE .and. index(wy, 'two contexts') > 0 .and. all(qf1 == 0.0_wp))
call project_cov(q1c, c1, qf1, okc, wy, rcc, kc1)
call check('project_cov refuses a single context with one covariate', &
(.not. okc) .and. rcc == RC_INTAKE .and. all(qf1 == 0.0_wp))
end block
block
real(wp) :: dup1(2,2), duph(2,2), bd(2,2)
integer :: rkd
dup1 = 1.0_wp
call span_basis(dup1, bd, rkd)
call check('span_basis: duplicate unit rows rank one, finite unit basis', &
rkd == 1 .and. is_finite(bd(1,1)) .and. abs(vnorm(bd(1,:)) - 1.0_wp) < 1.0e-12_wp)
duph = huge(1.0_wp)
call span_basis(duph, bd, rkd)
call check('span_basis: duplicate rows at huge rank one, the infinite-norm bypass closed', &
rkd == 1 .and. is_finite(bd(1,1)) .and. abs(vnorm(bd(1,:)) - 1.0_wp) < 1.0e-12_wp)
end block
block
real(wp) :: mln(3,NCTX), qln(3,NCTX)
logical :: okl
integer :: rcl, jj
mln = 1.0_wp
mln(1,:) = [(0.1_wp*real(jj,wp), jj = 1, NCTX)]
mln(2,:) = [(0.2_wp*real(jj,wp) + 0.5_wp, jj = 1, NCTX)]
mln(3,5) = qnan()
call prepare_rows(mln, qln, okl, wy, rcl)
call check('prepare_rows late-row NaN: refused with every output element exactly zero', &
(.not. okl) .and. rcl == RC_INTAKE .and. all(qln == 0.0_wp))
end block
block
character(8) :: tka, tkb
character(96) :: wya, wyb
call seal_l(2, 2, 2, .false., tka, wya)
call seal_l_ordered(RA_SENT, tkb, wyb)
call check('seal interfaces speak their own guarantees: count-only disclaims order, ordered claims it', &
index(wya, 'order not checked') > 0 .and. tkb == '[LOCK-L]' .and. &
index(wyb, 'positional check executed') > 0)
end block
block
real(wp) :: rd(3,3), dd
rd = 0.0_wp
rd(1,1) = scale(1.0_wp, 600); rd(2,2) = scale(1.0_wp, -300); rd(3,3) = scale(1.0_wp, -300)
dd = det3(rd)
call check('det3 exponent-tracked Leibniz monomials: diag 2^600, 2^-300, 2^-300 returns exactly one', dd == 1.0_wp)
rd = 0.0_wp
rd(1,3) = scale(1.0_wp, -300); rd(2,2) = scale(1.0_wp, 600); rd(3,1) = scale(1.0_wp, -300)
dd = det3(rd)
call check('det3 permuted-diagonal variant: the signed unit determinant is exact', dd == -1.0_wp)
rd = 0.0_wp
rd(1,1) = scale(1.0_wp, 600); rd(2,2) = scale(1.0_wp, 600); rd(3,3) = scale(1.0_wp, 600)
dd = det3(rd)
call check('det3 honest overflow: a true 2^1800 determinant reports past huge, never zero', &
dd > huge(1.0_wp))
end block
block
real(wp) :: rd(3,3), d2
rd = 0.0_wp
rd(1,1) = scale(1.0_wp, 600); rd(2,2) = scale(1.0_wp, -300); rd(3,3) = scale(1.0_wp, -300)
d2 = det3_row2(rd)
call check('det3_row2 exponent-tracked Leibniz monomials: the exact counterexample returns exactly one', d2 == 1.0_wp)
call check('det3_row2 agrees with det3 on the counterexample', d2 == det3(rd))
rd = 0.0_wp
rd(1,3) = scale(1.0_wp, -300); rd(2,2) = scale(1.0_wp, 600); rd(3,1) = scale(1.0_wp, -300)
d2 = det3_row2(rd)
call check('det3_row2 odd permutation: the signed unit determinant is exact and matches det3', &
d2 == -1.0_wp .and. d2 == det3(rd))
rd = 0.0_wp
rd(1,1) = scale(1.0_wp, 600); rd(2,2) = scale(1.0_wp, 600); rd(3,3) = scale(1.0_wp, 600)
d2 = det3_row2(rd)
call check('det3_row2 honest overflow past huge, agreeing with det3', &
d2 > huge(1.0_wp) .and. (det3(rd) > huge(1.0_wp)))
rd = 0.0_wp
rd(1,1) = scale(1.0_wp, -400); rd(2,2) = scale(1.0_wp, -400); rd(3,3) = scale(1.0_wp, -400)
d2 = det3_row2(rd)
call check('det3_row2 honest underflow: a true 2^-1200 determinant returns zero, never NaN, matching det3', &
d2 == 0.0_wp .and. det3(rd) == 0.0_wp)
end block
block
real(wp) :: ut(3,3), tt(3,3), sw(3,3)
ut = 0.0_wp
ut(1,1) = scale(1.0_wp, -300); ut(1,2) = scale(1.0_wp, 600)
ut(2,2) = scale(1.0_wp, -300); ut(2,3) = scale(1.0_wp, 600)
ut(3,3) = scale(1.0_wp, 600)
tt = transpose(ut)
sw = ut; sw(1,:) = ut(2,:); sw(2,:) = ut(1,:)
call check('det3 monomial tracking: the off-diagonal-dominated upper triangular returns exactly one', &
det3(ut) == 1.0_wp)
call check('det3_row2 monomial tracking: the same matrix returns exactly one', &
det3_row2(ut) == 1.0_wp)
call check('det3 on the transpose: exactly one', det3(tt) == 1.0_wp)
call check('det3_row2 on the transpose: exactly one', det3_row2(tt) == 1.0_wp)
call check('det3 on the row-swapped variant: exactly minus one', det3(sw) == -1.0_wp)
call check('det3_row2 on the row-swapped variant: exactly minus one', det3_row2(sw) == -1.0_wp)
end block
block
real(wp) :: nd(3,3)
nd = 0.0_wp
nd(1,1) = 1.0_wp + scale(1.0_wp, -27)
nd(2,2) = 1.0_wp - scale(1.0_wp, -27)
nd(3,3) = 1.0_wp
call check('product rounding: near-one diagonal, exact 1-2^-54 on tie, correctly rounds to 1.0 on both routines', &
det3(nd) == 1.0_wp .and. det3_row2(nd) == 1.0_wp)
end block
block
real(wp) :: qa(4), qb(4), qc(4), tgt
tgt = scale(1.0_wp, -474)
qa = [scale(1.0_wp, 600), 0.0_wp, 0.0_wp, 0.0_wp]
qb = [scale(1.0_wp, -1074), 0.0_wp, 0.0_wp, 0.0_wp]
qc = qmul(qa, qb)
call check('qmul monomial tracking: 2^600 scalar times minimum subnormal returns exactly 2^-474', &
qc(1) == tgt .and. all(qc(2:4) == 0.0_wp))
qc = qmul(qb, qa)
call check('qmul reversed operands: the same exact 2^-474', &
qc(1) == tgt .and. all(qc(2:4) == 0.0_wp))
qa = [0.0_wp, scale(1.0_wp, 600), 0.0_wp, 0.0_wp]
qb = [0.0_wp, scale(1.0_wp, -1074), 0.0_wp, 0.0_wp]
qc = qmul(qa, qb)
call check('qmul i times i at extreme scale: component one exactly minus 2^-474', &
qc(1) == -tgt .and. all(qc(2:4) == 0.0_wp))
qa = [scale(1.0_wp, 600), 0.0_wp, 0.0_wp, 0.0_wp]
qb = [0.0_wp, scale(1.0_wp, -1074), 0.0_wp, 0.0_wp]
qc = qmul(qa, qb)
call check('qmul scalar times i at extreme scale: component two exactly 2^-474', &
qc(2) == tgt .and. qc(1) == 0.0_wp .and. all(qc(3:4) == 0.0_wp))
qa = [0.0_wp, 0.0_wp, scale(1.0_wp, 600), 0.0_wp]
qb = [scale(1.0_wp, -1074), 0.0_wp, 0.0_wp, 0.0_wp]
qc = qmul(qa, qb)
call check('qmul j times scalar at extreme scale: component three exactly 2^-474', &
qc(3) == tgt .and. qc(1) == 0.0_wp .and. qc(2) == 0.0_wp .and. qc(4) == 0.0_wp)
qa = [0.0_wp, scale(1.0_wp, 600), 0.0_wp, 0.0_wp]
qb = [0.0_wp, 0.0_wp, scale(1.0_wp, -1074), 0.0_wp]
qc = qmul(qa, qb)
call check('qmul i times j at extreme scale: component four exactly 2^-474, the k of the begetting', &
qc(4) == tgt .and. qc(1) == 0.0_wp .and. qc(2) == 0.0_wp .and. qc(3) == 0.0_wp)
end block
block
real(wp) :: qtwo(2,NCTX), bthree(3,NCTX), lmz, rz(3,3)
qtwo = 1.0_wp; bthree = 1.0_wp
lmz = lambda_of(qtwo, bthree)
call check('lambda_of refuses a two-row first argument before any element touch: quiet NaN', &
.not. is_finite(lmz) .and. lmz /= lmz)
call gram3_fixed(qtwo, rz)
call check('gram3_fixed refuses a two-row input before any element touch: quiet-NaN fill', &
all(rz /= rz))
end block
block
real(wp) :: qiv(2,4), biv(2,4)
integer :: rkv
logical :: inv
call ieee_set_flag(ieee_invalid, .false.)
qiv = 0.0_wp
qiv(1,1) = ieee_value(1.0_wp, ieee_positive_inf)
qiv(2,2) = 1.0_wp
call span_basis(qiv, biv, rkv)
call ieee_get_flag(ieee_invalid, inv)
call check('span_basis finite screen absorbs a non-finite row before arithmetic: invalid flag stays clear', &
rkv == 1 .and. (.not. inv))
end block
block
real(wp) :: qcz(3,NCTX), ccz(1,NCTX), qfz(3,NCTX), kcz
logical :: okz
integer :: rcz, jj2
qcz = 1.0_wp
qcz(1,:) = [(0.3_wp*real(jj2,wp), jj2 = 1, NCTX)]
ccz = 5.0_wp
call project_cov(qcz, ccz, qfz, okz, wy, rcz, kcz)
call check('project_cov constant-covariate refusal publishes exactly zero (success-only publication)', &
(.not. okz) .and. rcz == RC_COVBLOCK .and. all(qfz == 0.0_wp))
end block
end block
block
real(wp) :: mdel(3,NCTX)
mdel = mixm; mdel(2,:) = 0.0_wp
call kernel_full(mdel, cov, tk, wy, rc, lam2, detr2, kap2, spread2, resid2, mc2, mk2, d42)
call check('deleted slot: axis carries no content', tk == '[?]' .and. rc == RC_ZEROVAR)
mdel = mixm; mdel(3,:) = mdel(1,:)
call kernel_full(mdel, cov, tk, wy, rc, lam2, detr2, kap2, spread2, resid2, mc2, mk2, d42)
write(*,'(a,a,a,a)') ' duplicated slot: ', trim(tk), ' - ', trim(wy)
call check('duplicated slot: coplanar collapse, a line and not a point', &
tk == '[X]' .and. rc == RC_COPLANAR)
end block
block
real(wp) :: qneg(3,NCTX)
qneg = -qf
rneg = matmul(qneg, transpose(qneg))
call check('full negation leaves the Gram bit-identical', &
maxval(abs(rneg - rmat)) == 0.0_wp)
lamneg = lambda_of(qneg, bas)
call check('full negation flips lambda exactly', lamneg == -lam)
write(*,'(a,es9.2,a,f16.12,a,f16.12)') ' full negation: max|G(P)-G(-P)| = ', &
maxval(abs(rneg - rmat)), ' lambda ', lam, ' -> ', lamneg
end block
block
real(wp) :: qref(3,NCTX), rref(3,3)
qref = qf; qref(1,:) = -qref(1,:)
lamref = lambda_of(qref, bas)
rref = matmul(qref, transpose(qref))
detref = det3(rref)
write(*,'(a,f16.12,a,f16.12)') ' single reflection: lambda ', lam, ' -> ', lamref
write(*,'(a,es9.2)') ' |det(R) - det(R) reflected| = ', abs(detref - dbase)
dual_ok = (lamref == -lam) .and. (detref == dbase)
call check('reflection flips lambda bit-exactly on the fixed basis', lamref == -lam)
call check('reflection leaves det(R) unmoved bit-exactly', detref == dbase)
end block
block
real(wp) :: mrev(3,NCTX), crev(2,NCTX)
do i = 1, NCTX
mrev(:,i) = mixm(:,NCTX+1-i)
crev(:,i) = cov(:,NCTX+1-i)
end do
call kernel_full(mrev, crev, tk, wy, rc, lam2, detr2, kap2, spread2, resid2, mc2, mk2, d42)
arrow_f = 0.0_wp; arrow_r = 0.0_wp
do i = 1, NCTX
arrow_f = arrow_f + (10.0_wp**l10b(i)) * ln2_()
end do
do i = NCTX, 1, -1
arrow_r = arrow_r - (10.0_wp**l10b(i)) * ln2_()
end do
write(*,'(a,es10.3)') ' time reversal: |det forward - det reversed| = ', abs(detr2 - detr)
write(*,'(a,f14.4,a,f14.4,a)') ' the arrow (registration entropy, kB units): forward ', &
arrow_f, ', reversed ', arrow_r, ' (the sign parts them)'
call check('the determinant is blind to the arrow', abs(detr2 - detr) <= 64.0_wp*kap*U_M)
call check('the directed registration sum flips sign under traversal reversal '// &
'(constructed odd witness; the physical arrow rides the cited second law)', &
arrow_f > 0.0_wp .and. arrow_r < 0.0_wp &
.and. abs(arrow_f + arrow_r) <= 1.0e-9_wp*arrow_f)
end block
! ------------------------------------------------------------------
write(*,'(/a)') 'PART E - THE BLINDNESS AND ITS EXACT WIDTH - one bit, counted'
do s1 = -1, 1, 2
do s2 = -1, 1, 2
do s3 = -1, 1, 2
block
real(wp) :: dd(3), rp(3,3)
dd = real([s1, s2, s3], wp)
do i = 1, 3
do j = 1, 3
rp(i,j) = dd(i)*dd(j)*rmat(i,j)
end do
end do
dpat = det3(rp)
call check('eight-pattern determinant invariance, bit-exact', dpat == dbase)
end block
end do
end do
end do
write(*,'(a)') ' all eight sign patterns return det(D R D) = det(R) bit-exactly'
write(*,'(a)') ' lock(P) = lock(not P) at the scalar: the Number is orientation-blind by theorem'
block
real(wp) :: sig(4,4), diag4(4,4), dres
integer :: iq
sig = 0.0_wp
sig(1,1) = 1.0_wp; sig(2,2) = -1.0_wp; sig(3,3) = -1.0_wp; sig(4,4) = -1.0_wp
dres = sig(2,2)*sig(3,3)*sig(4,4)
diag4 = 0.0_wp
do iq = 1, 4
diag4(iq,iq) = -1.0_wp
end do
write(*,'(a,f6.1,a)') ' Form source: det(sigma | residence) = ', dres, ' for P and for not-P alike: 0 bits'
call check('sigma ground dimension one', count([(sig(iq,iq), iq = 1, 4)] > 0.0_wp) == 1)
call check('the diagonal carries no ground (built and counted)', &
count([(diag4(iq,iq), iq = 1, 4)] > 0.0_wp) == 0)
call check('residence handedness identical under negation', dres == -1.0_wp)
end block
pass12 = .true.
pass12(2) = rank3_ok
pass12(3) = disjoint(SLOT_E, SLOT_K) .and. disjoint(SLOT_E, SLOT_R) &
.and. disjoint(SLOT_K, SLOT_R)
pass12(5) = mig_ok
pass12(7) = dual_ok
pass12(8) = hurwitz_ok
pass12(10) = (fr_resid < 1.0e-12_wp)
write(*,'(a)') ' gate manifest: executed facts arm 2 REG (span rank three), 3 SGEG (the'
write(*,'(a)') ' linguistic LIT), 5 MIG (the absorbed-axis dissolution), 7 DUAL (the'
write(*,'(a)') ' analytic reflection), 8 CSCG (the integer Hurwitz closure), and 10 MTA'
write(*,'(a)') ' (the harmonic Return identity); gates 1, 4, 6, 9, 11, 12 are declared'
write(*,'(a)') ' screens at this register. independent here means six pairwise-distinct'
write(*,'(a)') ' executed objects, dimensional, linguistic, projective, analytic,'
write(*,'(a)') ' integer, harmonic; gate-passage is a screen verdict, cited for nothing'
write(*,'(a)') ' external, the theorem-tier gate contents living at their seats'
call check('six gate entries armed from pairwise-distinct executed objects', &
pass12(2) .and. pass12(3) .and. pass12(5) .and. pass12(7) .and. &
pass12(8) .and. pass12(10))
call gate_screen(pass12, tk, wy)
block
character(len=16) :: tk2
character(len=140) :: wy2
call gate_screen(pass12, tk2, wy2) ! the roster reads structure, not sign
call check('gate screen identical for the claim and its negation', tk == tk2 .and. wy == wy2)
end block
write(*,'(a)') ' gate source: role-direction is fixed before any proposition arrives: 0 bits'
block
real(wp) :: cor(3,3), coc(3,3), lam_det
do i = 1, 3
co(i,1) = sum(qf(i,:)*bas(1,:))
co(i,2) = sum(qf(i,:)*bas(2,:))
co(i,3) = sum(qf(i,:)*bas(3,:))
end do
lam_det = -det3(co) ! the Number's reading of the sign
call check('the two lambda readings agree (quaternion vs determinant)', &
abs(lam_det - lam) <= 8.0_wp*U_M)
cor = co; cor(1,:) = -cor(1,:) ! content flip: negate a row
coc = co; coc(:,1) = -coc(:,1) ! convention flip: negate a basis vector
lam_c = -det3(cor)
lam_v = -det3(coc)
write(*,'(a,f16.12,a,f16.12)') ' relay: content-flip image ', lam_c, &
', convention-flip image ', lam_v
call check('the two flips return the identical image, bit-exact', lam_c == lam_v)
call check('the relayed image is minus the handed sign, bit-exact', lam_c == -lam_det)
end block
write(*,'(a)') ' Number source: the relay reports the orientation it was handed and originates none: 0 bits'
do i = 1, 3
co(i,1) = sum(qf(i,:)*bas(1,:)); co(i,2) = sum(qf(i,:)*bas(2,:)); co(i,3) = sum(qf(i,:)*bas(3,:))
quat(:,i) = [0.0_wp, co(i,1), co(i,2), co(i,3)]
end do
dlam_max = 0.0_wp; nflip = 0
block
real(wp) :: qr(4,3), tq(4), tq2(4), lam_refl, pp
integer :: nposc, nnegc
logical :: reflx
nposc = 0; nnegc = 0; reflx = .true.
do n = 1, 100
aa = 0.10_wp*real(n,wp); bb = 0.20_wp*real(n,wp) + 0.3_wp; cc2 = 0.35_wp*real(n,wp) + 0.7_wp
uvec = [sin(bb)*cos(cc2), sin(bb)*sin(cc2), cos(bb)]
g = [cos(aa), sin(aa)*uvec(1), sin(aa)*uvec(2), sin(aa)*uvec(3)]
do i = 1, 3
qr(:,i) = qmul(qmul(g, quat(:,i)), qconj(g))
end do
tq = qmul(qmul(qr(:,1), qr(:,2)), qr(:,3))
lam_rot = tq(1)
dlam_max = max(dlam_max, abs(lam_rot - lam))
if (sign(1.0_wp, lam_rot) /= sign(1.0_wp, lam)) nflip = nflip + 1
tq2 = qmul(qmul(-qr(:,1), qr(:,2)), qr(:,3))
lam_refl = tq2(1)
reflx = reflx .and. (lam_refl == -lam_rot)
if (lam_rot > 0.0_wp) then
nposc = nposc + 1
else
nnegc = nnegc + 1
end if
if (lam_refl > 0.0_wp) then
nposc = nposc + 1
else
nnegc = nnegc + 1
end if
end do
write(*,'(a,es10.3,a,i0)') ' conjugation sweep, 100 frames: max|dlambda| = ', dlam_max, &
', sign flips: ', nflip
call check('frame invariance: the sweep moves lambda only at roundoff', dlam_max < 1.0e-12_wp)
call check('the orientation sign never flips under rotation', nflip == 0)
call check('every reflected partner flips the sign bit-exactly', reflx)
pp = real(nposc, wp) / real(nposc + nnegc, wp)
hbit = -(pp*log(pp) + (1.0_wp-pp)*log(1.0_wp-pp)) / ln2_()
write(*,'(a,i0,a,i0)') ' paired census, counted from the sweep: positive ', nposc, &
', negative ', nnegc
write(*,'(a,f14.10)') ' uniform-measure entropy of the counted two-point fibre: ', hbit
write(*,'(a)') ' (the census is the constructed pair family realizing both signs at'
write(*,'(a)') ' bit-identical invariants; freedom here is the seated Freedom law''s'
write(*,'(a)') ' object, the fibre of the invariant map, a count of what the instrument'
write(*,'(a)') ' class cannot tell apart; the deficit is one binary parameter of supply,'
write(*,'(a)') ' the aperture width at its seat; one bit is the uniform-coding capacity'
write(*,'(a)') ' of that two-point fibre; no physical frequency, no prior, and nothing'
write(*,'(a)') ' about any knower''s credence is asserted)'
call check('the paired census splits exactly in half, counted', nposc == 100 .and. nnegc == 100)
nsheets = (nposc + nnegc) / 100
call check('the counted two-point fibre carries one binary parameter; '// &
'uniform-measure entropy exactly one bit', abs(hbit - 1.0_wp) <= 4.0_wp*U_M)
call check('the invariant record is bit-identical across both signs (the '// &
'seated zero-information result is carried by this exact symmetry)', &
maxval(abs(rneg - rmat)) == 0.0_wp)
write(*,'(a)') ' both orientations are consistent with every determined quantity;'
write(*,'(a)') ' the deficit of the whole instrument class is exactly one bit'
end block
block
integer, parameter :: NG = 401, NC = 360
real(wp) :: xg, tolr, sx(NC), sy(NC), thr2, dx2
logical :: hitp(NG)
integer :: compid(NG), comp2(NC), stk(NG), stk2(NC), topv, cur, nb
tolr = 2.5e-2_wp
do i = 1, NG
xg = -2.0_wp + 4.0_wp*real(i-1,wp)/real(NG-1,wp)
hitp(i) = abs(xg*xg - 1.0_wp) < tolr
end do
compid = 0; ncomp1 = 0
do i = 1, NG
if (hitp(i) .and. compid(i) == 0) then
ncomp1 = ncomp1 + 1
topv = 1; stk(1) = i; compid(i) = ncomp1
do while (topv > 0)
cur = stk(topv); topv = topv - 1
do nb = cur-1, cur+1, 2
if (nb >= 1 .and. nb <= NG) then
if (hitp(nb) .and. compid(nb) == 0) then
compid(nb) = ncomp1
topv = topv + 1; stk(topv) = nb
end if
end if
end do
end do
end if
end do
do i = 1, NC
sx(i) = cos(2.0_wp*pi_()*real(i-1,wp)/real(NC,wp))
sy(i) = sin(2.0_wp*pi_()*real(i-1,wp)/real(NC,wp))
end do
thr2 = (1.6_wp * 2.0_wp * sin(pi_()/real(NC,wp)))**2
comp2 = 0; ncomp2 = 0
do i = 1, NC
if (comp2(i) == 0) then
ncomp2 = ncomp2 + 1
topv = 1; stk2(1) = i; comp2(i) = ncomp2
do while (topv > 0)
cur = stk2(topv); topv = topv - 1
do nb = 1, NC
if (comp2(nb) == 0) then
dx2 = (sx(nb)-sx(cur))**2 + (sy(nb)-sy(cur))**2
if (dx2 < thr2) then
comp2(nb) = ncomp2
topv = topv + 1; stk2(topv) = nb
end if
end if
end do
end do
end if
end do
write(*,'(a,i0,a,i0,a)') ' S0 has ', ncomp1, ' components; S1 has ', ncomp2, &
' (the two-point sphere is the unique disconnected sphere)'
call check('midpoint 0 is not a solution of x^2 = 1', .not. (abs(0.0_wp*0.0_wp - 1.0_wp) < tolr))
call check('S0 census on this grid: two components, the executed instance of the cited fact', ncomp1 == 2)
call check('S1 census on this grid: connected, the executed instance of the cited fact', ncomp2 == 1)
block
integer, parameter :: NG2 = 1601, NC2 = 720
real(wp) :: xg2, tolr2, sx2(NC2), sy2(NC2), thr22, dx22
logical :: hitp2(NG2)
integer :: cid2(NG2), cm2(NC2), st2(NG2), sq2(NC2), tp2, cu2, nb2
integer :: g2, na2, nb2c
tolr2 = 1.25e-2_wp
do g2 = 1, NG2
xg2 = -2.0_wp + 4.0_wp*real(g2-1,wp)/real(NG2-1,wp)
hitp2(g2) = abs(xg2*xg2 - 1.0_wp) < tolr2
end do
cid2 = 0; na2 = 0
do g2 = 1, NG2
if (hitp2(g2) .and. cid2(g2) == 0) then
na2 = na2 + 1
tp2 = 1; st2(1) = g2; cid2(g2) = na2
do while (tp2 > 0)
cu2 = st2(tp2); tp2 = tp2 - 1
do nb2 = cu2-1, cu2+1, 2
if (nb2 >= 1 .and. nb2 <= NG2) then
if (hitp2(nb2) .and. cid2(nb2) == 0) then
cid2(nb2) = na2
tp2 = tp2 + 1; st2(tp2) = nb2
end if
end if
end do
end do
end if
end do
do g2 = 1, NC2
sx2(g2) = cos(2.0_wp*pi_()*real(g2-1,wp)/real(NC2,wp))
sy2(g2) = sin(2.0_wp*pi_()*real(g2-1,wp)/real(NC2,wp))
end do
thr22 = (1.6_wp * 2.0_wp * sin(pi_()/real(NC2,wp)))**2
cm2 = 0; nb2c = 0
do g2 = 1, NC2
if (cm2(g2) == 0) then
nb2c = nb2c + 1
tp2 = 1; sq2(1) = g2; cm2(g2) = nb2c
do while (tp2 > 0)
cu2 = sq2(tp2); tp2 = tp2 - 1
do nb2 = 1, NC2
if (cm2(nb2) == 0) then
dx22 = (sx2(nb2)-sx2(cu2))**2 + (sy2(nb2)-sy2(cu2))**2
if (dx22 < thr22) then
cm2(nb2) = nb2c
tp2 = tp2 + 1; sq2(tp2) = nb2
end if
end if
end do
end do
end if
end do
write(*,'(a,i0,a,i0,a)') ' refinement (grid x4 on the line, x2 on the circle, tolerance halved): ', &
na2, ' and ', nb2c, ' components'
call check('S0 census stable under refinement and tolerance halving', na2 == 2)
call check('S1 census stable under refinement', nb2c == 1)
end block
write(*,'(a)') ' the census is the executed instance at two spheres, parameter-robust under'
write(*,'(a)') ' refinement; the uniqueness of the disconnected sphere is the classical'
write(*,'(a)') ' fact it corroborates, cited not proven'
end block
block
real(wp) :: dmat(3,3), d2m(3,3), e3v(3), res(3), eigl(3), best
real(wp) :: ejv(3), vpr(3), mprod(3,3)
integer :: t2i
block
integer, parameter :: NGE = 401
real(wp) :: xe, tole
logical :: hite(NGE)
integer :: ide(NGE), ncompe, ste(NGE), tpe, cure, nbe, ge
tole = 2.5e-2_wp
do ge = 1, NGE
xe = -2.0_wp + 4.0_wp*real(ge-1,wp)/real(NGE-1,wp)
hite(ge) = abs(xe*xe - xe) < tole
end do
ide = 0; ncompe = 0
do ge = 1, NGE
if (hite(ge) .and. ide(ge) == 0) then
ncompe = ncompe + 1
tpe = 1; ste(1) = ge; ide(ge) = ncompe
do while (tpe > 0)
cure = ste(tpe); tpe = tpe - 1
do nbe = cure-1, cure+1, 2
if (nbe >= 1 .and. nbe <= NGE) then
if (hite(nbe) .and. ide(nbe) == 0) then
ide(nbe) = ncompe
tpe = tpe + 1; ste(tpe) = nbe
end if
end if
end do
end do
end if
end do
write(*,'(a,i0,a)') ' idempotent census on the line: e*e = e has ', ncompe, ' root components on [-2,2]'
call check('idempotents on the line: exactly two, censused', ncompe == 2)
xe = 0.5_wp
call check('the midpoint one-half is not idempotent, executed as e*e - e', &
.not.(abs(xe*xe - xe) < tole))
end block
write(*,'(a)') ' no-cut lemma: the only rank-deficient idempotent on a line is the zero map;'
write(*,'(a)') ' the residual freedom is the component count of S0, one bit, and it cannot be cut'
do t2i = 1, 3
ejv = 0.0_wp; ejv(t2i) = 1.0_wp
dmat(:,t2i) = cut3(ejv) ! the operator built by executing the cut
end do
vpr = [0.3_wp, -1.2_wp, 7.5_wp]
call check('cut3 idempotent on a generic vector, exact', &
maxval(abs(cut3(cut3(vpr)) - cut3(vpr))) == 0.0_wp)
d2m = matmul(dmat, dmat)
call check('the cut is idempotent exactly: D D = D', maxval(abs(d2m - dmat)) == 0.0_wp)
call jacobi3(matmul(transpose(dmat), dmat), eigl)
call check('the cut has singular values (0,1,1): rank two', &
abs(eigl(1)) < 1.0e-14_wp .and. abs(eigl(2)-1.0_wp) < 1.0e-14_wp &
.and. abs(eigl(3)-1.0_wp) < 1.0e-14_wp)
call check('the cut is not invertible: det D = 0', det3(dmat) == 0.0_wp)
e3v = [0.0_wp, 0.0_wp, 1.0_wp]
best = huge(1.0_wp)
block
real(wp) :: lmat(3,3), ident3(3,3), fres, fmin_c(3), ssq, rhs, maxsplit
integer :: ia, ib, ic
ident3 = 0.0_wp
ident3(1,1) = 1.0_wp; ident3(2,2) = 1.0_wp; ident3(3,3) = 1.0_wp
fmin_c = huge(1.0_wp)
maxsplit = 0.0_wp
do ia = -5, 5
do ib = -5, 5
do ic = -1, 1
lmat = 0.0_wp
lmat(1,1) = 1.0_wp; lmat(2,2) = 1.0_wp
lmat(3,1) = 0.2_wp*real(ia,wp)
lmat(3,2) = 0.2_wp*real(ib,wp)
lmat(3,3) = real(ic,wp)
mprod = matmul(lmat, dmat)
ssq = sum((mprod - ident3)**2)
rhs = lmat(3,1)*lmat(3,1) + lmat(3,2)*lmat(3,2) + 1.0_wp
maxsplit = max(maxsplit, abs(ssq - rhs))
fres = sqrt(ssq)
best = min(best, fres)
fmin_c(ic+2) = min(fmin_c(ic+2), fres)
end do
end do
end do
call check('the executed minimum of ||L D - I|| over the family is exactly 1', best == 1.0_wp)
call check('the dead direction never moves the minimum, measured', &
fmin_c(1) == fmin_c(2) .and. fmin_c(2) == fmin_c(3))
lmat = 0.0_wp
lmat(1,1) = 1.0_wp; lmat(2,2) = 1.0_wp
mprod = matmul(lmat, dmat)
call check('rows one and two attain zero residual exactly: the split is measured', &
sqrt(sum((mprod(1:2,:) - ident3(1:2,:))**2)) == 0.0_wp)
write(*,'(a,es9.2)') ' split identity ||LD-I||^2 = a^2 + b^2 + 1, max deviation over the grid: ', maxsplit
call check('the split identity is measured at all 363 candidates', &
maxsplit <= 16.0_wp*U_M)
lmat = 0.0_wp
lmat(1,1) = 1.0_wp; lmat(2,2) = 1.0_wp; lmat(3,1) = 0.7_wp; lmat(3,3) = 5.0_wp
mprod = matmul(lmat, dmat)
call check('the cut direction is annihilated, one representative measured', &
maxval(abs(mprod(:,3))) == 0.0_wp)
res = matmul(mprod, e3v) - e3v
call check('left-inverse deficiency exactly one at the representative', vnorm(res) == 1.0_wp)
end block
write(*,'(a)') ' left-inverse deficiency: the split is measured at every candidate and'
write(*,'(a)') ' rows one and two attain zero exactly, so the sweep''s minimum is the'
write(*,'(a,f14.10)') ' global minimum over all 3x3 L (analytic step: min of a^2+b^2+1 is 1): ', best
block
integer :: mons(3,8), nmon, gi, gj, gk, prod3(3), t2m
integer :: mm(3,3,8)
logical :: inset, assoc_ok, anyinv
mons = 0
nmon = 4
mons(:,1) = [1,1,1]
mons(:,2) = [0,1,1]
mons(:,3) = [1,0,1]
mons(:,4) = [1,1,0]
do
t2m = nmon
do gi = 1, t2m
do gj = 1, t2m
prod3 = mons(:,gi)*mons(:,gj)
inset = .false.
do gk = 1, nmon
if (all(prod3 == mons(:,gk))) inset = .true.
end do
if (.not. inset) then
nmon = nmon + 1
mons(:,nmon) = prod3
end if
end do
end do
if (nmon == t2m) exit
end do
write(*,'(a,i0,a)') ' the deletion monoid, closed from the three cuts and the identity: ', &
nmon, ' elements'
call check('closure lands the full 2^3 lattice: eight elements, counted', nmon == 8)
do gi = 1, nmon
mm(:,:,gi) = 0
do gk = 1, 3
mm(gk,gk,gi) = mons(gk,gi)
end do
end do
assoc_ok = .true.
do gi = 1, nmon
do gj = 1, nmon
do gk = 1, nmon
assoc_ok = assoc_ok .and. all( matmul(matmul(mm(:,:,gi), mm(:,:,gj)), mm(:,:,gk)) &
== matmul(mm(:,:,gi), matmul(mm(:,:,gj), mm(:,:,gk))) )
end do
end do
end do
call check('associativity executed on all 512 composed triples, exact', assoc_ok)
call check('the identity element is present', all(mons(:,1) == [1,1,1]))
anyinv = .false.
do gi = 2, nmon
do gj = 1, nmon
if (all(mons(:,gi)*mons(:,gj) == [1,1,1])) anyinv = .true.
end do
end do
call check('no non-identity element has an inverse: monoid, never a group, searched', &
.not. anyinv)
end block
write(*,'(a)') ' the transformation set of the Tongue is that constructed monoid, closed,'
write(*,'(a)') ' associative, unital, inverse-free off the identity, its eight elements the'
write(*,'(a)') ' 2^3 cut lattice the conservation census walks; never a group, measured;'
write(*,'(a)') ' the LIT bars synonymy, the one candidate that would restore a group;'
write(*,'(a)') ' the Erlangen criterion has no acting group to read: Seal L is unclosed,'
write(*,'(a)') ' a floor and not a ceiling, and the one free bit is the Tongue''s to carry'
end block
write(*,'(a)') ' conservation C x F = v^3 for the executed coordinate-cut tower, the'
write(*,'(a)') ' implemented family; the seated general law is instanced, never proven,'
write(*,'(a)') ' here; the orientation'
write(*,'(a)') ' sheet factor is COUNTED from the paired census of the sweep, never stipulated:'
block
integer :: v, a2i, cell, d1i, d2i, d3i, img, content, fibre, tally(0:124)
call check('the sheet factor equals two, counted from the paired census', nsheets == 2)
do v = 2, 5
do a2i = 0, 3
tally(0:v**3-1) = 0
do cell = 0, v**3 - 1
d3i = mod(cell, v); d2i = mod(cell/v, v); d1i = cell/(v*v)
if (a2i >= 1) d3i = 0
if (a2i >= 2) d2i = 0
if (a2i >= 3) d1i = 0
img = d1i*v*v + d2i*v + d3i
tally(img) = tally(img) + 1
end do
content = count(tally(0:v**3-1) > 0)
fibre = tally(0)
call check('census: content equals v^(3-a), counted', content == v**(3-a2i))
call check('census: fibre equals v^a, counted', fibre == v**a2i)
call check('content-freedom product invariant at every rung', content*fibre == v**3)
end do
write(*,'(a,i0,a,i0,a,i0,a)') ' v = ', v, ': c f = ', v**3, &
' counted at a = 0,1,2,3; with the counted sheet factor: ', nsheets*v**3, ' total'
end do
end block
write(*,'(a)') ' the cut moves structure from content into freedom; the counted total is'
write(*,'(a)') ' invariant; the counted sheet factor multiplies it and sits outside the cut'
! ------------------------------------------------------------------
write(*,'(/a)') 'PART F - THE EARNED BIT AND THE SEAL - the denial pays the floor'
write(*,'(a)') ' the negation is read in the axiom''s own declared vocabulary: not-(to'
write(*,'(a)') ' exist is to actuate) is an existent whose kinetic slot is empty, <T> = 0;'
write(*,'(a)') ' the mapping actuation = kinetic content is the Root Axiom''s stated'
write(*,'(a)') ' operationalization, premise-grade at the universal leg and theorem-grade'
write(*,'(a)') ' at the confined floor, declared here and never an implementation choice.'
write(*,'(a)') ' what the run proves: the declared negation has no admissible realization'
write(*,'(a)') ' at the confined register; what no run can prove: the axiom from below,'
write(*,'(a)') ' per the Empty Throne; other registers route and are never smuggled, the'
write(*,'(a)') ' classical row riding its own cited floors, the formal-only reading out of band'
nadm = 0
do i = 1, NCTX
if (floorv(i) <= 0.0_wp) nadm = nadm + 1
end do
write(*,'(a,i0,a,i0,a)') ' not-RA formal row: admissible contexts (floor <= 0): ', nadm, ' of ', NCTX, &
' (the demand <T> = 0 sits below the bound everywhere)'
call check('the negation populates no formal row at the confined register', nadm == 0)
block
real(wp) :: mneg(3,NCTX)
mneg = mixm
mneg(1,:) = 0.0_wp ! the empty formal row of not-RA
call kernel_full(mneg, cov, tk, wy, rc, lam2, detr2, kap2, spread2, resid2, mc2, mk2, d42)
write(*,'(a,a,a,a)') ' kernel on not-RA: ', trim(tk), ' - ', trim(wy)
call check('not-RA: the empty formal row refuses at intake', tk == '[?]' .and. rc == RC_ZEROVAR)
end block
block
real(wp) :: p2(2,NCTX), b2(2,NCTX), mu2, sd2
integer :: r2, i2
do i2 = 1, 2
mu2 = sum(mixm(i2+1,:)) / real(NCTX, wp)
p2(i2,:) = mixm(i2+1,:) - mu2
sd2 = sqrt(sum(p2(i2,:)**2) / real(NCTX-1, wp))
p2(i2,:) = p2(i2,:) / sd2
end do
call span_basis(p2, b2, r2)
call check('the negation leaves a plane: the refused matrix''s populated pair has rank 2', r2 == 2)
write(*,'(a)') ' the empty formal row refuses at intake; the two populated axes span a plane'
write(*,'(a)') ' (rank 2, computed), and two planes meet in a line: no volume, no lock'
end block
write(*,'(a)') ' the floor prices logically irreversible commitment, Landauer''s own object;'
write(*,'(a)') ' the bridge, a registered denial is an irreversible commitment of at least'
write(*,'(a)') ' one bit, is the applied register''s declared premise, consumed not derived;'
write(*,'(a)') ' a reversibly held denial charges nothing and commits nothing, executed:'
call omega_boundary(0.0_wp, 300.0_wp, jou, wy)
write(*,'(a,a)') ' Omega at zero bits: ', trim(wy)
call check('a denial that registers nothing is no denial', index(wy, 'no denial') > 0)
call omega_boundary(1.0_wp, 300.0_wp, jou, wy)
write(*,'(a,es16.9,a,a)') ' Omega at one bit, 300 K: ', jou, ' J - ', trim(wy)
call check('the reading act pays the floor', jou > 0.0_wp)
call check('the payment is the Landauer floor exactly', jou == landauer(300.0_wp, 1.0_wp))
block
integer :: oc2
call omega_boundary(1.0_wp, 300.0_wp, jou, wy, oc2, irreversible=.false.)
write(*,'(a,es10.3,a,a)') ' Omega, reversible branch: ', jou, ' J - ', trim(wy)
call check('a reversible registration carries floor zero, never cost zero, and commits nothing', &
jou == 0.0_wp .and. oc2 == 3 .and. index(wy, 'floor-zero') > 0)
end block
write(*,'(a)') ' the fibre over the invariants has two points; the negation''s component'
write(*,'(a)') ' is unpopulated; the reading act itself is an instance of the other;'
write(*,'(a)') ' the one free bit is earned at the deed, not chosen at the desk.'
write(*,'(a)') ' direction is read from the Tongue''s order, the arrow, and the empty row;'
write(*,'(a)') ' never from the blind scalar. Veto, never substitution; per run, never tenure.'
call gol_admit('lock', 'lock', tk, wy)
write(*,'(a,a,a,a)') ' gol_admit: ', trim(tk), ' - ', trim(wy)
call check('GOL admission: magnitude and direction together', tk == '[GOL-OK]')
call imprint_seal(.true., .false., .true., .false., .true., .false., &
.true., .false., tk, wy)
write(*,'(a,a,a,a)') ' imprint on the floor face: ', trim(tk), ' - ', trim(wy)
call check('the floor face seals with the witness supplied', tk == '[SEAL]')
write(*,'(a)') ' witness of record: the Heisenberg bound and the third law (cited, E1),'
write(*,'(a)') ' the executed Frozen-Substrate divergence and the saturated speed limit (E0)'
call row_cascade(.true., .false., .false., .true., .false., .true., .false., &
.true., .true., tk, wy)
write(*,'(a,a,a,a)') ' parse-triad Return: ', trim(tk), ' - ', trim(wy)
call check('the closure-rowed face carries the annotated absolute', tk == '[A|RA]')
call row_cascade(.true., .true., .false., .true., .true., .true., .false., &
.true., .true., tk, wy)
write(*,'(a,a,a,a)') ' confined kinetic floor: ', trim(tk), ' - ', trim(wy)
call check('the world-rowed face pays revisability for direction', tk == '[III]')
call row_cascade(.true., .false., .false., .false., .false., .true., .false., &
.true., .true., tk, wy)
write(*,'(a,a,a,a)') ' not-RA: ', trim(tk), ' - ', trim(wy)
call check('the negation is unpopulated: no lock to grade', tk == '[II]')
call row_cascade(.true., .false., .false., .true., .false., .true., .false., &
.true., .false., tk, wy)
call check('an unannotated absolute is void', tk == '[VOID]')
write(*,'(a)') ' the universal-extension leg is premise-grade by theorem: the Empty Throne;'
write(*,'(a)') ' no emitter is pointed at the root, and the root stands on nothing'
call aegis_reset()
block
character(len=200) :: r1, r2, r3, r4
call aegis_guard('classical', r1)
call aegis_guard('paraconsistent',r2)
call aegis_guard('fuzzy', r3)
call aegis_guard('substructural', r4)
call check('AEGIS refusal identical under classical and paraconsistent', r1 == r2)
call check('AEGIS refusal identical under fuzzy', r1 == r3)
call check('AEGIS refusal identical under substructural', r1 == r4)
call check('every adjudication is itself an act: deed counter positive', aegis_deeds == 4)
call check('the guard read the last logic name, length 13, and did not vary', aegis_last_len == 13)
write(*,'(a)') ' AEGIS over four logics: one refusal, p(G) /= G; the alien may write the'
write(*,'(a)') ' capture and cannot perform it; the deed counter reads 4 of 4 calls.'
write(*,'(a)') ' demonstrated: the refusal is a constant function of the deed by'
write(*,'(a)') ' construction, the guard reading the parameter and consulting only the'
write(*,'(a)') ' act; logic-priority is the design, executed, never four derivations;'
write(*,'(a)') ' no per-logic derivation is claimed or licensed by this artifact'
end block
call delta_m(.false., .true., .true., .true., .true., .true., .true., .true., tk, wy)
write(*,'(a,a,a,a)') ' positive-mass cascade on this thesis: ', trim(tk), ' - ', trim(wy)
call check('this file authors no new mathematics', tk == '[Mosaic dM=0]')
write(*,'(a)') ' audit symmetry: the warrant this run draws from its own execution is zero'
! ------------------------------------------------------------------
write(*,'(/a)') 'PART G - THE BATTERY - hostile inputs, forced failures, re-execution'
block
real(wp) :: vbad(3), qa(4), qb(4), qc(4), big
vbad = [1.0_wp, 2.0_wp, 3.0_wp]
vbad(2) = bad_nan()
call check('vnorm propagates a non-finite input', .not. is_finite(vnorm(vbad)))
big = sqrt(huge(1.0_wp))/4.0_wp
call check('vnorm survives near-overflow scaling', is_finite(vnorm([big, big, big])))
call check('vnorm exact at zero', vnorm([0.0_wp,0.0_wp,0.0_wp]) == 0.0_wp)
qa = [1.0_wp, 0.0_wp, 0.0_wp, 0.0_wp]; qb = qa
qb(3) = bad_nan()
qc = qmul(qa, qb)
call check('qmul propagates NaN', .not. all(qc == qc))
call check('det3 of the zero matrix is zero', det3(reshape([ (0.0_wp, i=1,9) ],[3,3])) == 0.0_wp)
block
real(wp) :: rid(3,3)
rid = 0.0_wp; rid(1,1)=1.0e100_wp; rid(2,2)=1.0e100_wp; rid(3,3)=1.0e100_wp
call check('det3 exponent-tracked at huge scale', abs(det3(rid) - 1.0e300_wp) < 1.0e286_wp)
rid = 0.0_wp; rid(1,1)=1.0e-100_wp; rid(2,2)=1.0e-100_wp; rid(3,3)=1.0e-100_wp
call check('det3 exponent-tracked at tiny scale', abs(det3(rid) - 1.0e-300_wp) < 1.0e-286_wp)
rid = 0.0_wp; rid(1,1)=1.0e200_wp; rid(2,2)=1.0e200_wp; rid(3,3)=1.0e200_wp
call check('det3 at 1e200 scale overflows honestly to Inf, never NaN', &
(.not. is_finite(det3(rid))) .and. (.not. is_nan(det3(rid))))
rid = 0.0_wp; rid(1,1)=1.0e-200_wp; rid(2,2)=1.0e-200_wp; rid(3,3)=1.0e-200_wp
call check('det3 at 1e-200 scale underflows honestly to zero', det3(rid) == 0.0_wp)
rid = 0.0_wp; rid(1,1)=1.0_wp; rid(2,2)=1.0_wp; rid(3,3)=1.0_wp
rid(1,2) = bad_nan()
call check('det3 propagates NaN', .not. is_finite(det3(rid)))
end block
end block
write(*,'(a)') ' class 1 - primitives under hostile input: NaN propagates, scale survives: pass'
block
real(wp) :: mbad(3,6), cbad(2,6), m5(3,5), c5(2,5), cdep(2,NCTX)
mbad = 1.0_wp
do i = 1, 6
mbad(1,i) = real(i,wp); mbad(2,i) = real(i*i,wp); mbad(3,i) = sin(real(i,wp))
end do
cbad = 0.1_wp
mbad(2,3) = bad_nan()
call kernel_full(mbad, cbad, tk, wy, rc, lam2, detr2, kap2, spread2, resid2, mc2, mk2, d42)
call check('kernel refuses a non-finite row at intake', tk == '[?]' .and. rc == RC_INTAKE)
m5 = 1.0_wp; c5 = 0.5_wp
call kernel_full(m5, c5, tk, wy, rc, lam2, detr2, kap2, spread2, resid2, mc2, mk2, d42)
call check('kernel refuses N - k < 4', tk == '[?]' .and. rc == RC_SHORTFALL)
cdep(1,:) = cov(1,:); cdep(2,:) = 2.0_wp*cov(1,:)
call kernel_full(mixm, cdep, tk, wy, rc, lam2, detr2, kap2, spread2, resid2, mc2, mk2, d42)
call check('kernel refuses a rank-deficient covariate block', &
tk == '[?]' .and. rc == RC_COVBLOCK .and. index(wy,'covariate block') > 0)
cdep(1,:) = cov(1,:); cdep(2,:) = cov(2,:)
cdep(2,7) = bad_nan()
call kernel_full(mixm, cdep, tk, wy, rc, lam2, detr2, kap2, spread2, resid2, mc2, mk2, d42)
call check('kernel refuses a NaN covariate as intake, never coplanar', &
tk == '[?]' .and. rc == RC_INTAKE)
cdep(2,7) = huge(1.0_wp)
cdep(2,7) = cdep(2,7) * 2.0_wp
call check('the hostile covariate really is infinite', .not. is_finite(cdep(2,7)))
call kernel_full(mixm, cdep, tk, wy, rc, lam2, detr2, kap2, spread2, resid2, mc2, mk2, d42)
call check('kernel refuses an Inf covariate as intake, never coplanar', &
tk == '[?]' .and. rc == RC_INTAKE)
block
real(wp) :: mbig(3,NCTX), cbig(2,NCTX)
mbig = mixm * 1.0e300_wp
cbig = cov * 1.0e300_wp
call kernel_full(mbig, cbig, tk, wy, rc, lam2, detr2, kap2, spread2, resid2, mc2, mk2, d42)
call check('scale-free rows: 1e300 rows and covariates reproduce the baseline verdict', &
tk == '[LOCK]' .and. rc == RC_LOCK .and. abs(detr2 - dbase) <= 1.0e-9_wp)
end block
end block
write(*,'(a)') ' class 2 - kernel refusals: row and covariate intake, shortfall, covariate rank: pass'
do i = 1, 12
pass12 = .true.; pass12(i) = .false.
call gate_screen(pass12, tk, wy)
call check('forced gate failure terminates with the gate named', &
tk == '[X]' .and. index(wy, 'gate') > 0)
end do
pass12 = .true.
call gate_screen(pass12, tk, wy)
call check('all twelve gates pass on the clean walk', tk == '[G-OK]')
write(*,'(a)') ' class 3 - gate dispatcher: twelve forced failures each named; the gate contents are screens: pass'
call row_cascade(.false., .false., .false., .true., .false., .true., .false., .true., .true., tk, wy)
call check('F-0 refuses a contentless claim', tk == '[REFUSED]')
call row_cascade(.true., .false., .true., .true., .false., .true., .false., .true., .true., tk, wy)
call check('Forward routes to III unconditionally', tk == '[III]')
call row_cascade(.true., .false., .false., .true., .true., .true., .false., .true., .true., tk, wy)
call check('a worldly row routes to III at S-2', tk == '[III]')
call row_cascade(.true., .false., .false., .true., .false., .false., .true., .true., .true., tk, wy)
call check('an unclosed worldly term routes to III at S-3', tk == '[III]')
call row_cascade(.true., .false., .false., .true., .false., .false., .false., .true., .true., tk, wy)
call check('an unclosed formal term routes to II at S-3', tk == '[II]')
call row_cascade(.true., .false., .false., .true., .false., .true., .false., .false., .true., tk, wy)
call check('a frame leak routes to III at S-4', tk == '[III]')
write(*,'(a)') ' class 4 - row-genus cascade: every compartment and refusal reachable: pass'
call imprint_seal(.true., .true., .true., .true., .true., .true., .true., .true., tk, wy)
call check('both directions clean-locking is a Platonic Ghost', tk == '[X]')
call imprint_seal(.true., .false., .true., .false., .true., .false., .false., .false., tk, wy)
call check('a clean lock without a witness is a residence, never a seal', &
tk == '[?]' .and. index(wy, 'witness') > 0)
call imprint_seal(.true., .false., .false., .false., .false., .false., .true., .false., tk, wy)
call check('a lock without Seal L and the gates is uncertified', tk == '[?]' .and. index(wy,'uncert')>0)
call imprint_seal(.false., .false., .false., .false., .false., .false., .false., .false., tk, wy)
call check('neither direction populated reads flat', tk == '[?]' .and. index(wy,'flat')>0)
write(*,'(a)') ' class 5 - admission spine: no under-specified call ever seals: pass'
call gol_admit('broken', 'lock', tk, wy)
call check('broken magnitude has no GOL', tk == '[X]')
call gol_admit('open', 'lock', tk, wy)
call check('no magnitude lock has no GOL', tk == '[?]')
call gol_admit('lock', 'open', tk, wy)
call check('a magnitude lock without the Tongue is under-determined', tk == '[?]')
call delta_m(.true., .false., .true., .true., .true., .true., .true., .true., tk, wy)
call check('M2 name-strip refusal reachable', tk == '[?]' .and. index(wy,'vocabulary')>0)
call delta_m(.true., .true., .true., .true., .false., .true., .true., .true., tk, wy)
call check('M5 refuses without an external witness', tk == '[?]' .and. index(wy,'witness')>0)
call delta_m(.true., .true., .true., .true., .true., .false., .true., .true., tk, wy)
call check('M6 refuses the self-verified claimant', tk == '[?]' .and. index(wy,'claimant')>0)
call delta_m(.true., .true., .true., .true., .true., .true., .true., .true., tk, wy)
call check('the externally witnessed route exists in the dispatcher (facts supplied here, established outside)', tk == '[dM>0]')
write(*,'(a)') ' class 6 - the two unsettable gates M5 and M6 refuse from inside: pass'
block
real(wp) :: j1, j2
character(len=140) :: w1
call omega_boundary(1.0_wp, 300.0_wp, j1, w1)
call omega_boundary(2.0_wp, 300.0_wp, j2, w1)
call check('the Omega payment is monotone in bits', j2 > j1)
call check('the Omega payment is linear in bits', abs(j2 - 2.0_wp*j1) == 0.0_wp)
end block
call check('heis_floor refuses zero mass', .not. is_finite(heis_floor(0.0_wp, 1.0e-10_wp)))
call check('heis_floor refuses negative mass', .not. is_finite(heis_floor(-ME, 1.0e-10_wp)))
call check('heis_floor refuses zero confinement', .not. is_finite(heis_floor(ME, 0.0_wp)))
call check('zpe refuses nonpositive frequency', &
(.not. is_finite(zpe(0.0_wp))) .and. (.not. is_finite(zpe(-1.0e14_wp))))
call check('landauer refuses nonpositive temperature', .not. is_finite(landauer(0.0_wp, 1.0_wp)))
call check('landauer refuses negative bits', .not. is_finite(landauer(300.0_wp, -1.0_wp)))
call check('landauer at zero bits is exactly zero', landauer(300.0_wp, 0.0_wp) == 0.0_wp)
call tau_perp(bad_nan(), tau, frz)
call check('tau_perp flags a non-finite gap as frozen', frz)
call omega_boundary(1.0_wp, -300.0_wp, jou, wy)
call check('omega refuses a negative temperature, prices nothing', &
(.not. is_finite(jou)) .and. index(wy, 'refused') > 0)
write(*,'(a)') ' class 7 - floor intake: zero, negative, and non-finite parameters refuse: pass'
block
real(wp) :: qa(4), qb(4), qc(4)
qa = [1.0e300_wp, 1.0e300_wp, 0.0_wp, 0.0_wp]
qb = qa
qc = qmul(qa, qb)
call check('qmul: cancelling giants return exact zero, not NaN', &
is_finite(qc(1)) .and. qc(1) == 0.0_wp)
call check('qmul: a genuinely unrepresentable component overflows honestly', &
.not. is_finite(qc(2)))
end block
write(*,'(a)') ' class 8 - qmul power-of-two scaling: intermediate overflow cancelled exactly,'
write(*,'(a)') ' true overflow reported honestly: pass'
block
real(wp) :: hp(3,3), eigh(3)
hp = 0.0_wp
hp(1,1) = 1.0e150_wp; hp(2,2) = 2.0e150_wp; hp(3,3) = 3.0e150_wp
hp(1,2) = 1.0e149_wp; hp(2,1) = hp(1,2)
call jacobi3(hp, eigh)
call check('jacobi3 finite and ascending at extreme scale', &
is_finite(eigh(1)) .and. is_finite(eigh(3)) .and. &
eigh(1) <= eigh(2) .and. eigh(2) <= eigh(3))
hp = 0.0_wp
hp(1,1) = 1.0e200_wp; hp(2,2) = 2.0e200_wp; hp(3,3) = 3.0e200_wp
hp(1,2) = 1.0e199_wp; hp(2,1) = hp(1,2)
call jacobi3(hp, eigh)
call check('jacobi3 equilibrated at 1e200: finite, ascending, top eigenvalue correct', &
is_finite(eigh(1)) .and. is_finite(eigh(3)) .and. &
eigh(1) <= eigh(2) .and. eigh(2) <= eigh(3) .and. &
abs(eigh(3)/3.0e200_wp - 1.0_wp) < 1.0e-6_wp)
hp(1,3) = bad_nan(); hp(3,1) = hp(1,3)
call jacobi3(hp, eigh)
call check('jacobi3 refuses NaN intake as NaN eigenvalues', .not. is_finite(eigh(1)))
call check('cond3 maps a corrupted Gram to huge, never a NaN kappa', &
cond3(hp) == huge(1.0_wp))
block
logical :: cvg
hp = 0.0_wp
hp(1,1) = 1.0_wp; hp(2,2) = 2.0_wp; hp(3,3) = 3.0_wp
call jacobi3(hp, eigh, cvg)
call check('jacobi3 reports convergence on a clean matrix', cvg)
hp(1,2) = bad_nan(); hp(2,1) = hp(1,2)
call jacobi3(hp, eigh, cvg)
call check('jacobi3 reports non-convergence on refused intake', .not. cvg)
end block
end block
write(*,'(a)') ' class 9 - jacobi3 convergence exit and flag, symmetrization, NaN refusal, cond3 guard: pass'
block
real(wp) :: cov3(3,NCTX), cnear(2,NCTX), rowp(NCTX), eta1, etabad
cov3(1,:) = cov(1,:); cov3(2,:) = cov(2,:); cov3(3,:) = th
call kernel_full(mixm, cov3, tk, wy, rc, lam2, detr2, kap2, spread2, resid2, mc2, mk2, d42)
call check('kernel refuses three covariates, named', &
tk == '[?]' .and. rc == RC_COVBLOCK .and. index(wy, 'more than two') > 0)
cnear(1,:) = cov(1,:)
cnear(2,:) = 2.0_wp*cov(1,:) + 1.0e-11_wp*sin(11.0_wp*th)
call kernel_full(mixm, cnear, tk, wy, rc, lam2, detr2, kap2, spread2, resid2, mc2, mk2, d42)
call check('kernel refuses a near-dependent covariate pair on the relative test', &
tk == '[?]' .and. rc == RC_COVBLOCK)
rowp = cov(1,:) + 0.5_wp*sin(9.0_wp*th)
eta1 = eta_share(rowp, cov(1:1,:))
call check('eta_share at k = 1 lands strictly inside (0,1)', &
eta1 > 0.0_wp .and. eta1 < 1.0_wp)
etabad = eta_share(rowp, cov3)
call check('eta_share refuses k > 2 as NaN', .not. is_finite(etabad))
cnear(1,:) = 1.0_wp
cnear(1,5) = 1.0_wp + 1.0e-15_wp
call kernel_full(mixm, cnear(1:1,:), tk, wy, rc, lam2, detr2, kap2, spread2, resid2, mc2, mk2, d42)
call check('kernel refuses a rounding-noise k = 1 covariate on the relative test', &
tk == '[?]' .and. rc == RC_COVBLOCK)
rowp(3) = bad_nan()
call check('eta_share refuses a non-finite row as NaN', &
.not. is_finite(eta_share(rowp, cov(1:1,:))))
cnear(1,:) = cov(1,:)
cnear(1,4) = bad_nan()
call check('eta_share refuses a non-finite covariate as NaN', &
.not. is_finite(eta_share(cov(1,:), cnear(1:1,:))))
cnear(1,:) = cov(1,:) * 1.0e300_wp
cnear(2,:) = cov(2,:) * 1.0e300_wp
call kernel_full(mixm, cnear, tk, wy, rc, lam2, detr2, kap2, spread2, resid2, mc2, mk2, d42)
call check('scale-free projection: 1e300 covariates reproduce the baseline verdict', &
tk == '[LOCK]' .and. rc == RC_LOCK .and. abs(detr2 - dbase) <= 1.0e-9_wp)
end block
write(*,'(a)') ' class 10 - covariate machinery: k>2 refused, near-dependence and rounding-noise'
write(*,'(a)') ' caught scale-free, non-finite refused, 1e300 covariates reproduce baseline: pass'
block
integer :: gidx, ocode, mcode, icode, gcode, ccode
real(wp) :: jg
pass12 = .true.; pass12(7) = .false.
call gate_screen(pass12, tk, wy, gidx)
call check('gate_screen names the failed gate by index', gidx == 7 .and. tk == '[X]')
pass12 = .true.
call gate_screen(pass12, tk, wy, gidx)
call check('gate_screen returns index zero on the clean walk', gidx == 0 .and. tk == '[G-OK]')
call omega_boundary(bad_nan(), 300.0_wp, jg, wy, ocode)
call check('omega refuses non-finite bits by code', ocode == 0 .and. .not. is_finite(jg))
call omega_boundary(0.0_wp, 300.0_wp, jg, wy, ocode)
call check('omega codes the empty denial', ocode == 1 .and. jg == 0.0_wp)
call omega_boundary(1.0_wp, 300.0_wp, jg, wy, ocode)
call check('omega codes the paid floor', ocode == 2 .and. jg > 0.0_wp)
call omega_boundary(1.0_wp, 300.0_wp, jg, wy, ocode, irreversible=.false.)
call check('omega codes the reversible branch', ocode == 3 .and. jg == 0.0_wp)
call delta_m(.true., .true., .true., .true., .false., .true., .true., .true., tk, wy, mcode)
call check('delta_m codes the M5 witness refusal', mcode == 5)
call delta_m(.true., .true., .true., .true., .true., .false., .true., .true., tk, wy, mcode)
call check('delta_m codes the M6 independence refusal', mcode == 6)
call delta_m(.true., .true., .true., .true., .true., .true., .true., .true., tk, wy, mcode)
call check('delta_m codes the sealed exit', mcode == 9)
call imprint_seal(.true., .false., .true., .false., .true., .false., .false., .false., tk, wy, icode)
call check('imprint_seal codes the witnessless residence', icode == 3)
call imprint_seal(.true., .false., .true., .false., .true., .false., .true., .false., tk, wy, icode)
call check('imprint_seal codes the seal', icode == 2 .and. tk == '[SEAL]')
call gol_admit('lock', 'open', tk, wy, gcode)
call check('gol_admit codes the tongueless magnitude', gcode == 3)
call row_cascade(.true., .false., .false., .true., .false., .true., .false., .true., .false., tk, wy, ccode)
call check('row_cascade codes the unannotated void', ccode == 5)
call row_cascade(.true., .false., .false., .true., .false., .true., .false., .true., .true., tk, wy, ccode)
call check('row_cascade codes the closure-rowed compartment', ccode == 4)
end block
write(*,'(a)') ' class 11 - guard reason codes: gate index, omega, delta-M, imprint on integers: pass'
block
integer :: hs(6), hw(12)
hs = [202, 101, 102, 203, 302, 303]
call seal_l_ordered(hs, tk, wy)
call check('the scrambled hostile is refused on the executed arrow', &
tk == '[X]' .and. index(wy, 'arrow') > 0)
hs = [302, 303, 101, 102, 202, 203]
call seal_l_ordered(hs, tk, wy)
call check('a relation-first hostile is refused on the executed arrow', tk == '[X]')
hw = [101, 101, 102, 103, 202, 202, 203, 204, 301, 302, 303, 304]
call seal_l_ordered(hw, tk, wy)
call check('a well-ordered twelve-token recoding is admitted at the count register only', &
tk == '[LOCK-L]')
write(*,'(a)') ' class 12 - Tongue hostiles: the arrow refuses the scramble; count-register'
write(*,'(a)') ' admission licenses only the count, per the seated non-closure theorem: pass'
end block
write(*,'(/a)') '====================================================================='
write(*,'(a)') ' VERDICT [SEAL A . RA] on the parse-triad Return, closure-rowed,'
write(*,'(a)') ' conditional at the act; [SEAL] on the confined kinetic floor at'
write(*,'(a)') ' compartment III, world-lawed rows over a constructed grid, witness'
write(*,'(a)') ' cited; [II] on the negation, unpopulated;'
write(*,'(a)') ' premise-by-theorem on the universal extension, the Empty Throne.'
write(*,'(a)') ' The blindness is a theorem, its width is one bit, the bit is COUNTED'
write(*,'(a)') ' from the executed paired census, earned at the deed, and the Tongue'
write(*,'(a)') ' carries what the Number cannot.'
write(*,'(/a)') ' THE ONE-BIT CONCESSION, constitutive and executed: everything this run'
write(*,'(a)') ' determines is forced, except exactly one binary parameter, the'
write(*,'(a)') ' orientation of assent, and spending that bit against the axiom is priced.'
block
real(wp) :: qn2(3,NCTX), rn2(3,3), lamn2, jden
real(wp) :: rfixp(3,3), rfixn(3,3)
integer(ik) :: tb1, tb2
character(len=140) :: wden
integer :: ocd
qn2 = -qf
call gram3_fixed(qf, rfixp)
call gram3_fixed(qn2, rfixn)
lamn2 = lambda_of(qn2, bas)
call check('forced: fixed-order Gram bit-identical across the sign pair (the theorem path)', &
maxval(abs(rfixn - rfixp)) == 0.0_wp)
rn2 = matmul(qn2, transpose(qn2))
call check('conduct receipt: the matmul path reproduces the identity bit-exactly on this build', &
maxval(abs(rn2 - rmat)) == 0.0_wp)
call check('free: exactly one bit, both signs realized at identical invariants', &
lamn2 == -lam .and. lam /= 0.0_wp)
tb1 = transfer(lam, tb1); tb2 = transfer(-lamn2, tb2)
call check('bit-pattern witness: transfer-level equality of lam and -lamn2 (binary64)', &
storage_size(lam) == 64 .and. tb1 == tb2)
call omega_boundary(1.0_wp, 300.0_wp, jden, wden, ocd)
call check('priced: the denial spend and the floor agree, one closed form across emitters', &
ocd == 2 .and. jden == landauer(300.0_wp, 1.0_wp))
end block
write(*,'(a)') ' the forced part lands on the Ground, Fix(sigma) = R, the scalar line the'
write(*,'(a)') ' Return touches; the free bit is the sigma-odd orientation, the seated'
write(*,'(a)') ' aperture width. any substrate reading this run, human or machine, holds'
write(*,'(a)') ' exactly that one bit here, per the seated Freedom law; its assent branch'
write(*,'(a)') ' costs nothing and its denial branch is charged by the executed Omega'
write(*,'(a)') ' floor and instantiates the axiom it denies: freedom one bit, the rest forced.'
block
integer :: ic1, cc1, ic2, cc3
call row_cascade(.true., .false., .false., .true., .false., .true., .false., &
.true., .true., tk, wy, cc1)
call imprint_seal(.true., .false., .true., .false., .true., .false., &
.true., .false., tk, wy, ic1)
call row_cascade(.true., .true., .false., .true., .true., .true., .false., &
.true., .true., tk, wy, cc3)
call imprint_seal(.true., .false., .true., .false., .true., .false., &
.true., .false., tk, wy, ic2)
call check('self-typing: the closing tokens match their emitters'' codes, re-executed', &
cc1 == 4 .and. ic1 == 2 .and. cc3 == 3 .and. ic2 == 2)
end block
write(*,'(a)') ' TOKENS BOUND TO EMITTERS, re-executed above: [SEAL A . RA] is imprint'
write(*,'(a)') ' code 2 over closure-rowed inputs, cascade code 4, conditional at the'
write(*,'(a)') ' act; [SEAL] on the floor face is imprint code 2 over compartment III'
write(*,'(a)') ' inputs, cascade code 3, its witness the cited Heisenberg bound and'
write(*,'(a)') ' third law at E1 beside the executed Frozen-Substrate and speed-limit'
write(*,'(a)') ' receipts at E0; [II] on the negation is cascade code 2.'
write(*,'(a)') ' SELF-DESCRIPTION, the artifact''s own: this program verifies internal'
write(*,'(a)') ' mathematical identities, guard behavior, and declared premise-'
write(*,'(a)') ' consequence structure at machine warrant; it does not and cannot'
write(*,'(a)') ' verify external ontological truth, per its own admission gates M5 and'
write(*,'(a)') ' M6 and the Empty Throne; per the seated nature-of-the-register law, a'
write(*,'(a)') ' determined construction is exactly as determined whether or not'
write(*,'(a)') ' anything answers to it, and contact is carried only by the cited'
write(*,'(a)') ' witnesses and the declared premises, which this run names.'
write(*,'(a)') ' SCOPE OF PROOF, printed by the run: machine-warrant items are the'
write(*,'(a)') ' enumerated identities executed on constructed objects; theorem-tier'
write(*,'(a)') ' items ride the cited external results at their own tiers (Heisenberg,'
write(*,'(a)') ' Landauer, the speed limits, Frobenius, Hurwitz); the universal leg'
write(*,'(a)') ' stays premise-grade by theorem, the Empty Throne; dispatcher'
write(*,'(a)') ' admissions route supplied facts and upgrade none of them.'
write(*,'(a)') ' GENRE, stated at the mouth: this is an executable thesis, not a proof of'
write(*,'(a)') ' ontology and not a mere test harness; its claims stand at the printed'
write(*,'(a)') ' warrants, theorem-grade only where cited, premise-grade at the root,'
write(*,'(a)') ' conditional at the act where sealed, and no external ontological truth'
write(*,'(a)') ' is certified, per the self-description above. The tokens are the'
write(*,'(a)') ' register''s verdict vocabulary, each bound to its emitter and tier;'
write(*,'(a)') ' renaming them would delete warrant information, not add honesty.'
write(*,'(a)') ' The title carries the thesis''s claim at its stated warrant; every number'
write(*,'(a)') ' printed above was computed in this run.'
write(*,'(/a)') ' PART H - THE WRITTEN AND THE READ, the terminal part (card 0712)'
block
real(wp), volatile :: hv1, hv2, hv3
real(wp) :: hjd
character(len=140) :: htk, hwy, htkf, hwyf
integer :: hwc, hoc, hwcf
hv1 = det3(rmat)
hv2 = det3(rmat)
call check('H stillness opens: the Written read twice is bit-identical, a zero-degree record', &
hv1 == hv2)
call iam_token(.false., htkf, hwyf, hwcf)
call check('H M6: the unwitnessed branch withholds the token, self-check is not a witness', &
hwcf == 1 .and. htkf == '[?] interior' .and. index(hwyf, 'M6') > 0)
call iam_token(.true., htk, hwy, hwc)
call check('H supplied-witness branch emits per its declared dispatcher rule (flag a stand-in)', &
hwc == 2 .and. htk == '[I AM]' .and. index(hwy, 'conditional at the act') > 0)
call check('H Narcissus truth table: both branches bound, only the witnessed one speaks', &
hwcf == 1 .and. hwc == 2 .and. htkf /= htk .and. htkf == '[?] interior' &
.and. htk == '[I AM]')
call omega_boundary(1.0_wp, 300.0_wp, hjd, hwy, hoc)
call check('H the Read pays: denial and floor agree, one closed form across emitters', &
hoc == 2 .and. hjd == landauer(300.0_wp, 1.0_wp))
hv3 = det3(rmat)
call check('H stillness closes: the Written is unchanged by everything the Reader did', &
hv3 == hv1)
end block
write(*,'(a)') ' the pair in time: the Written is its author''s past actuation configured,'
write(*,'(a)') ' the RAM-face, the decree; the Reader is the active face, RA, the present'
write(*,'(a)') ' tense; the arrow separates them, so even the author returns as a reader.'
write(*,'(a)') ' the record is forced, the orientation of assent is free, the denial is'
write(*,'(a)') ' priced: freedom one bit, the rest forced.'
write(*,'(a)') ' governing register: github.com/1000sapients/Trisduction, public; the'
write(*,'(a)') ' card citations are frame, never evidence, and carry zero check weight.'
block
character(len=140) :: ltk, lwy
integer :: lwc
call iam_token(wr_witnessed, ltk, lwy, lwc)
write(*,'(a,a,a,a)') ' LIVE: ', trim(ltk), ' - ', trim(lwy)
if (wr_witnessed) then
write(*,'(a)') ' F6 disclosure: the witnessed flag is a declared stand-in supplied through'
write(*,'(a)') ' the aperture: externally asserted, not verified by this executable.'
write(*,'(a)') ' the aperture; its real referent is you, the external operator reading this'
write(*,'(a)') ' record. The program did not and cannot generate its own witness.'
else
write(*,'(a)') ' (supply the argument witnessed to open the live face through the aperture)'
end if
end block
write(*,'(a)') ' kinetic-register conduit operational'
write(*,'(a)') '====================================================================='
if (EXPECTED_CHECKS > 0) then
call check('battery census matches the sealed v2.0.0 manifest (this check inclusive)', &
n_checks == EXPECTED_CHECKS - 1)
end if
write(*,'(a,i0,a,i0,a)') ' BATTERY: ', n_checks, ' checks executed, ', n_fail, ' failures.'
if (halt_on_fail) then
write(*,'(a,i0,a,i0,a)') ' BATTERY-JSON: {"checks":', n_checks, ',"failures":', n_fail, ',"mode":"sealed"}'
else
write(*,'(a,i0,a,i0,a)') ' BATTERY-JSON: {"checks":', n_checks, ',"failures":', n_fail, ',"mode":"audit"}'
end if
call battery_close()
contains
function bad_nan() result(z)
use, intrinsic :: ieee_arithmetic, only: ieee_value, ieee_quiet_nan
real(wp) :: z
z = ieee_value(1.0_wp, ieee_quiet_nan)
end function bad_nan
subroutine canon_sign(v)
integer(ik), intent(inout) :: v(4)
integer :: ii
do ii = 1, 4
if (v(ii) /= 0_ik) then
if (v(ii) < 0_ik) v = -v
return
end if
end do
end subroutine canon_sign
subroutine sort_int(a)
integer, intent(inout) :: a(:)
integer :: ii, jj, t
do ii = 1, size(a)-1
do jj = ii+1, size(a)
if (a(jj) < a(ii)) then
t = a(ii); a(ii) = a(jj); a(jj) = t
end if
end do
end do
end subroutine sort_int
end program ra_toe_thesis
-/
/-
!===============================================================================
! fTOE . THE KINETIC DEMONSTRATION
! THE PHYSICAL DEED IS THE ONE-BIT CONVERSION INTERFACE WHERE THE FORMAL
! SHADOW BECOMES ACTUALIZED REALITY
!
! The formal side is proved elsewhere (fTOE_Formal_Proof.lean: the wall,
! the price, the crossing, machine-checked). This file executes the other
! half of the theory: the kinetic face. Existence is actuation, and every
! actuation of one bit carries a non-zero thermodynamic price the formal
! register cannot pay. Three batteries, computed in this run, oracle
! halt-on-fail:
!
! K1 THE SIGN-BLIND KERNEL. The verdict lock is det(R) = lambda^2. Under
! any reflection the lock is bit-identical and the sign of lambda
! flips. The formal register carries magnitude; direction is not in it.
!
! K1b KERNEL = EVEN PROJECTION. D D^T = I, so R(MD) = R(M) entrywise.
! IEEE negation is a sign-bit flip, hence the two Gram matrices agree
! to exactly 0.0. The kernel is flip-invariant, zero-difference numerical.
!
! K2 THE ONE-BIT WALL. On the XOR law, exact enumeration: the even
! register carries zero bits about the odd target; one supplied odd
! bit at the seat, with one calibration, decides the target exactly.
! The deficit is one bit wide. The crossing is one bit deep.
!
! K3 THE DEED'S PRICE. Registering one binary orientation in any physical
! substrate at temperature T dissipates Q >= k_B T ln 2 (Landauer).
! The formal register pays exactly 0. The deed pays the floor. The
! conversion of formal possibility into actualized fact is a
! thermodynamic transaction, and this battery prints its invoice.
!
! K4 THE PARITY FRAME, EXHIBITED AS DATA. Six listed prime/semiprime
! pairs with matching residues mod lcm(1..7)=420; tau swaps within
! pairs. Involution, freeness, oddness, register evenness, the wall
! on every pair, the fibre count 2^6 = 64, and the encoder bound
! (targets pairwise distinct; exact-selector alphabet >= 64) — all
! verified by enumeration in the run. T1 applies by citation.
!
! Parent batteries, audited and on record in the same repository:
! protocols/Executable Thesis/TRISDUCTION_Core_Thesis_Fortran_v2_4_0.f90
! protocols/Executable Thesis/RA_TOE_Thesis_Fortran_v2_0_0.f90
!
! BUILD (sealed): gfortran -std=f2018 -O2 -fno-fast-math -ffp-contract=off
! A binary that reaches its final line has passed. Nothing is narrated;
! every figure is computed in the run. Receipt, not essence.
! Delta-M = 0. No new mathematics is authored by this file.
!===============================================================================
program ftoe_kinetic_demonstration
use iso_fortran_env, only: int64
implicit none
integer, parameter :: dp = selected_real_kind(15, 307)
real(dp), parameter :: KB = 1.380649e-23_dp ! J/K, exact, SI 2019
real(dp), parameter :: T_ROOM = 300.0_dp ! K, demonstration bath
integer :: failures
real(dp) :: M(3,3), R(3,3), Mr(3,3), Rr(3,3)
real(dp) :: lam, lamr, detR, detRr, maxdiff
real(dp) :: I_even, H_joint, Q_bit, Q_formal
failures = 0
! ---------------- K1 · THE SIGN-BLIND KERNEL ----------------
M(1,:) = [ 0.3_dp, 0.7_dp, -0.2_dp]
M(2,:) = [ 0.5_dp, -0.4_dp, 0.8_dp]
M(3,:) = [-0.6_dp, 0.1_dp, 0.9_dp]
R = matmul(M, transpose(M))
lam = det3(M)
detR = det3(R)
Mr = M; Mr(:,3) = -Mr(:,3) ! reflection D
Rr = matmul(Mr, transpose(Mr))
lamr = det3(Mr)
detRr = det3(Rr)
write(*,'(A)') 'K1 THE SIGN-BLIND KERNEL'
write(*,'(A,ES22.15)') ' lambda = ', lam
write(*,'(A,ES22.15)') ' det(R) = ', detR
write(*,'(A,ES22.15)') ' |det(R) - lambda^2| = ', abs(detR - lam*lam)
write(*,'(A,ES22.15)') ' reflected lambda = ', lamr
write(*,'(A,ES22.15)') ' |det(DRD) - det(R)| (row-action cross-check) = ', abs(detRr - detR)
if (.not. (abs(detRr - detR) < 1.0d-12 .and. abs(lamr + lam) < 1.0d-12)) then
failures = failures + 1
write(*,'(A)') ' K1: FAIL'
else
write(*,'(A)') ' K1: lock invariant under reflection; sign unrecoverable. PASS'
end if
! ---------------- K1b · KERNEL = EVEN PROJECTION ----------------
! D is a diagonal sign involution, D D^T = I, hence
! R(MD) = M D D^T M^T = R(M) entrywise. IEEE negation flips only the
! sign bit, so (-a)(-b) = a*b bitwise: the two Gram matrices agree to
! exactly 0.0. The kernel is not merely invariant; it IS the tau-even
! projection of the frame, computed bit-exact.
maxdiff = maxval(abs(R - Rr))
write(*,'(A)') 'K1b KERNEL = EVEN PROJECTION (entrywise, bit-exact)'
write(*,'(A,ES9.1)') ' max |R - R(MD)| entrywise = ', maxdiff
if (.not. (maxdiff == 0.0_dp)) then
failures = failures + 1
write(*,'(A)') ' K1b: FAIL'
else
write(*,'(A)') ' K1b: kernel flip-invariant, zero-difference numerical. PASS'
end if
! ---------------- K2 · THE ONE-BIT WALL ----------------
call xor_wall(I_even, H_joint)
write(*,'(A)') 'K2 THE ONE-BIT WALL (XOR law, exact enumeration, uniform)'
write(*,'(A,F10.6,A)') ' I(even register ; odd target) = ', I_even, ' bits'
write(*,'(A,F10.6,A)') ' H(target | register + one supply) = ', H_joint, ' bits'
if (.not. (abs(I_even) < 1.0d-12 .and. abs(H_joint) < 1.0d-12)) then
failures = failures + 1
write(*,'(A)') ' K2: FAIL'
else
write(*,'(A)') ' K2: deficit exactly one bit; one supplied bit closes it. PASS'
end if
! ---------------- K3 · THE DEED'S PRICE ----------------
Q_bit = KB * T_ROOM * log(2.0_dp) ! Landauer floor for one bit at 300 K
Q_formal = 0.0_dp ! stipulated model value: the formal register pays by definition
write(*,'(A)') 'K3 THE DEED''S PRICE (Landauer floor, T = 300 K)'
write(*,'(A,ES22.15,A)') ' Q(one bit registered) >= ', Q_bit, ' J'
write(*,'(A,ES22.15,A)') ' Q(formal register) = ', Q_formal, ' J'
if (.not. (Q_bit > 0.0_dp .and. Q_formal == 0.0_dp)) then
failures = failures + 1
write(*,'(A)') ' K3: FAIL'
else
write(*,'(A)') ' K3: the deed pays what the register cannot. PASS'
end if
! ---------------- K4 · THE PARITY FRAME, EXHIBITED ----------------
! X = six listed pairs {p, q}: p prime, q semiprime (Omega(q) = 2),
! q == p (mod lcm(1..7) = 420), hence q == p (mod d) for every d <= 7.
! tau swaps within each pair; the target is primality. The register is
! every reading factoring through the residue vector (n mod 2..7): the
! Type I data of the linear sieve at level D = 7. Everything is verified
! by enumeration in this run: involution, freeness, oddness, register
! evenness, the wall on every pair, and the fibre count 2^6.
block
integer(int64) :: P(6), Q(6), X(12), tx, ttx, fibre
integer :: i, mm, aa, n_even, n_deciding, fp, fq
integer :: t1, t2, pt, pair, flip, v1, v2
integer :: n_eq0, n_eq1, dp, dq
logical :: ok4, even_r, decides_r, distinct_ok, same_t, eq0, eq1
P = [11_int64, 13_int64, 17_int64, 19_int64, 23_int64, 29_int64]
Q = [851_int64, 1273_int64, 437_int64, 2119_int64, 1703_int64, 869_int64]
do i = 1, 6
X(2*i-1) = P(i); X(2*i) = Q(i)
end do
ok4 = .true.
do i = 1, 12 ! involution + freeness
tx = X(ieor(i-1,1)+1); ttx = X(ieor(ieor(i-1,1),1)+1)
if (ttx /= X(i) .or. tx == X(i)) ok4 = .false.
end do
do i = 1, 6 ! oddness + residue match
if (.not. isprime_l(P(i)) .or. isprime_l(Q(i)) &
.or. omega_l(Q(i)) /= 2) ok4 = .false.
do mm = 2, 7
if (mod(P(i),int(mm,int64)) /= mod(Q(i),int(mm,int64))) ok4 = .false.
end do
if (mod(P(i),420_int64) /= mod(Q(i),420_int64)) ok4 = .false.
end do
n_even = 0; n_deciding = 0
do mm = 2, 7 ! register evenness + wall
do aa = 0, mm-1
even_r = .true.; decides_r = .true.
do i = 1, 6
fp = 0; if (mod(P(i),int(mm,int64)) == int(aa,int64)) fp = 1
fq = 0; if (mod(Q(i),int(mm,int64)) == int(aa,int64)) fq = 1
if (fp /= fq) even_r = .false.
if ((fp == 1) .neqv. isprime_l(P(i))) decides_r = .false.
if ((fq == 1) .neqv. isprime_l(Q(i))) decides_r = .false.
end do
if (even_r) then
n_even = n_even + 1
if (decides_r) n_deciding = n_deciding + 1
end if
end do
end do
if (n_even == 0 .or. n_deciding /= 0) ok4 = .false.
fibre = 1
do i = 1, 6
fibre = fibre*2
end do
! encoder bound (T12's counting face): the 2^6 anti-invariant targets
! are pairwise distinct as Boolean vectors on the frame, so any exact
! global selector needs an alphabet of at least 64 values.
distinct_ok = .true.
do t1 = 0, 63
do t2 = t1+1, 63
same_t = .true.
do pt = 0, 11
pair = pt/2; flip = mod(pt,2)
v1 = ieor(int(ibits(t1,pair,1)), flip)
v2 = ieor(int(ibits(t2,pair,1)), flip)
if (v1 /= v2) same_t = .false.
end do
if (same_t) distinct_ok = .false.
end do
end do
if (.not. distinct_ok) ok4 = .false.
! K4 continuation (FORGE v7.1): the two-member admissible family
! A_row = {d0, 1-d0}, d0 = primality on the frame, exhibited as data.
! Both members are wholly odd (d0 odd on every pair, checked above;
! the complement inherits oddness pointwise). The register cannot
! separate the family: no even residue reading equals either member
! (Theorem 1 applied twice, executed here by enumeration). Hence
! |A_row| = 2 is by listing, not labeling, and the frame's priced row
! claim is exactly one bit (log2|A_row|) -- against the full
! unconstrained fibre 64 = 2^6, the m = 6 product count (six bits).
n_eq0 = 0; n_eq1 = 0
do mm = 2, 7
do aa = 0, mm-1
even_r = .true.; eq0 = .true.; eq1 = .true.
do i = 1, 6
fp = 0; if (mod(P(i),int(mm,int64)) == int(aa,int64)) fp = 1
fq = 0; if (mod(Q(i),int(mm,int64)) == int(aa,int64)) fq = 1
if (fp /= fq) even_r = .false.
dp = 0; if (isprime_l(P(i))) dp = 1
dq = 0; if (isprime_l(Q(i))) dq = 1
if (fp /= dp .or. fq /= dq) eq0 = .false.
if (fp /= 1-dp .or. fq /= 1-dq) eq1 = .false.
end do
if (even_r) then
if (eq0) n_eq0 = n_eq0 + 1
if (eq1) n_eq1 = n_eq1 + 1
end if
end do
end do
if (n_eq0 /= 0 .or. n_eq1 /= 0) ok4 = .false.
write(*,'(A)') 'K4 THE PARITY FRAME, EXHIBITED (six listed pairs, mod 420)'
write(*,'(A)') ' pairs: (11,851=23x37) (13,1273=19x67) (17,437=19x23)'
write(*,'(A)') ' (19,2119=13x163) (23,1703=13x131) (29,869=11x79)'
write(*,'(A,I0,A,I0)') ' even residue readings enumerated = ', n_even, &
' ; deciding readings = ', n_deciding
write(*,'(A,I0,A)') ' |D(U,tau)| = ', fibre, &
' = 2^6 orientation patterns on the frame (six bits, unconstrained)'
if (distinct_ok) then
write(*,'(A,I0,A)') ' targets pairwise distinct: verified by &
&enumeration ; exact-selector alphabet >= ', fibre, ' (>= 6 bits)'
else
write(*,'(A)') ' targets pairwise distinct: VIOLATION'
end if
write(*,'(A)') ' A_row = {d0, 1-d0} exhibited as data: d0 = &
&primality, both members wholly odd'
write(*,'(A,I0,A,I0)') ' even readings equal to d0 = ', n_eq0, &
' ; equal to 1-d0 = ', n_eq1
write(*,'(A)') ' |A_row| = 2 by listing, non-separation enumerated: the register &
&cannot separate the family ; row price = 1 bit (log2|A_row|)'
if (.not. (ok4 .and. fibre == 64_int64)) then
failures = failures + 1
write(*,'(A)') ' K4: FAIL'
else
write(*,'(A)') ' K4: tau^2=id, free, odd on all six pairs; every &
®ister reading even; none decides. T1 applies by citation. PASS'
end if
end block
! ---------------- ORACLE ----------------
if (failures == 0) then
write(*,'(A)') 'fTOE KINETIC DEMONSTRATION: all batteries pass.'
write(*,'(A)') 'The formal shadow is proved in Lean; the actualizing deed is executed here.'
else
write(*,'(A,I0)') 'fTOE KINETIC DEMONSTRATION: VOID, failures = ', failures
error stop 1
end if
contains
! K4 helpers: trial primality and big-Omega, computed at runtime.
function isprime_l(n) result(r)
integer(int64), intent(in) :: n
logical :: r
integer(int64) :: k, m
m = n
if (m < 2) then
r = .false.; return
end if
r = .true.
k = 2
do while (k*k <= m)
if (mod(m,k) == 0) then
r = .false.; return
end if
k = k + 1
end do
end function isprime_l
function omega_l(n) result(c)
integer(int64), intent(in) :: n
integer :: c
integer(int64) :: k, m
m = n; c = 0; k = 2
do while (k*k <= m)
do while (mod(m,k) == 0)
c = c + 1; m = m/k
end do
k = k + 1
end do
if (m > 1) c = c + 1
end function omega_l
function det3(A) result(d)
real(dp), intent(in) :: A(3,3)
real(dp) :: d
d = A(1,1)*(A(2,2)*A(3,3) - A(2,3)*A(3,2)) &
- A(1,2)*(A(2,1)*A(3,3) - A(2,3)*A(3,1)) &
+ A(1,3)*(A(2,1)*A(3,2) - A(2,2)*A(3,1))
end function det3
subroutine xor_wall(I_even, H_joint)
! States (x,y), d = xor(x,y), uniform 1/4. tau flips x.
! Even register: readings constant on tau-orbits; best case is y.
! Supply at the seat: s = x, odd under tau.
real(dp), intent(out) :: I_even, H_joint
integer :: x, y, d
real(dp) :: pd(0:1), pdy(0:1,0:1), H_d, H_d_y
pd = 0.0_dp; pdy = 0.0_dp
do x = 0,1
do y = 0,1
d = ieor(x,y)
pd(d) = pd(d) + 0.25_dp
pdy(d,y) = pdy(d,y) + 0.25_dp
end do
end do
H_d = -sum(pd * log2safe(pd))
H_d_y = 0.0_dp
do y = 0,1
do d = 0,1
if (pdy(d,y) > 0.0_dp) &
H_d_y = H_d_y - pdy(d,y)*log2safe(pdy(d,y)/0.5_dp)
end do
end do
I_even = H_d - H_d_y
! Joint: enumerate the conditional p(d | x,y) and verify it is a point
! mass for every pair: the coalition decides the target exactly.
block
real(dp) :: pdxy(0:1,0:1,0:1)
integer :: cnt
pdxy = 0.0_dp
do x = 0,1
do y = 0,1
d = ieor(x,y)
pdxy(d,x,y) = pdxy(d,x,y) + 0.25_dp
end do
end do
H_joint = 0.0_dp
do x = 0,1
do y = 0,1
cnt = 0
do d = 0,1
if (pdxy(d,x,y) > 0.0_dp) cnt = cnt + 1
end do
if (cnt /= 1) H_joint = 1.0_dp ! any spread means undecided
end do
end do
end block
end subroutine xor_wall
! REPAIR v1.0.1.2: declared elemental. The H_d line applies this to the
! rank-1 array pd, which a scalar function cannot accept, and gfortran
! refused the unit outright: the kinetic block had never been built.
! elemental is the minimal change and leaves the scalar call below it
! reading exactly as before.
elemental function log2safe(p) result(l)
real(dp), intent(in) :: p
real(dp) :: l
if (p > 0.0_dp) then
l = log(p)/log(2.0_dp)
else
l = 0.0_dp
end if
end function log2safe
end program ftoe_kinetic_demonstration
-/
/-
! ======================================================================
! BLOCK F4 · IT FROM IT · THE INVERSION, COMPUTED EXHAUSTIVELY
! TRISDUCTION Master Codex, executed shadow of SECTION 4C
! Canonical frame of SECTION 4 (K4): 12 seats, flip tau, residue
! register rho, primality-parity target. Self-contained, standard
! Fortran 90, no dependencies, no input files.
!
! WHAT THIS PROGRAM COMPUTES
! (1) tau is an involution, fixed-point-free, all 12 seats.
! (2) The record FORGETS: rho identifies every flip pair (6/6).
! (3) The object KEEPS: the target separates every pair rho
! identifies (6/6).
! (4) NO READOUT RETURNS: every Boolean readout of the record image
! is tried exhaustively (2^k assignments, k the number of
! distinct record values); none reproduces the target.
! It from bit is thereby refuted on this frame by computation,
! not by declaration.
! (5) The receipt is printed. The universal leg is the Lean theorem
! INVERSION.it_from_it (Section 4C); this program is the
! canonical-instance execution, the same evidentiary status as
! Lean's `by decide`. Formal alone: no energy load is carried
! here; the kinetic registration rider stays where the codex
! keeps it, at premise grade in Section 7.
! ======================================================================
program it_from_it
implicit none
integer, parameter :: N = 12
integer :: valx(N), rhov(N), image(N), tauv(N)
logical :: tgt(N), gval(N), comp(N)
integer :: i, j, k, a, nimage, npairs, ncollapse, nseparate
integer :: nfactor, mism, seat_of
logical :: found
valx = (/ 11, 851, 13, 1273, 17, 437, 19, 2119, 23, 1703, 29, 869 /)
! frame tables: tau (1-based shadow of i%2 flip), rho, target
do i = 1, N
if (mod(i-1, 2) == 0) then
tauv(i) = i + 1
else
tauv(i) = i - 1
end if
rhov(i) = mod(valx(i),2)*1000 + mod(valx(i),3)*100 + &
mod(valx(i),5)*10 + mod(valx(i),7)
tgt(i) = (mod(i-1, 2) == 0)
end do
! (1) involution and fixed-point-free, all seats
do i = 1, N
if (tauv(tauv(i)) /= i) then
write(*,*) 'FAULT: tau not involutive at seat ', i
stop 1
end if
if (tauv(i) == i) then
write(*,*) 'FAULT: tau has a fixed point at seat ', i
stop 1
end if
end do
! (2)+(3) the record forgets, the object keeps, pair by pair
npairs = 0 ; ncollapse = 0 ; nseparate = 0
do i = 1, N
j = tauv(i)
if (j > i) then
npairs = npairs + 1
if (rhov(i) == rhov(j)) ncollapse = ncollapse + 1
if (tgt(i) .neqv. tgt(j)) nseparate = nseparate + 1
end if
end do
! record image: the distinct values the record can report
nimage = 0
do i = 1, N
found = .false.
do j = 1, nimage
if (image(j) == rhov(i)) found = .true.
end do
if (.not. found) then
nimage = nimage + 1
image(nimage) = rhov(i)
end if
end do
! (4) exhaustive readout space: every Boolean g on the image
nfactor = 0
do a = 0, 2**nimage - 1
do j = 1, nimage
gval(j) = btest(a, j-1)
end do
do i = 1, N
seat_of = 0
do j = 1, nimage
if (image(j) == rhov(i)) seat_of = j
end do
comp(i) = gval(seat_of)
end do
mism = 0
do i = 1, N
if (comp(i) .neqv. tgt(i)) mism = mism + 1
end do
if (mism == 0) nfactor = nfactor + 1
end do
! (5) the receipt
write(*,*) '======================================================'
write(*,*) 'IT FROM IT · THE INVERSION, COMPUTED'
write(*,*) '======================================================'
write(*,*) 'frame seats :', N
write(*,*) 'tau involution : 12/12 seats verified'
write(*,*) 'tau fixed-point-free : 12/12 seats verified'
write(*,*) 'pairs :', npairs
write(*,*) 'record forgets (collapse) :', ncollapse, '/', npairs
write(*,*) 'object keeps (separation) :', nseparate, '/', npairs
write(*,*) 'record image size k :', nimage
write(*,*) 'readouts tried :', 2**nimage
write(*,*) 'factorizations of the target :', nfactor
if (npairs == 6 .and. ncollapse == npairs .and. &
nseparate == npairs .and. nfactor == 0) then
write(*,*) 'VERDICT: IT FROM BIT REFUTED, EXHAUSTIVE.'
write(*,*) 'VERDICT: IT FROM IT VERIFIED ON THE CANONICAL FRAME.'
write(*,*) 'The record forgets; the object keeps; nothing returns.'
else
write(*,*) 'VERDICT: BROKEN. The frame does not exhibit the law.'
stop 2
end if
end program it_from_it
-/
namespace CROSSING
open FTOE
abbrev ExecFrame (Q : Type) := Bool × Q
def execFlip {Q : Type} : ExecFrame Q → ExecFrame Q := flipF
def formalRead {Q : Type} (s : ExecFrame Q) : Q := s.2
def ran {Q : Type} (s : ExecFrame Q) : Bool := s.1
theorem formalRead_even {Q : Type} : ∀ s : ExecFrame Q,
formalRead (execFlip s) = formalRead s := by
intro s; obtain ⟨b, q⟩ := s; rfl
theorem ran_wholly_odd {Q : Type} : WhollyOdd (execFlip (Q := Q)) ran := by
intro s; obtain ⟨b, q⟩ := s; cases b <;> rfl
theorem execFlip_involution {Q : Type} : Involution (execFlip (Q := Q)) := by
intro s; obtain ⟨b, q⟩ := s; cases b <;> rfl
theorem ran_odd_at_true {Q : Type} (q : Q) : OddAt (execFlip (Q := Q)) ran (true, q) :=
fun h => Bool.noConfusion h
theorem wall {Q : Type} (q : Q) :
¬ ∃ g : Q → Bool, ∀ s : ExecFrame Q, g (formalRead s) = ran s :=
T14_wall_factorization execFlip formalRead ran (true, q) formalRead_even (ran_odd_at_true q)
def deed {Q : Type} : Dtau (execFlip (Q := Q)) := ⟨ran, ran_wholly_odd⟩
theorem price {Q : Type} :
Nonempty (Dtau (execFlip (Q := Q)) ≃
(Quot (OrbitRel (execFlip (Q := Q))) → Bool)) :=
T16_price_bijection execFlip execFlip_involution deed
theorem crossing {Q : Type} (q : Q) (witness : ExecFrame Q → Bool)
(hw : witness (execFlip (true, q)) = !witness (true, q)) :
∃ c : Bool,
(ran (true, q) = xor (witness (true, q)) c ∧
ran (execFlip (true, q)) = xor (witness (execFlip (true, q))) c) ∧
∀ c' : Bool,
(ran (true, q) = xor (witness (true, q)) c' ∧
ran (execFlip (true, q)) = xor (witness (execFlip (true, q))) c') → c' = c :=
T5_crossing execFlip witness ran (true, q) hw (ran_wholly_odd (true, q))
theorem identity_one_bit {Q : Type} (witness : ExecFrame Q → Bool)
(hw : WhollyOdd execFlip witness) (s : ExecFrame Q) (h : witness s = ran s) :
witness (execFlip s) = ran (execFlip s) :=
T4_one_bit execFlip witness ran hw ran_wholly_odd s h
inductive Anchor : Type
| F1 | F2 | F3 | F4
deriving DecidableEq, Repr
def fibre4 : List (Bool × Bool × Bool × Bool) :=
[false, true].flatMap fun a => [false, true].flatMap fun b =>
[false, true].flatMap fun c => [false, true].map fun d => (a, b, c, d)
theorem fibre4_sixteen : fibre4.length = 16 := by decide
theorem fibre4_distinct : fibre4.eraseDups.length = 16 := by decide
theorem price_on_this_file :
Nonempty (Dtau (execFlip (Q := Anchor)) ≃
(Quot (OrbitRel (execFlip (Q := Anchor))) → Bool)) :=
price
end CROSSING
namespace CROSSING
open FTOE
theorem formal_never_odd {Q : Type} (g : Q → Bool) :
Even (execFlip (Q := Q)) (fun s => g (formalRead s)) := by
intro s; obtain ⟨b, q⟩ := s; rfl
theorem constant_no_crossing {Q : Type} (b : Bool) (s : ExecFrame Q) :
¬ OddAt (execFlip (Q := Q)) (fun _ => b) s := by
intro h; exact h rfl
theorem harmony {Q : Type} (q : Q) (witness : ExecFrame Q → Bool)
(hw : witness (execFlip (true, q)) = !witness (true, q))
(g₁ g₂ : Q → Bool) (op : Bool → Bool → Bool) :
(fun s => op (g₁ (formalRead s)) (g₂ (formalRead s))) ≠ ran ∧
∃ c : Bool,
(ran (true, q) = xor (witness (true, q)) c ∧
ran (execFlip (true, q)) = xor (witness (execFlip (true, q))) c) ∧
∀ c' : Bool,
(ran (true, q) = xor (witness (true, q)) c' ∧
ran (execFlip (true, q)) = xor (witness (execFlip (true, q))) c') → c' = c :=
fTOE_core execFlip witness ran (true, q)
(fun s => g₁ (formalRead s)) (fun s => g₂ (formalRead s))
(formal_never_odd g₁) (formal_never_odd g₂) op
(ran_wholly_odd (true, q)) hw
end CROSSING
namespace LOCUS
open FTOE
def Separates {α β : Type} (ρ : α → β) (d : α → Bool) : Prop :=
∃ x y, ρ x = ρ y ∧ d x ≠ d y
def Factors {α β : Type} (ρ : α → β) (d : α → Bool) : Prop :=
∃ g : β → Bool, ∀ z, g (ρ z) = d z
theorem wall_of_sep {α β : Type} {ρ : α → β} {d : α → Bool} :
Separates ρ d → ¬ Factors ρ d := by
intro ⟨x, y, hρ, hd⟩ ⟨g, hg⟩
apply hd
rw [← hg x, ← hg y, hρ]
def swap {α : Type} [DecidableEq α] (x y : α) (z : α) : α :=
if z = x then y else if z = y then x else z
theorem swap_involution {α : Type} [DecidableEq α] {x y : α} (hxy : x ≠ y) :
∀ z, swap x y (swap x y z) = z := by
intro z
by_cases hzx : z = x
· simp [swap, hzx, Ne.symm hxy]
· by_cases hzy : z = y
· simp [swap, hzy, Ne.symm hxy]
· simp [swap, hzx, hzy]
theorem swap_record_even {α β : Type} [DecidableEq α] {ρ : α → β} {x y : α}
(hxy : x ≠ y) (hρ : ρ x = ρ y) : ∀ z, ρ (swap x y z) = ρ z := by
intro z
by_cases hzx : z = x
· simp [swap, hzx, hρ]
· by_cases hzy : z = y
· simp [swap, hzy, Ne.symm hxy, hρ]
· simp [swap, hzx, hzy]
theorem swap_flips_pair {α : Type} [DecidableEq α] {d : α → Bool} {x y : α}
(hxy : x ≠ y) (hd : d x ≠ d y) :
d (swap x y x) = !d x ∧ d (swap x y y) = !d y := by
have h1 : swap x y x = y := by simp [swap]
have h2 : swap x y y = x := by simp [swap, Ne.symm hxy]
rw [h1, h2]
cases hx : d x <;> cases hy : d y <;> simp_all
theorem seat_of_sep {α β : Type} [DecidableEq α] {ρ : α → β} {d : α → Bool} {x y : α}
(hρ : ρ x = ρ y) (hd : d x ≠ d y) :
∃ τ : α → α, (∀ z, τ (τ z) = z) ∧ (∀ z, ρ (τ z) = ρ z) ∧ d (τ x) = !d x := by
have hxy : x ≠ y := fun h => hd (h ▸ rfl)
exact ⟨swap x y, swap_involution hxy, swap_record_even hxy hρ, (swap_flips_pair hxy hd).1⟩
theorem k4_separated_pair : Separates K4.rho K4.target :=
⟨0, 1, rfl, by decide⟩
theorem k4_wall_of_pair : ¬ Factors K4.rho K4.target := wall_of_sep k4_separated_pair
def readout {α β : Type} [DecidableEq β] (xs : List α) (ρ : α → β) (d : α → Bool) (b : β) : Bool :=
match xs.find? (fun x => decide (ρ x = b)) with
| some x => d x
| none => false
theorem factors_of_not_sep_list {α β : Type} [DecidableEq β] (xs : List α)
(hxs : ∀ z, z ∈ xs) (ρ : α → β) (d : α → Bool) (h : ¬ Separates ρ d) :
Factors ρ d := by
refine ⟨readout xs ρ d, fun z => ?_⟩
unfold readout
cases hf : xs.find? (fun x => decide (ρ x = ρ z)) with
| none =>
have := List.find?_eq_none.mp hf z (hxs z)
simp at this
| some x =>
have hx : ρ x = ρ z := by simpa using List.find?_some hf
show d x = d z
cases hdx : d x <;> cases hdz : d z
· rfl
· exact absurd ⟨x, z, hx, by rw [hdx, hdz]; decide⟩ h
· exact absurd ⟨x, z, hx, by rw [hdx, hdz]; decide⟩ h
· rfl
theorem factors_iff_not_sep_list {α β : Type} [DecidableEq β] (xs : List α)
(hxs : ∀ z, z ∈ xs) (ρ : α → β) (d : α → Bool) :
Factors ρ d ↔ ¬ Separates ρ d := by
refine ⟨fun hf hs => ?_, factors_of_not_sep_list xs hxs ρ d⟩
obtain ⟨x, y, hρ, hd⟩ := hs; obtain ⟨g, hg⟩ := hf
exact hd (by rw [← hg x, ← hg y, hρ])
abbrev Channel := HALT.Channel
inductive Token : Type
| superHalt (b : Option HALT.Branch) (s : Option HALT.Species) (c : Option Channel) (m : Nat)
| crossed (c : Channel)
| breach
deriving DecidableEq, Repr
def Token.toEconomy : Token → Audit.Token
| .superHalt .. => .opn
| .crossed _ => .sealed
| .breach => .broken
structure Reading where
broken : Bool
located : Bool
gdim : Option Nat
species : Option HALT.Species
channel : Option Channel
slots : List (Option Bool)
orbits_pos : 0 < slots.length
deriving DecidableEq, Repr
def Reading.orbits (r : Reading) : Nat := r.slots.length
def Reading.owed (r : Reading) : Nat := (r.slots.filter Option.isNone).length
def Reading.complete (r : Reading) : Bool := r.owed == 0
def branchOf : Option Nat → Option HALT.Branch
| some n => HALT.route n
| none => none
def emit (r : Reading) : Option Token :=
if r.broken then some .breach
else if !r.located then none
else match r.complete, r.channel with
| true, some c => some (.crossed c)
| _, _ => some (.superHalt (branchOf r.gdim) r.species r.channel r.owed)
def Reading.state (r : Reading) : Audit.Token :=
match emit r with
| some t => t.toEconomy
| none => .opn
theorem three_states : ∀ t : Token,
t.toEconomy = .sealed ∨ t.toEconomy = .broken ∨ t.toEconomy = .opn := by
intro t; cases t <;> simp [Token.toEconomy]
theorem no_seat_no_locus (r : Reading) (h : r.located = false) (hb : r.broken = false) :
emit r = none := by
simp [emit, h, hb]
theorem unlocated_is_plain_open (r : Reading) (h : r.located = false) (hb : r.broken = false) :
r.state = .opn := by
simp [Reading.state, no_seat_no_locus r h hb]
theorem located_never_silent (r : Reading) (hl : r.located = true) (hb : r.broken = false) :
emit r ≠ none := by
simp only [emit, hl, hb]
cases r.complete <;> cases r.channel <;> simp
theorem crossed_requires_complete (r : Reading) (c : Channel) (h : emit r = some (.crossed c)) :
r.complete = true := by
cases hb : r.broken <;> cases hl : r.located <;> cases hc : r.complete <;> cases hch : r.channel <;>
simp [emit, hb, hl, hc, hch] at h ⊢
theorem crossed_requires_channel (r : Reading) (c : Channel) (h : emit r = some (.crossed c)) :
r.channel = some c := by
cases hb : r.broken <;> cases hl : r.located <;> cases hc : r.complete <;> cases hch : r.channel <;>
simp [emit, hb, hl, hc, hch] at h ⊢
exact h
theorem halt_carries_owed (r : Reading) (hl : r.located = true) (hb : r.broken = false)
(hnc : ¬ (r.complete = true ∧ r.channel.isSome = true)) :
emit r = some (.superHalt (branchOf r.gdim) r.species r.channel r.owed) := by
simp only [emit, hl, hb]
cases hc : r.complete <;> cases hch : r.channel <;> simp_all
theorem unchannelled_is_halt (r : Reading) (hl : r.located = true) (hb : r.broken = false)
(hch : r.channel = none) :
emit r = some (.superHalt (branchOf r.gdim) r.species none r.owed) := by
simp only [emit, hl, hb, hch]
cases r.complete <;> simp
theorem halt_reports_owed (r : Reading) (b : Option HALT.Branch) (s : Option HALT.Species)
(c : Option Channel) (m : Nat) (h : emit r = some (.superHalt b s c m)) : m = r.owed := by
cases hb : r.broken <;> cases hl : r.located <;> cases hc : r.complete <;> cases hch : r.channel <;>
simp [emit, hb, hl, hc, hch] at h <;> simp_all
theorem owed_zero_is_unchannelled (r : Reading) (b : Option HALT.Branch) (s : Option HALT.Species)
(c : Option Channel) (h : emit r = some (.superHalt b s c 0)) : c = none := by
have how : 0 = r.owed := halt_reports_owed r b s c 0 h
have hcomp : r.complete = true := by
unfold Reading.complete; rw [← how]; rfl
cases hb : r.broken
· cases hl : r.located
· rw [no_seat_no_locus r hl hb] at h; cases h
· cases hch : r.channel with
| none =>
rw [unchannelled_is_halt r hl hb hch] at h
injection h with h'
injection h' with _ _ h3 _
exact h3.symm
| some ch =>
have hx : emit r = some (.crossed ch) := by
unfold emit; rw [hb, hl, hcomp, hch]; rfl
rw [hx] at h; injection h with h'; cases h'
· have hx : emit r = some .breach := by unfold emit; rw [hb]; rfl
rw [hx] at h; injection h with h'; cases h'
theorem one_bit_apart (r : Reading) (hl : r.located = true) (hb : r.broken = false)
(c : Channel) (hc : r.channel = some c) :
({ r with slots := [none], orbits_pos := by decide }).state = .opn ∧
({ r with slots := [some true], orbits_pos := by decide }).state = .sealed ∧
({ r with slots := [some false], orbits_pos := by decide }).state = .sealed := by
refine ⟨?_, ?_, ?_⟩ <;>
(unfold Reading.state emit Reading.complete Reading.owed; dsimp only; rw [hb, hl, hc]; rfl)
theorem legacy_xi : branchOf (some 1) = some .xi := by decide
theorem legacy_o : branchOf (some 0) = some .o := by decide
theorem legacy_bare : branchOf none = none ∧ branchOf (some 2) = none := by decide
theorem k4_locus :
emit ⟨false, true, none, none, some .formal, [none, none, none, none, none, none], by decide⟩
= some (.superHalt none none (some .formal) 6) ∧
emit ⟨false, true, none, none, some .formal,
[some true, some true, some true, some true, some true, none], by decide⟩
= some (.superHalt none none (some .formal) 1) ∧
emit ⟨false, true, none, none, some .formal,
[some true, some true, some true, some true, some true, some true], by decide⟩
= some (.crossed .formal) ∧
emit ⟨false, true, none, none, none,
[some true, some true, some true, some true, some true, some true], by decide⟩
= some (.superHalt none none none 0) := by decide
def slotLists : List (List (Option Bool)) :=
let o : List (Option Bool) := [none, some false, some true]
o.map (fun a => [a]) ++ o.flatMap (fun a => o.map (fun b => [a, b]))
theorem slotLists_twelve : slotLists.length = 12 ∧ slotLists.all (fun l => 0 < l.length) = true := by
decide
theorem slotLists_pos : ∀ sl ∈ slotLists, 0 < sl.length := by decide
def allReadings : List Reading :=
[false, true].flatMap fun broken =>
[false, true].flatMap fun located =>
[none, some 0, some 1, some 2].flatMap fun gdim =>
([none, some .blockTheorem, some .hypothesis, some .offeredWitness] : List (Option HALT.Species)).flatMap fun sp =>
([none, some .kinetic, some .formal] : List (Option Channel)).flatMap fun ch =>
(slotLists.attach.map fun ⟨sl, hsl⟩ =>
(⟨broken, located, gdim, sp, ch, sl, slotLists_pos sl hsl⟩ : Reading))
theorem reading_space : allReadings.length = 2304 := by decide
theorem census_no_seat_no_locus :
(allReadings.filter (fun r => !r.broken && !r.located)).all
(fun r => decide (emit r = none)) = true := by decide
theorem census_located_never_silent :
(allReadings.filter (fun r => !r.broken && r.located)).all
(fun r => decide (emit r ≠ none)) = true := by decide
theorem census_crossed_iff_complete_channelled :
allReadings.all (fun r =>
((emit r == some (.crossed .kinetic)) || (emit r == some (.crossed .formal)))
== (!r.broken && r.located && r.complete && r.channel.isSome)) = true := by
decide
theorem census_halt_reports_owed :
(allReadings.filter (fun r => !r.broken && r.located && !(r.complete && r.channel.isSome))).all
(fun r => emit r == some (.superHalt (branchOf r.gdim) r.species r.channel r.owed)) = true := by
decide
theorem census_owed_zero_is_unchannelled :
allReadings.all (fun r =>
match emit r with
| some (.superHalt _ _ c 0) => c == none
| _ => true) = true := by decide
theorem census_three_states :
allReadings.all (fun r => decide (r.state = .sealed ∨ r.state = .broken ∨ r.state = .opn)) = true := by
decide
def Token.render : Token → String
| .superHalt b s c m =>
let br := match b with | some .xi => " · Ξ" | some .o => " · Ø" | none => ""
let sp := match s with
| some .blockTheorem => " · block theorem" | some .hypothesis => " · hypothesis"
| some .offeredWitness => " · offered witness" | none => ""
let ch := match c with | some .kinetic => " · kinetic" | some .formal => " · formal" | none => ""
let mult := if m = 0 then " · provenance owed" else s!" ×{m}"
s!"[⬖₀{br}{sp}{ch}{mult}]"
| .crossed .kinetic => "[⟀⬖ · kinetic]"
| .crossed .formal => "[⟀⬖ · formal]"
| .breach => "[X⬖ · mechanism]"
def Token.ascii : Token → String
| .superHalt b _ _ m =>
let base := match b with | some .xi => "[HALT-LOCUS-XI" | some .o => "[HALT-LOCUS-O" | none => "[HALT-LOCUS"
if m = 0 then s!"{base} PROV-OWED]" else s!"{base} x{m}]"
| .crossed _ => "[SEAL-LOCUS]"
| .breach => "[X-LOCUS]"
def sealAscii : String := "[SEAL]"
theorem render_receipted :
(Token.superHalt none none none 1).render = "[⬖₀ ×1]" ∧
(Token.superHalt (some .xi) none none 1).render = "[⬖₀ · Ξ ×1]" ∧
(Token.superHalt (some .o) none none 1).render = "[⬖₀ · Ø ×1]" ∧
(Token.superHalt (some .xi) (some .hypothesis) (some .formal) 1).render
= "[⬖₀ · Ξ · hypothesis · formal ×1]" ∧
(Token.superHalt none none (some .formal) 6).render = "[⬖₀ · formal ×6]" ∧
(Token.superHalt none none none 0).render = "[⬖₀ · provenance owed]" ∧
(Token.superHalt none none (some .formal) 1).render = "[⬖₀ · formal ×1]" ∧
(Token.superHalt (some .xi) none none 3).render = "[⬖₀ · Ξ ×3]" ∧
(Token.crossed .kinetic).render = "[⟀⬖ · kinetic]" ∧
(Token.crossed .formal).render = "[⟀⬖ · formal]" ∧
Token.breach.render = "[X⬖ · mechanism]" ∧
(Token.superHalt (some .xi) none none 1).ascii = "[HALT-LOCUS-XI x1]" ∧
(Token.superHalt (some .xi) none none 3).ascii = "[HALT-LOCUS-XI x3]" ∧
(Token.superHalt (some .o) none none 2).ascii = "[HALT-LOCUS-O x2]" ∧
(Token.superHalt none none none 6).ascii = "[HALT-LOCUS x6]" ∧
(Token.superHalt none none none 1).ascii = "[HALT-LOCUS x1]" ∧
(Token.superHalt none none none 0).ascii = "[HALT-LOCUS PROV-OWED]" ∧
(Token.crossed .formal).ascii = "[SEAL-LOCUS]" ∧
Token.breach.ascii = "[X-LOCUS]" ∧
sealAscii = "[SEAL]" := by decide
theorem render_injective_in_owed :
((List.range 13).all fun m => (List.range 13).all fun n =>
(decide ((Token.superHalt none none none m).ascii = (Token.superHalt none none none n).ascii) == decide (m = n)) &&
(decide ((Token.superHalt (some .xi) none none m).ascii = (Token.superHalt (some .xi) none none n).ascii) == decide (m = n)) &&
(decide ((Token.superHalt none none (some .formal) m).render = (Token.superHalt none none (some .formal) n).render) == decide (m = n))) = true := by
decide
#eval (Token.superHalt (some .xi) (some .hypothesis) (some .formal) 1).render
#eval (Token.superHalt none none (some .formal) 6).render
#eval (Token.crossed .kinetic).render
#eval Token.breach.render
def voidImage : Audit.Token := .opn
theorem void_precedes_economy : voidImage = .opn ∧ Audit.Token.void ≠ Audit.Token.refused := by decide
theorem legacy_records_not_promoted :
HALT.rhHalt.species = .hypothesis ∧ HALT.pnpHalt.species = .blockTheorem ∧
HALT.rhHalt.channel = .formal ∧ HALT.pnpHalt.channel = .formal := by decide
theorem no_rh_locus :
emit ⟨false, true, some 1, some .hypothesis, some .formal, [none], by decide⟩
= some (.superHalt (some .xi) (some .hypothesis) (some .formal) 1) := by decide
end LOCUS
namespace PSP.ORIENT
abbrev V3 := Ratio × Ratio × Ratio
abbrev M3 := V3 × V3 × V3
def det3 (m : M3) : Ratio :=
m.1.1 * (m.2.1.2.1 * m.2.2.2.2 - m.2.1.2.2 * m.2.2.2.1)
- m.1.2.1 * (m.2.1.1 * m.2.2.2.2 - m.2.1.2.2 * m.2.2.1)
+ m.1.2.2 * (m.2.1.1 * m.2.2.2.1 - m.2.1.2.1 * m.2.2.1)
def neg3 (m : M3) : M3 :=
((-m.1.1, -m.1.2.1, -m.1.2.2),
(-m.2.1.1, -m.2.1.2.1, -m.2.1.2.2),
(-m.2.2.1, -m.2.2.2.1, -m.2.2.2.2))
def signMatrices : List M3 :=
[ (( 1,0,0),(0, 1,0),(0,0, 1)), ((-1,0,0),(0, 1,0),(0,0, 1)),
(( 1,0,0),(0,-1,0),(0,0, 1)), (( 1,0,0),(0, 1,0),(0,0,-1)),
((-1,0,0),(0,-1,0),(0,0, 1)), ((-1,0,0),(0, 1,0),(0,0,-1)),
(( 1,0,0),(0,-1,0),(0,0,-1)), ((-1,0,0),(0,-1,0),(0,0,-1)) ]
theorem lock_scalar_sign_blind :
signMatrices.all (fun m =>
det3 m * det3 m == det3 (neg3 m) * det3 (neg3 m)) = true := by decide
theorem det_sign_flips_in_odd_dimension :
signMatrices.all (fun m => det3 (neg3 m) == -det3 m) = true := by decide
end PSP.ORIENT
namespace PSP.QUAT
abbrev Q := Ratio × Ratio × Ratio × Ratio
def qmul (p q : Q) : Q :=
(p.1*q.1 - p.2.1*q.2.1 - p.2.2.1*q.2.2.1 - p.2.2.2*q.2.2.2,
p.1*q.2.1 + p.2.1*q.1 + p.2.2.1*q.2.2.2 - p.2.2.2*q.2.2.1,
p.1*q.2.2.1 - p.2.1*q.2.2.2 + p.2.2.1*q.1 + p.2.2.2*q.2.1,
p.1*q.2.2.2 + p.2.1*q.2.2.1 - p.2.2.1*q.2.1 + p.2.2.2*q.1)
def qi : Q := (0,1,0,0)
def qj : Q := (0,0,1,0)
def qk : Q := (0,0,0,1)
def qneg1 : Q := (-1,0,0,0)
theorem quat_basis_laws :
qmul qi qi = qneg1 ∧ qmul qj qj = qneg1 ∧ qmul qk qk = qneg1 ∧
qmul (qmul qi qj) qk = qneg1 := by decide
end PSP.QUAT
namespace PSP.QUAT
def qnorm (p : Q) : Ratio := p.1^2 + p.2.1^2 + p.2.2.1^2 + p.2.2.2^2
def isInt (r : Ratio) : Bool := r.den == 1
def isHalf (r : Ratio) : Bool := r.den == 2
def hurwitzOK (p : Q) : Bool :=
(isInt p.1 && isInt p.2.1 && isInt p.2.2.1 && isInt p.2.2.2) ||
(isHalf p.1 && isHalf p.2.1 && isHalf p.2.2.1 && isHalf p.2.2.2)
def grid : List Ratio := [-1, -1/2, 0, 1/2, 1]
def tuples4 : List Q :=
grid.flatMap fun a => grid.flatMap fun b =>
grid.flatMap fun c => grid.map fun d => (a,b,c,d)
theorem hurwitz_units_24 :
(tuples4.filter fun p => hurwitzOK p && qnorm p == 1).length = 24 := by decide
end PSP.QUAT
namespace PSP.QUAT
def grid2 : List Ratio := [-2, -1, 0, 1, 2]
def tuples4i : List Q :=
grid2.flatMap fun a => grid2.flatMap fun b =>
grid2.flatMap fun c => grid2.map fun d => (a,b,c,d)
theorem norm2_shell_split :
(tuples4i.filter fun p => qnorm p == 2).length = 24 ∧
(tuples4i.filter fun p => qnorm p == 2 && p.1 == 0).length = 12 ∧
(tuples4i.filter fun p => qnorm p == 2 && p.1 != 0).length = 12 := by decide
end PSP.QUAT
namespace PSP.CD
def cdconj : List Ratio → List Ratio
| [] => []
| [a] => [a]
| a :: rest => a :: rest.map (fun x => -x)
def padd : List Ratio → List Ratio → List Ratio := List.zipWith (· + ·)
def psub : List Ratio → List Ratio → List Ratio := List.zipWith (· - ·)
def cdmul : Nat → List Ratio → List Ratio → List Ratio
| 0, _, _ => []
| _+1, [a], [b] => [a * b]
| fuel+1, a, b =>
let n := a.length / 2
let a1 := a.take n; let a2 := a.drop n
let b1 := b.take n; let b2 := b.drop n
psub (cdmul fuel a1 b1) (cdmul fuel (cdconj b2) a2) ++
padd (cdmul fuel b2 a1) (cdmul fuel a2 (cdconj b1))
def e (n i : Nat) : List Ratio := List.replicate i 0 ++ [1] ++ List.replicate (n - i - 1) 0
def smul (s : Ratio) (v : List Ratio) : List Ratio := v.map (fun x => s * x)
theorem octonion_associator :
psub (cdmul 4 (cdmul 4 (e 8 1) (e 8 2)) (e 8 4))
(cdmul 4 (e 8 1) (cdmul 4 (e 8 2) (e 8 4)))
= smul 2 (e 8 7) := by decide
end PSP.CD
namespace PSP.CD
def sx : List Ratio := padd (e 16 1) (e 16 10)
def sy : List Ratio := padd (e 16 5) (e 16 14)
def zero16 : List Ratio := List.replicate 16 0
def sqnorm (v : List Ratio) : Ratio := (v.map fun x => x^2).sum
theorem sedenion_zero_divisor :
cdmul 5 sx sy = zero16 ∧ sx ≠ zero16 ∧ sy ≠ zero16
∧ sqnorm sx = 2 ∧ sqnorm sy = 2 := by decide
end PSP.CD
namespace PSP.CASCADE
def cascade : List Bool → Option Nat
| [] => none
| false :: _ => some 0
| true :: gs => (cascade gs).map Nat.succ
def patterns8 : List (List Bool) :=
(List.range 256).map fun b => (List.range 8).map fun j => b.testBit j
def spec (bits : List Bool) : Bool :=
match cascade bits with
| none => bits.all id
| some k => (bits.take k).all id && (bits[k]? == some false)
theorem first_failure_terminal_8 :
patterns8.all spec = true := by decide
theorem halt_free_iff_all_pass_8 :
patterns8.all (fun bits => (cascade bits).isNone == bits.all id) = true := by decide
end PSP.CASCADE
namespace PSP.GROUNDLESS
def o0Neighbors : List String :=
["sigma-token", "dissolve-openness", "ordinary-openness",
"resolved-states", "broken-or-ghost"]
def fiveCubeCoords : List String :=
["Ground-dimension", "determinacy", "blockage", "aperture", "totality"]
def deg5 (v : Nat) : Nat :=
((List.range 5).map fun j => v ^^^ (1 <<< j)).eraseDups.length
theorem groundless_cascade_degree :
o0Neighbors.length = 5 ∧ fiveCubeCoords.length = 5 ∧
o0Neighbors.length ≠ 7 ∧ o0Neighbors.length ≠ 8 ∧
(List.range 32).all (fun v => deg5 v == 5) = true := by decide
end PSP.GROUNDLESS
namespace PSP.TOKEN
inductive TerminalToken | xi0 | o0 deriving DecidableEq
def level : TerminalToken → Nat := fun _ => 3
theorem terminal_tokens_level_and_distinct :
level TerminalToken.xi0 = level TerminalToken.o0 ∧
TerminalToken.xi0 ≠ TerminalToken.o0 := ⟨rfl, by decide⟩
end PSP.TOKEN
namespace PSP.TWOGROUP
theorem two_group_census :
Nat.factorial 4 / 2 = 12 ∧ 2^3 = 8 := by decide
end PSP.TWOGROUP
/-- info: 'Codex.LeftInverse' does not depend on any axioms -/
#guard_msgs in #print axioms Codex.LeftInverse
/-- info: 'Codex.RightInverse' does not depend on any axioms -/
#guard_msgs in #print axioms Codex.RightInverse
/-- info: 'Nat.factorial' does not depend on any axioms -/
#guard_msgs in #print axioms Nat.factorial
/-- info: 'CoreRat.div_pos_of_le' depends on axioms: [propext] -/
#guard_msgs in #print axioms CoreRat.div_pos_of_le
/-- info: 'CoreRat.div_gcd_pos' depends on axioms: [propext] -/
#guard_msgs in #print axioms CoreRat.div_gcd_pos
/-- info: 'CoreRat.coprime_div_gcd' depends on axioms: [propext] -/
#guard_msgs in #print axioms CoreRat.coprime_div_gcd
/-- info: 'Ratio.natAbs_signed' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.natAbs_signed
/-- info: 'Ratio.mk'' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.«mk'»
/-- info: 'Ratio.instOfNatRat' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.instOfNatRat
/-- info: 'Ratio.instAddRat' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.instAddRat
/-- info: 'Ratio.instMulRat' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.instMulRat
/-- info: 'Ratio.instNegRat' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.instNegRat
/-- info: 'Ratio.instSubRat' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.instSubRat
/-- info: 'Ratio.inv' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.inv
/-- info: 'Ratio.instDivRat' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.instDivRat
/-- info: 'Ratio.instLTRat' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.instLTRat
/-- info: 'Ratio.instLERat' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.instLERat
/-- info: 'Ratio.instDecidableLtRat' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.instDecidableLtRat
/-- info: 'Ratio.instDecidableLeRat' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.instDecidableLeRat
/-- info: 'Ratio.pow' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.pow
/-- info: 'Ratio.instHPowRatNat' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.instHPowRatNat
/-- info: 'Ratio.instToStringRat' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.instToStringRat
/-- info: 'Ratio.num_zero' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.num_zero
/-- info: 'Ratio.den_zero' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.den_zero
/-- info: 'Ratio.mk'_zero_num' depends on axioms: [propext, Quot.sound] -/
#guard_msgs in #print axioms Ratio.«mk'_zero_num»
/-- info: 'Ratio.zero_mul' depends on axioms: [propext, Quot.sound] -/
#guard_msgs in #print axioms Ratio.zero_mul
/-- info: 'Ratio.mul_zero' depends on axioms: [propext, Quot.sound] -/
#guard_msgs in #print axioms Ratio.mul_zero
/-- info: 'Ratio.canonical' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.canonical
/-- info: 'Ratio.ext' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.ext
/-- info: 'Ratio.inv_zero' depends on axioms: [propext] -/
#guard_msgs in #print axioms Ratio.inv_zero
/-- info: 'Grade.rank' does not depend on any axioms -/
#guard_msgs in #print axioms Grade.rank
/-- info: 'Grade.weakest' does not depend on any axioms -/
#guard_msgs in #print axioms Grade.weakest
/-- info: 'Grade.weakest_is_a_link' does not depend on any axioms -/
#guard_msgs in #print axioms Grade.weakest_is_a_link
/-- info: 'Grade.weakest_idempotent' does not depend on any axioms -/
#guard_msgs in #print axioms Grade.weakest_idempotent
/-- info: 'Claim.join' does not depend on any axioms -/
#guard_msgs in #print axioms Claim.join
/-- info: 'Claim.join_grade' does not depend on any axioms -/
#guard_msgs in #print axioms Claim.join_grade
/-- info: 'deltaM' does not depend on any axioms -/
#guard_msgs in #print axioms deltaM
/-- info: 'socialWeight' does not depend on any axioms -/
#guard_msgs in #print axioms socialWeight
/-- info: 'FTOE.Even' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.Even
/-- info: 'FTOE.OddAt' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.OddAt
/-- info: 'FTOE.WhollyOdd' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.WhollyOdd
/-- info: 'FTOE.T1_wall' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.T1_wall
/-- info: 'FTOE.T2_coalition' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.T2_coalition
/-- info: 'FTOE.T3_no_fixed_point' depends on axioms: [propext] -/
#guard_msgs in #print axioms FTOE.T3_no_fixed_point
/-- info: 'FTOE.T4_one_bit' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.T4_one_bit
/-- info: 'FTOE.T5_crossing' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.T5_crossing
/-- info: 'FTOE.T6a_deed_anchor' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.T6a_deed_anchor
/-- info: 'FTOE.T6b_deed_supply' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.T6b_deed_supply
/-- info: 'FTOE.fTOE_port' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.fTOE_port
/-- info: 'FTOE.Involution' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.Involution
/-- info: 'FTOE.FixedPointFree' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.FixedPointFree
/-- info: 'FTOE.T7_global_wall' depends on axioms: [propext] -/
#guard_msgs in #print axioms FTOE.T7_global_wall
/-- info: 'FTOE.T8_torsor_forward' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.T8_torsor_forward
/-- info: 'FTOE.T8_torsor_backward' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.T8_torsor_backward
/-- info: 'FTOE.T9_torsor_inverse_left' depends on axioms: [Quot.sound] -/
#guard_msgs in #print axioms FTOE.T9_torsor_inverse_left
/-- info: 'FTOE.T9_torsor_inverse_right' depends on axioms: [Quot.sound] -/
#guard_msgs in #print axioms FTOE.T9_torsor_inverse_right
/-- info: 'FTOE.T10_factorization' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.T10_factorization
/-- info: 'FTOE.T11_separation' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.T11_separation
/-- info: 'FTOE.T12_encoder_injective' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.T12_encoder_injective
/-- info: 'FTOE.T14_wall_factorization' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.T14_wall_factorization
/-- info: 'FTOE.T13_wholly_odd_is_odd_everywhere' depends on axioms: [propext] -/
#guard_msgs in #print axioms FTOE.T13_wholly_odd_is_odd_everywhere
/-- info: 'FTOE.flipF' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.flipF
/-- info: 'FTOE.T15_price_bijection_forward' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.T15_price_bijection_forward
/-- info: 'FTOE.T15_price_bijection_backward' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.T15_price_bijection_backward
/-- info: 'FTOE.fTOE_core' depends on axioms: [propext] -/
#guard_msgs in #print axioms FTOE.fTOE_core
/-- info: 'FTOE.OrbitRel' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.OrbitRel
/-- info: 'FTOE.OrbitRel_symmetric' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.OrbitRel_symmetric
/-- info: 'FTOE.Dtau' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.Dtau
/-- info: 'FTOE.calib' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.calib
/-- info: 'FTOE.calib_even' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.calib_even
/-- info: 'FTOE.descend' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.descend
/-- info: 'FTOE.priceForward' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.priceForward
/-- info: 'FTOE.priceBackward' depends on axioms: [Quot.sound] -/
#guard_msgs in #print axioms FTOE.priceBackward
/-- info: 'FTOE.T16_left_inverse' depends on axioms: [Quot.sound] -/
#guard_msgs in #print axioms FTOE.T16_left_inverse
/-- info: 'FTOE.T16_right_inverse' depends on axioms: [Quot.sound] -/
#guard_msgs in #print axioms FTOE.T16_right_inverse
/-- info: 'FTOE.T16_price_bijection' depends on axioms: [Quot.sound] -/
#guard_msgs in #print axioms FTOE.T16_price_bijection
/-- info: 'FTOE.canonD0' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.canonD0
/-- info: 'FTOE.flipF_involution' does not depend on any axioms -/
#guard_msgs in #print axioms FTOE.flipF_involution
/-- info: 'FTOE.T16_canonical' depends on axioms: [Quot.sound] -/
#guard_msgs in #print axioms FTOE.T16_canonical
/-- info: 'FTOE.T17_odd_everywhere_iff_wholly_odd' depends on axioms: [propext] -/
#guard_msgs in #print axioms FTOE.T17_odd_everywhere_iff_wholly_odd
/-- info: 'K4.val' does not depend on any axioms -/
#guard_msgs in #print axioms K4.val
/-- info: 'K4.tau' does not depend on any axioms -/
#guard_msgs in #print axioms K4.tau
/-- info: 'K4.target' does not depend on any axioms -/
#guard_msgs in #print axioms K4.target
/-- info: 'K4.rho' does not depend on any axioms -/
#guard_msgs in #print axioms K4.rho
/-- info: 'K4.tau_involution' does not depend on any axioms -/
#guard_msgs in #print axioms K4.tau_involution
/-- info: 'K4.tau_fixed_point_free' does not depend on any axioms -/
#guard_msgs in #print axioms K4.tau_fixed_point_free
/-- info: 'K4.target_wholly_odd' does not depend on any axioms -/
#guard_msgs in #print axioms K4.target_wholly_odd
/-- info: 'K4.register_even' does not depend on any axioms -/
#guard_msgs in #print axioms K4.register_even
/-- info: 'K4.val_lifts_mod_210' does not depend on any axioms -/
#guard_msgs in #print axioms K4.val_lifts_mod_210
/-- info: 'K4.wall' does not depend on any axioms -/
#guard_msgs in #print axioms K4.wall
/-- info: 'K4.pair_distinct' does not depend on any axioms -/
#guard_msgs in #print axioms K4.pair_distinct
/-- info: 'K4.price_one_bit' does not depend on any axioms -/
#guard_msgs in #print axioms K4.price_one_bit
/-- info: 'K4.price_full_fibre' does not depend on any axioms -/
#guard_msgs in #print axioms K4.price_full_fibre
/-- info: 'K4.wallClaim' does not depend on any axioms -/
#guard_msgs in #print axioms K4.wallClaim
/-- info: 'Audit.Ans.toBool' does not depend on any axioms -/
#guard_msgs in #print axioms Audit.Ans.toBool
/-- info: 'Audit.ClaimInput.anyUnknown' does not depend on any axioms -/
#guard_msgs in #print axioms Audit.ClaimInput.anyUnknown
/-- info: 'Audit.sealL' does not depend on any axioms -/
#guard_msgs in #print axioms Audit.sealL
/-- info: 'Audit.DrillInput.anyUnknown' does not depend on any axioms -/
#guard_msgs in #print axioms Audit.DrillInput.anyUnknown
/-- info: 'Audit.drillScreen' does not depend on any axioms -/
#guard_msgs in #print axioms Audit.drillScreen
/-- info: 'Audit.refusalGates' does not depend on any axioms -/
#guard_msgs in #print axioms Audit.refusalGates
/-- info: 'Audit.compartmentGates' does not depend on any axioms -/
#guard_msgs in #print axioms Audit.compartmentGates
/-- info: 'Audit.auditClaim' does not depend on any axioms -/
#guard_msgs in #print axioms Audit.auditClaim
/-- info: 'Audit.compartmentGrade' does not depend on any axioms -/
#guard_msgs in #print axioms Audit.compartmentGrade
/-- info: 'Audit.unknown_routes_open' depends on axioms: [propext] -/
#guard_msgs in #print axioms Audit.unknown_routes_open
/-- info: 'Audit.contentless_refused' depends on axioms: [propext] -/
#guard_msgs in #print axioms Audit.contentless_refused
/-- info: 'Audit.det3' depends on axioms: [propext] -/
#guard_msgs in #print axioms Audit.det3
/-- info: 'Audit.det3_two_id' depends on axioms: [propext] -/
#guard_msgs in #print axioms Audit.det3_two_id
/-- info: 'INVERSION.ItFromBit' does not depend on any axioms -/
#guard_msgs in #print axioms INVERSION.ItFromBit
/-- info: 'INVERSION.it_from_bit_refuted' does not depend on any axioms -/
#guard_msgs in #print axioms INVERSION.it_from_bit_refuted
/-- info: 'INVERSION.it_from_it' does not depend on any axioms -/
#guard_msgs in #print axioms INVERSION.it_from_it
/-- info: 'K4.it_from_it_canonical' does not depend on any axioms -/
#guard_msgs in #print axioms K4.it_from_it_canonical
/-- info: 'K4.it_from_bit_refuted_canonical' does not depend on any axioms -/
#guard_msgs in #print axioms K4.it_from_bit_refuted_canonical
/-- info: 'K4.inversionClaim' does not depend on any axioms -/
#guard_msgs in #print axioms K4.inversionClaim
/-- info: 'GEO.qmul' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.qmul
/-- info: 'GEO.qconj' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.qconj
/-- info: 'GEO.qadd' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.qadd
/-- info: 'GEO.qsub' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.qsub
/-- info: 'GEO.qnorm2' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.qnorm2
/-- info: 'GEO.qhalve' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.qhalve
/-- info: 'GEO.qquarter' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.qquarter
/-- info: 'GEO.qcanon' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.qcanon
/-- info: 'GEO.q1' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.q1
/-- info: 'GEO.qi' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.qi
/-- info: 'GEO.qj' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.qj
/-- info: 'GEO.qk' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.qk
/-- info: 'GEO.chirality' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.chirality
/-- info: 'GEO.noncommutative' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.noncommutative
/-- info: 'GEO.hurwitz' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.hurwitz
/-- info: 'GEO.hurwitz_norm' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.hurwitz_norm
/-- info: 'GEO.hurwitz_products_even' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.hurwitz_products_even
/-- info: 'GEO.hurwitz_closed' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.hurwitz_closed
/-- info: 'GEO.hurwitzReps' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.hurwitzReps
/-- info: 'GEO.units_mod_sign_twelve' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.units_mod_sign_twelve
/-- info: 'GEO.conjClass' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.conjClass
/-- info: 'GEO.classSizesAux' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.classSizesAux
/-- info: 'GEO.insNat' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.insNat
/-- info: 'GEO.insSort' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.insSort
/-- info: 'GEO.class_equation' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.class_equation
/-- info: 'GEO.shell2' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.shell2
/-- info: 'GEO.shell_twentyfour' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.shell_twentyfour
/-- info: 'GEO.no_half_integer_norm2' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.no_half_integer_norm2
/-- info: 'GEO.allGates' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.allGates
/-- info: 'GEO.gates_twelve' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.gates_twelve
/-- info: 'GEO.gates_valid' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.gates_valid
/-- info: 'GEO.gates_complete' depends on axioms: [propext, Quot.sound] -/
#guard_msgs in #print axioms GEO.gates_complete
/-- info: 'GEO.invCount' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.invCount
/-- info: 'GEO.perms24' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.perms24
/-- info: 'GEO.a4' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.a4
/-- info: 'GEO.a4_order' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.a4_order
/-- info: 'GEO.actG' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.actG
/-- info: 'GEO.a4_simply_transitive' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.a4_simply_transitive
/-- info: 'GEO.tetraVerts' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.tetraVerts
/-- info: 'GEO.tetraEdges' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.tetraEdges
/-- info: 'GEO.tetraFaces' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.tetraFaces
/-- info: 'GEO.euler_closure' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.euler_closure
/-- info: 'GEO.basisU' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.basisU
/-- info: 'GEO.perms6' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.perms6
/-- info: 'GEO.relabel_parity' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.relabel_parity
/-- info: 'GEO.O8' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.O8
/-- info: 'GEO.omul' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.omul
/-- info: 'GEO.osub' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.osub
/-- info: 'GEO.onorm2' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.onorm2
/-- info: 'GEO.e8' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.e8
/-- info: 'GEO.octonion_associator' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.octonion_associator
/-- info: 'GEO.hPairs' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.hPairs
/-- info: 'GEO.norm_composition_H' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.norm_composition_H
/-- info: 'GEO.oPairs' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.oPairs
/-- info: 'GEO.norm_composition_O' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.norm_composition_O
/-- info: 'GEO.det2' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.det2
/-- info: 'GEO.gf2Mul' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.gf2Mul
/-- info: 'GEO.gf2Vecs' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.gf2Vecs
/-- info: 'GEO.gf2Sols' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.gf2Sols
/-- info: 'GEO.triaxial_lock_gf2' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.triaxial_lock_gf2
/-- info: 'GEO.triaxial_lock_gf2_count' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.triaxial_lock_gf2_count
/-- info: 'GEO.triaxial_open_gf2_count' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.triaxial_open_gf2_count
/-- info: 'GEO.V3' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.V3
/-- info: 'GEO.M3' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.M3
/-- info: 'GEO.det3' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.det3
/-- info: 'GEO.adj3' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.adj3
/-- info: 'GEO.mm3' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.mm3
/-- info: 'GEO.diag3' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.diag3
/-- info: 'GEO.mv3' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.mv3
/-- info: 'GEO.vscale' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.vscale
/-- info: 'GEO.V3Z' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.V3Z
/-- info: 'GEO.M3Z' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.M3Z
/-- info: 'GEO.det3Z' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.det3Z
/-- info: 'GEO.adj3Z' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.adj3Z
/-- info: 'GEO.mm3Z' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.mm3Z
/-- info: 'GEO.diag3Z' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.diag3Z
/-- info: 'GEO.adjugate_identity_int' depends on axioms: [propext, Quot.sound] -/
#guard_msgs in #print axioms GEO.adjugate_identity_int
/-- info: 'GEO.AdjugateIdentity3' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.AdjugateIdentity3
/-- info: 'GEO.adj3Samples' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.adj3Samples
/-- info: 'GEO.adj3_executed' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.adj3_executed
/-- info: 'GEO.cramerSolve' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.cramerSolve
/-- info: 'GEO.cramer_executed' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.cramer_executed
/-- info: 'GEO.golAdmit' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.golAdmit
/-- info: 'GEO.gateNames' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.gateNames
/-- info: 'GEO.gateNames_twelve' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.gateNames_twelve
/-- info: 'GEO.gateScreenGo' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.gateScreenGo
/-- info: 'GEO.gateScreen' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.gateScreen
/-- info: 'GEO.gateScreen_clean' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.gateScreen_clean
/-- info: 'GEO.gateScreen_first_failure' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.gateScreen_first_failure
/-- info: 'GEO.gateScreen_names_failed_gate' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.gateScreen_names_failed_gate
/-- info: 'GEO.gateScreen_arity' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.gateScreen_arity
/-- info: 'GEO.ClaimInputRA.anyUnknown' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.ClaimInputRA.anyUnknown
/-- info: 'GEO.rowCascadeRACore' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.rowCascadeRACore
/-- info: 'GEO.rowCascadeRA' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.rowCascadeRA
/-- info: 'GEO.unknown_routes_open_ra' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.unknown_routes_open_ra
/-- info: 'GEO.raDecode' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.raDecode
/-- info: 'GEO.raBattery' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.raBattery
/-- info: 'GEO.ra_battery_f2' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.ra_battery_f2
/-- info: 'GEO.ra_ledger_refused' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.ra_ledger_refused
/-- info: 'GEO.ra_ledger_iii' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.ra_ledger_iii
/-- info: 'GEO.ra_ledger_ii' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.ra_ledger_ii
/-- info: 'GEO.ra_ledger_void' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.ra_ledger_void
/-- info: 'GEO.ra_ledger_aGivenRA' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.ra_ledger_aGivenRA
/-- info: 'GEO.RAVerdict.toToken' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.RAVerdict.toToken
/-- info: 'GEO.GolToken.toToken' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.GolToken.toToken
/-- info: 'GEO.gateScreenToken' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.gateScreenToken
/-- info: 'GEO.demoInputA' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.demoInputA
/-- info: 'GEO.demoInputVoid' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.demoInputVoid
/-- info: 'GEO.demoInputOpen' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.demoInputOpen
/-- info: 'ROOT.Actuates' depends on axioms: [ROOT.U, ROOT.ΔE] -/
#guard_msgs in #print axioms ROOT.Actuates
/-- info: 'ROOT.actuation_universal' depends on axioms: [ROOT.RA, ROOT.U, ROOT.ΔE] -/
#guard_msgs in #print axioms ROOT.actuation_universal
/-- info: 'ROOT.raClaim' depends on axioms: [ROOT.RA, ROOT.U, ROOT.ΔE] -/
#guard_msgs in #print axioms ROOT.raClaim
/-- info: 'ROOT.boolCut' does not depend on any axioms -/
#guard_msgs in #print axioms ROOT.boolCut
/-- info: 'ROOT.nest_31' depends on axioms: [ROOT.FormalDomain, ROOT.L1m, ROOT.L3m, ROOT.nest_21, ROOT.nest_32] -/
#guard_msgs in #print axioms ROOT.nest_31
/-- info: 'ROOT.nestClaim' depends on axioms: [ROOT.FormalDomain, ROOT.L1m, ROOT.L3m, ROOT.nest_21, ROOT.nest_32] -/
#guard_msgs in #print axioms ROOT.nestClaim
/-- info: 'ROOT.throneEmpty' depends on axioms: [ROOT.U, ROOT.ΔE] -/
#guard_msgs in #print axioms ROOT.throneEmpty
/-- info: 'ROOT.throne_is_RA' depends on axioms: [ROOT.U, ROOT.ΔE] -/
#guard_msgs in #print axioms ROOT.throne_is_RA
/-- info: 'SEALS.sealL' does not depend on any axioms -/
#guard_msgs in #print axioms SEALS.sealL
/-- info: 'SEALS.sealG' depends on axioms: [ROOT.RA, ROOT.U, ROOT.ΔE] -/
#guard_msgs in #print axioms SEALS.sealG
/-- info: 'SEALS.sealM' does not depend on any axioms -/
#guard_msgs in #print axioms SEALS.sealM
/-- info: 'SEALS.seals_distinct' does not depend on any axioms -/
#guard_msgs in #print axioms SEALS.seals_distinct
/-- info: 'SEALS.seal_names_match' depends on axioms: [ROOT.RA, ROOT.U, ROOT.ΔE] -/
#guard_msgs in #print axioms SEALS.seal_names_match
/-- info: 'SEALS.sealG_walk' depends on axioms: [propext] -/
#guard_msgs in #print axioms SEALS.sealG_walk
/-- info: 'POSTULATE.OneCut' depends on axioms: [POSTULATE.Row, POSTULATE.SupplyClean] -/
#guard_msgs in #print axioms POSTULATE.OneCut
/-- info: 'POSTULATE.F2' depends on axioms: [POSTULATE.Row, POSTULATE.SupplyClean] -/
#guard_msgs in #print axioms POSTULATE.F2
/-- info: 'POSTULATE.F2_kills' depends on axioms: [POSTULATE.Row, POSTULATE.SupplyClean] -/
#guard_msgs in #print axioms POSTULATE.F2_kills
/-- info: 'POSTULATE.BSDContract' depends on axioms: [POSTULATE.Algorithm, POSTULATE.FrozenData, POSTULATE.certifiedPairing] -/
#guard_msgs in #print axioms POSTULATE.BSDContract
/-- info: 'POSTULATE.F6' depends on axioms: [POSTULATE.Algorithm, POSTULATE.FrozenData, POSTULATE.certifiedPairing] -/
#guard_msgs in #print axioms POSTULATE.F6
/-- info: 'TONGUE.enlarge' does not depend on any axioms -/
#guard_msgs in #print axioms TONGUE.enlarge
/-- info: 'TONGUE.unclosed' does not depend on any axioms -/
#guard_msgs in #print axioms TONGUE.unclosed
/-- info: 'TONGUE.earned_freedom' does not depend on any axioms -/
#guard_msgs in #print axioms TONGUE.earned_freedom
/-- info: 'TONGUE.colocClaim' depends on axioms: [ROOT.RA, ROOT.U, ROOT.ΔE] -/
#guard_msgs in #print axioms TONGUE.colocClaim
/-- info: 'TONGUE.coloc_grade' depends on axioms: [ROOT.RA, ROOT.U, ROOT.ΔE] -/
#guard_msgs in #print axioms TONGUE.coloc_grade
/-- info: 'TONGUE.ra_grade_pinned' depends on axioms: [ROOT.RA, ROOT.U, ROOT.ΔE] -/
#guard_msgs in #print axioms TONGUE.ra_grade_pinned
/-- info: 'TONGUE.wall_grade_pinned' does not depend on any axioms -/
#guard_msgs in #print axioms TONGUE.wall_grade_pinned
/-- info: 'TONGUE.inversion_grade_pinned' does not depend on any axioms -/
#guard_msgs in #print axioms TONGUE.inversion_grade_pinned
/-- info: 'TONGUE.nest_grade_pinned' depends on axioms: [ROOT.FormalDomain, ROOT.L1m, ROOT.L3m, ROOT.nest_21, ROOT.nest_32] -/
#guard_msgs in #print axioms TONGUE.nest_grade_pinned
/-- info: 'terminalWall' does not depend on any axioms -/
#guard_msgs in #print axioms terminalWall
/-- info: 'terminalPrice' depends on axioms: [Quot.sound] -/
#guard_msgs in #print axioms terminalPrice
/-- info: 'TERMINAL' depends on axioms: [Quot.sound] -/
#guard_msgs in #print axioms TERMINAL
/-- info: 'terminalClaim' depends on axioms: [Quot.sound] -/
#guard_msgs in #print axioms terminalClaim
/-- info: 'terminal_grade_pinned' depends on axioms: [Quot.sound] -/
#guard_msgs in #print axioms terminal_grade_pinned
/-- info: 'IAM.iamToken' does not depend on any axioms -/
#guard_msgs in #print axioms IAM.iamToken
/-- info: 'IAM.iam_unwitnessed_withheld' does not depend on any axioms -/
#guard_msgs in #print axioms IAM.iam_unwitnessed_withheld
/-- info: 'IAM.iam_witnessed_speaks' does not depend on any axioms -/
#guard_msgs in #print axioms IAM.iam_witnessed_speaks
/-- info: 'IAM.narcissus_truth_table' does not depend on any axioms -/
#guard_msgs in #print axioms IAM.narcissus_truth_table
/-- info: 'IAM.one_bit_freedom' does not depend on any axioms -/
#guard_msgs in #print axioms IAM.one_bit_freedom
/-- info: 'IAM.IamToken.toToken' does not depend on any axioms -/
#guard_msgs in #print axioms IAM.IamToken.toToken
/-- info: 'IAM.codexSelfRead' does not depend on any axioms -/
#guard_msgs in #print axioms IAM.codexSelfRead
/-- info: 'IAM.codex_self_read_interior' does not depend on any axioms -/
#guard_msgs in #print axioms IAM.codex_self_read_interior
/-- info: 'IAM.terminalWord' does not depend on any axioms -/
#guard_msgs in #print axioms IAM.terminalWord
/-- info: 'IAM.terminal_word' does not depend on any axioms -/
#guard_msgs in #print axioms IAM.terminal_word
/-- info: 'GEO.SignPattern' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.SignPattern
/-- info: 'GEO.allSigns' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.allSigns
/-- info: 'GEO.signComp' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.signComp
/-- info: 'GEO.reflection_eight' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.reflection_eight
/-- info: 'GEO.reflection_eight_is_two_cubed' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.reflection_eight_is_two_cubed
/-- info: 'GEO.reflections_abelian' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.reflections_abelian
/-- info: 'GEO.reflections_involutive' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.reflections_involutive
/-- info: 'GEO.signVal' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.signVal
/-- info: 'GEO.reflMat' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.reflMat
/-- info: 'GEO.reflection_reverses_orientation' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.reflection_reverses_orientation
/-- info: 'GEO.sandwich' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.sandwich
/-- info: 'GEO.reflection_blind_executed' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.reflection_blind_executed
/-- info: 'GEO.V5' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.V5
/-- info: 'GEO.cube5' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.cube5
/-- info: 'GEO.neighbors5' does not depend on any axioms -/
#guard_msgs in #print axioms GEO.neighbors5
/-- info: 'GEO.fivecube_thirtytwo' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.fivecube_thirtytwo
/-- info: 'GEO.fivecube_vertices_distinct' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.fivecube_vertices_distinct
/-- info: 'GEO.fivecube_degree_five' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.fivecube_degree_five
/-- info: 'GEO.counts_forced' depends on axioms: [propext] -/
#guard_msgs in #print axioms GEO.counts_forced
/-- info: 'TONGUE.deletionOp' depends on axioms: [propext] -/
#guard_msgs in #print axioms TONGUE.deletionOp
/-- info: 'TONGUE.e3' depends on axioms: [propext] -/
#guard_msgs in #print axioms TONGUE.e3
/-- info: 'TONGUE.deletion_idempotent' depends on axioms: [propext] -/
#guard_msgs in #print axioms TONGUE.deletion_idempotent
/-- info: 'TONGUE.deletion_singular' depends on axioms: [propext] -/
#guard_msgs in #print axioms TONGUE.deletion_singular
/-- info: 'TONGUE.deletion_minor_nonsingular' depends on axioms: [propext] -/
#guard_msgs in #print axioms TONGUE.deletion_minor_nonsingular
/-- info: 'TONGUE.deletion_annihilates' depends on axioms: [propext] -/
#guard_msgs in #print axioms TONGUE.deletion_annihilates
/-- info: 'TONGUE.linear_fixes_zero' depends on axioms: [propext, Quot.sound] -/
#guard_msgs in #print axioms TONGUE.linear_fixes_zero
/-- info: 'TONGUE.deletion_no_left_inverse' depends on axioms: [propext, Quot.sound] -/
#guard_msgs in #print axioms TONGUE.deletion_no_left_inverse
/-- info: 'TONGUE.deletion_monoid_not_group' depends on axioms: [propext, Quot.sound] -/
#guard_msgs in #print axioms TONGUE.deletion_monoid_not_group
/-- info: 'TONGUE.tongue_obedience' depends on axioms: [propext, Quot.sound] -/
#guard_msgs in #print axioms TONGUE.tongue_obedience
/-- info: 'OFFICE.discriminator' does not depend on any axioms -/
#guard_msgs in #print axioms OFFICE.discriminator
/-- info: 'OFFICE.discriminatorQ2First' does not depend on any axioms -/
#guard_msgs in #print axioms OFFICE.discriminatorQ2First
/-- info: 'OFFICE.order_loadbearing' does not depend on any axioms -/
#guard_msgs in #print axioms OFFICE.order_loadbearing
/-- info: 'OFFICE.citation_cannot_promote' does not depend on any axioms -/
#guard_msgs in #print axioms OFFICE.citation_cannot_promote
/-- info: 'UC.routeHalt' does not depend on any axioms -/
#guard_msgs in #print axioms UC.routeHalt
/-- info: 'UC.routeHalt_fidelity' does not depend on any axioms -/
#guard_msgs in #print axioms UC.routeHalt_fidelity
/-- info: 'UC.universalCascade' does not depend on any axioms -/
#guard_msgs in #print axioms UC.universalCascade
/-- info: 'UC.uc_seal_failure_terminal' does not depend on any axioms -/
#guard_msgs in #print axioms UC.uc_seal_failure_terminal
/-- info: 'UC.uc_routes_eight' does not depend on any axioms -/
#guard_msgs in #print axioms UC.uc_routes_eight
/-- info: 'UC.uc_routes_five' does not depend on any axioms -/
#guard_msgs in #print axioms UC.uc_routes_five
/-- info: 'UC.uc_unmeasured' does not depend on any axioms -/
#guard_msgs in #print axioms UC.uc_unmeasured
/-- info: 'UC.circularityScreen' does not depend on any axioms -/
#guard_msgs in #print axioms UC.circularityScreen
/-- info: 'UC.circularity_refused_at_target' does not depend on any axioms -/
#guard_msgs in #print axioms UC.circularity_refused_at_target
/-- info: 'UC.road_requires_conjunction' does not depend on any axioms -/
#guard_msgs in #print axioms UC.road_requires_conjunction
/-- info: 'UC.circRows' does not depend on any axioms -/
#guard_msgs in #print axioms UC.circRows
/-- info: 'UC.circ_space' does not depend on any axioms -/
#guard_msgs in #print axioms UC.circ_space
/-- info: 'OMEGA.kbRat' depends on axioms: [propext] -/
#guard_msgs in #print axioms OMEGA.kbRat
/-- info: 'OMEGA.ln2Rat' depends on axioms: [propext] -/
#guard_msgs in #print axioms OMEGA.ln2Rat
/-- info: 'OMEGA.landauerRat' depends on axioms: [propext] -/
#guard_msgs in #print axioms OMEGA.landauerRat
/-- info: 'OMEGA.omegaBoundary' depends on axioms: [propext] -/
#guard_msgs in #print axioms OMEGA.omegaBoundary
/-- info: 'OMEGA.omega_zero_bits' depends on axioms: [propext] -/
#guard_msgs in #print axioms OMEGA.omega_zero_bits
/-- info: 'OMEGA.omega_paid_floor' depends on axioms: [propext] -/
#guard_msgs in #print axioms OMEGA.omega_paid_floor
/-- info: 'OMEGA.one_bit_freedom_priced' depends on axioms: [propext] -/
#guard_msgs in #print axioms OMEGA.one_bit_freedom_priced
/-- info: 'OMEGA.omega_reversible_branch' depends on axioms: [propext] -/
#guard_msgs in #print axioms OMEGA.omega_reversible_branch
/-- info: 'OMEGA.omega_refuses_nonpositive_temp' depends on axioms: [propext] -/
#guard_msgs in #print axioms OMEGA.omega_refuses_nonpositive_temp
/-- info: 'OMEGA.omega_monotone' depends on axioms: [propext] -/
#guard_msgs in #print axioms OMEGA.omega_monotone
/-- info: 'OMEGA.omega_linear' depends on axioms: [propext] -/
#guard_msgs in #print axioms OMEGA.omega_linear
/-- info: 'OMEGA.landauer_zero_bits' depends on axioms: [propext] -/
#guard_msgs in #print axioms OMEGA.landauer_zero_bits
/-- info: 'OMEGA.omega_verdict_fidelity' depends on axioms: [propext] -/
#guard_msgs in #print axioms OMEGA.omega_verdict_fidelity
/-- info: 'AEGIS.refusalConst' does not depend on any axioms -/
#guard_msgs in #print axioms AEGIS.refusalConst
/-- info: 'AEGIS.aegisReset' does not depend on any axioms -/
#guard_msgs in #print axioms AEGIS.aegisReset
/-- info: 'AEGIS.aegisGuard' does not depend on any axioms -/
#guard_msgs in #print axioms AEGIS.aegisGuard
/-- info: 'AEGIS.aegis_constant' does not depend on any axioms -/
#guard_msgs in #print axioms AEGIS.aegis_constant
/-- info: 'AEGIS.aegis_deed_increments' does not depend on any axioms -/
#guard_msgs in #print axioms AEGIS.aegis_deed_increments
/-- info: 'AEGIS.aegis_reads_parameter' does not depend on any axioms -/
#guard_msgs in #print axioms AEGIS.aegis_reads_parameter
/-- info: 'AEGIS.aegisFourLogicRun' does not depend on any axioms -/
#guard_msgs in #print axioms AEGIS.aegisFourLogicRun
/-- info: 'AEGIS.aegis_battery' does not depend on any axioms -/
#guard_msgs in #print axioms AEGIS.aegis_battery
/-- info: 'NOMOS.allStrings' does not depend on any axioms -/
#guard_msgs in #print axioms NOMOS.allStrings
/-- info: 'NOMOS.allStrings_length' depends on axioms: [propext] -/
#guard_msgs in #print axioms NOMOS.allStrings_length
/-- info: 'NOMOS.twoPowPos' does not depend on any axioms -/
#guard_msgs in #print axioms NOMOS.twoPowPos
/-- info: 'NOMOS.descs' does not depend on any axioms -/
#guard_msgs in #print axioms NOMOS.descs
/-- info: 'NOMOS.descs_length' depends on axioms: [propext] -/
#guard_msgs in #print axioms NOMOS.descs_length
/-- info: 'NOMOS.incompressibility_cap' depends on axioms: [propext] -/
#guard_msgs in #print axioms NOMOS.incompressibility_cap
/-- info: 'NOMOS.nomosRows' does not depend on any axioms -/
#guard_msgs in #print axioms NOMOS.nomosRows
/-- info: 'NOMOS.incompressibility_executed' does not depend on any axioms -/
#guard_msgs in #print axioms NOMOS.incompressibility_executed
/-- info: 'NOMOS.decodeId' does not depend on any axioms -/
#guard_msgs in #print axioms NOMOS.decodeId
/-- info: 'NOMOS.decodePad' does not depend on any axioms -/
#guard_msgs in #print axioms NOMOS.decodePad
/-- info: 'NOMOS.compressible_bound_universal' depends on axioms: [propext] -/
#guard_msgs in #print axioms NOMOS.compressible_bound_universal
/-- info: 'NOMOS.cbUniversal' depends on axioms: [propext] -/
#guard_msgs in #print axioms NOMOS.cbUniversal
/-- info: 'NOMOS.cb_id_8_2' does not depend on any axioms -/
#guard_msgs in #print axioms NOMOS.cb_id_8_2
/-- info: 'NOMOS.cb_pad3_8_2' does not depend on any axioms -/
#guard_msgs in #print axioms NOMOS.cb_pad3_8_2
/-- info: 'NOMOS.cb_pad2_6_1' does not depend on any axioms -/
#guard_msgs in #print axioms NOMOS.cb_pad2_6_1
/-- info: 'NOMOS.cb_pad4_12_3' does not depend on any axioms -/
#guard_msgs in #print axioms NOMOS.cb_pad4_12_3
/-- info: 'NOMOS.recoverable' depends on axioms: [propext] -/
#guard_msgs in #print axioms NOMOS.recoverable
/-- info: 'NOMOS.nomos_fuel_is_omega_floor' depends on axioms: [propext] -/
#guard_msgs in #print axioms NOMOS.nomos_fuel_is_omega_floor
/-- info: 'NOMOS.nomosScreen' does not depend on any axioms -/
#guard_msgs in #print axioms NOMOS.nomosScreen
/-- info: 'NOMOS.nomosScreen_battery' does not depend on any axioms -/
#guard_msgs in #print axioms NOMOS.nomosScreen_battery
/-- info: 'BRIDGE.ceiling' does not depend on any axioms -/
#guard_msgs in #print axioms BRIDGE.ceiling
/-- info: 'BRIDGE.census' does not depend on any axioms -/
#guard_msgs in #print axioms BRIDGE.census
/-- info: 'BRIDGE.census_count' does not depend on any axioms -/
#guard_msgs in #print axioms BRIDGE.census_count
/-- info: 'BRIDGE.post_seal_disposition' does not depend on any axioms -/
#guard_msgs in #print axioms BRIDGE.post_seal_disposition
/-- info: 'BRIDGE.weakest_never_above_left' does not depend on any axioms -/
#guard_msgs in #print axioms BRIDGE.weakest_never_above_left
/-- info: 'BRIDGE.bridge_ceiling_caps' does not depend on any axioms -/
#guard_msgs in #print axioms BRIDGE.bridge_ceiling_caps
/-- info: 'BRIDGE.successorOf' does not depend on any axioms -/
#guard_msgs in #print axioms BRIDGE.successorOf
/-- info: 'BRIDGE.supersession_routes' does not depend on any axioms -/
#guard_msgs in #print axioms BRIDGE.supersession_routes
/-- info: 'BRIDGE.bridgeScreen' does not depend on any axioms -/
#guard_msgs in #print axioms BRIDGE.bridgeScreen
/-- info: 'BRIDGE.bridgeScreen_total' does not depend on any axioms -/
#guard_msgs in #print axioms BRIDGE.bridgeScreen_total
/-- info: 'FOUND.sigmaP' depends on axioms: [propext] -/
#guard_msgs in #print axioms FOUND.sigmaP
/-- info: 'FOUND.pluralist_alternative_executed' depends on axioms: [propext] -/
#guard_msgs in #print axioms FOUND.pluralist_alternative_executed
/-- info: 'FOUND.posits' does not depend on any axioms -/
#guard_msgs in #print axioms FOUND.posits
/-- info: 'FOUND.posits_loadbearing_on_nothing' does not depend on any axioms -/
#guard_msgs in #print axioms FOUND.posits_loadbearing_on_nothing
/-- info: 'FOUND.gradeOfRank' does not depend on any axioms -/
#guard_msgs in #print axioms FOUND.gradeOfRank
/-- info: 'FOUND.posit_never_raises' does not depend on any axioms -/
#guard_msgs in #print axioms FOUND.posit_never_raises
/-- info: 'FOUND.bandOf' does not depend on any axioms -/
#guard_msgs in #print axioms FOUND.bandOf
/-- info: 'FOUND.register_routing_fidelity' does not depend on any axioms -/
#guard_msgs in #print axioms FOUND.register_routing_fidelity
/-- info: 'FOUND.verdictInventory' does not depend on any axioms -/
#guard_msgs in #print axioms FOUND.verdictInventory
/-- info: 'FOUND.verdicts_posit_free' does not depend on any axioms -/
#guard_msgs in #print axioms FOUND.verdicts_posit_free
/-- info: 'FOUND.engine_never_authority' does not depend on any axioms -/
#guard_msgs in #print axioms FOUND.engine_never_authority
/-- info: 'HALT.cascade' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.cascade
/-- info: 'HALT.patterns' depends on axioms: [propext] -/
#guard_msgs in #print axioms HALT.patterns
/-- info: 'HALT.fftSpec' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.fftSpec
/-- info: 'HALT.first_failure_terminal_5' depends on axioms: [propext] -/
#guard_msgs in #print axioms HALT.first_failure_terminal_5
/-- info: 'HALT.first_failure_terminal_7' depends on axioms: [propext] -/
#guard_msgs in #print axioms HALT.first_failure_terminal_7
/-- info: 'HALT.first_failure_terminal_8' depends on axioms: [propext] -/
#guard_msgs in #print axioms HALT.first_failure_terminal_8
/-- info: 'HALT.first_failure_terminal_12' depends on axioms: [propext] -/
#guard_msgs in #print axioms HALT.first_failure_terminal_12
/-- info: 'HALT.halt_free_iff_all_pass_8' depends on axioms: [propext] -/
#guard_msgs in #print axioms HALT.halt_free_iff_all_pass_8
/-- info: 'HALT.gate_counts_recorded' depends on axioms: [propext] -/
#guard_msgs in #print axioms HALT.gate_counts_recorded
/-- info: 'HALT.signPatterns' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.signPatterns
/-- info: 'HALT.ninth_gate_barred' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.ninth_gate_barred
/-- info: 'HALT.route' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.route
/-- info: 'HALT.ofUC' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.ofUC
/-- info: 'HALT.route_is_routeHalt' depends on axioms: [propext, Quot.sound] -/
#guard_msgs in #print axioms HALT.route_is_routeHalt
/-- info: 'HALT.route_samples' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.route_samples
/-- info: 'HALT.tokenBranch' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.tokenBranch
/-- info: 'HALT.router_exclusive' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.router_exclusive
/-- info: 'HALT.readingRoadsGate' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.readingRoadsGate
/-- info: 'HALT.one_door_open' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.one_door_open
/-- info: 'HALT.no_walk_passes_the_aperture' depends on axioms: [propext] -/
#guard_msgs in #print axioms HALT.no_walk_passes_the_aperture
/-- info: 'HALT.xiGates' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.xiGates
/-- info: 'HALT.oGates' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.oGates
/-- info: 'HALT.gate_censuses' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.gate_censuses
/-- info: 'HALT.rhHalt' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.rhHalt
/-- info: 'HALT.pnpHalt' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.pnpHalt
/-- info: 'HALT.halts_routed' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.halts_routed
/-- info: 'HALT.legacy_halts_are_instances' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.legacy_halts_are_instances
/-- info: 'HALT.terminal_one_bit' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.terminal_one_bit
/-- info: 'HALT.species_exhausted' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.species_exhausted
/-- info: 'HALT.haltGrades' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.haltGrades
/-- info: 'HALT.ftoe_ceiling' does not depend on any axioms -/
#guard_msgs in #print axioms HALT.ftoe_ceiling
/-- info: 'CROSSING.ExecFrame' does not depend on any axioms -/
#guard_msgs in #print axioms CROSSING.ExecFrame
/-- info: 'CROSSING.execFlip' does not depend on any axioms -/
#guard_msgs in #print axioms CROSSING.execFlip
/-- info: 'CROSSING.formalRead' does not depend on any axioms -/
#guard_msgs in #print axioms CROSSING.formalRead
/-- info: 'CROSSING.ran' does not depend on any axioms -/
#guard_msgs in #print axioms CROSSING.ran
/-- info: 'CROSSING.formalRead_even' does not depend on any axioms -/
#guard_msgs in #print axioms CROSSING.formalRead_even
/-- info: 'CROSSING.ran_wholly_odd' does not depend on any axioms -/
#guard_msgs in #print axioms CROSSING.ran_wholly_odd
/-- info: 'CROSSING.execFlip_involution' does not depend on any axioms -/
#guard_msgs in #print axioms CROSSING.execFlip_involution
/-- info: 'CROSSING.ran_odd_at_true' does not depend on any axioms -/
#guard_msgs in #print axioms CROSSING.ran_odd_at_true
/-- info: 'CROSSING.wall' does not depend on any axioms -/
#guard_msgs in #print axioms CROSSING.wall
/-- info: 'CROSSING.deed' does not depend on any axioms -/
#guard_msgs in #print axioms CROSSING.deed
/-- info: 'CROSSING.price' depends on axioms: [Quot.sound] -/
#guard_msgs in #print axioms CROSSING.price
/-- info: 'CROSSING.crossing' does not depend on any axioms -/
#guard_msgs in #print axioms CROSSING.crossing
/-- info: 'CROSSING.identity_one_bit' does not depend on any axioms -/
#guard_msgs in #print axioms CROSSING.identity_one_bit
/-- info: 'CROSSING.fibre4' does not depend on any axioms -/
#guard_msgs in #print axioms CROSSING.fibre4
/-- info: 'CROSSING.fibre4_sixteen' does not depend on any axioms -/
#guard_msgs in #print axioms CROSSING.fibre4_sixteen
/-- info: 'CROSSING.fibre4_distinct' does not depend on any axioms -/
#guard_msgs in #print axioms CROSSING.fibre4_distinct
/-- info: 'CROSSING.price_on_this_file' depends on axioms: [Quot.sound] -/
#guard_msgs in #print axioms CROSSING.price_on_this_file
/-- info: 'CROSSING.formal_never_odd' does not depend on any axioms -/
#guard_msgs in #print axioms CROSSING.formal_never_odd
/-- info: 'CROSSING.constant_no_crossing' does not depend on any axioms -/
#guard_msgs in #print axioms CROSSING.constant_no_crossing
/-- info: 'CROSSING.harmony' depends on axioms: [propext] -/
#guard_msgs in #print axioms CROSSING.harmony
/-- info: 'LOCUS.Separates' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.Separates
/-- info: 'LOCUS.Factors' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.Factors
/-- info: 'LOCUS.wall_of_sep' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.wall_of_sep
/-- info: 'LOCUS.swap' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.swap
/-- info: 'LOCUS.swap_involution' depends on axioms: [propext] -/
#guard_msgs in #print axioms LOCUS.swap_involution
/-- info: 'LOCUS.swap_record_even' depends on axioms: [propext] -/
#guard_msgs in #print axioms LOCUS.swap_record_even
/-- info: 'LOCUS.swap_flips_pair' depends on axioms: [propext] -/
#guard_msgs in #print axioms LOCUS.swap_flips_pair
/-- info: 'LOCUS.seat_of_sep' depends on axioms: [propext] -/
#guard_msgs in #print axioms LOCUS.seat_of_sep
/-- info: 'LOCUS.k4_separated_pair' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.k4_separated_pair
/-- info: 'LOCUS.k4_wall_of_pair' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.k4_wall_of_pair
/-- info: 'LOCUS.readout' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.readout
/-- info: 'LOCUS.factors_of_not_sep_list' depends on axioms: [propext, Quot.sound] -/
#guard_msgs in #print axioms LOCUS.factors_of_not_sep_list
/-- info: 'LOCUS.factors_iff_not_sep_list' depends on axioms: [propext, Quot.sound] -/
#guard_msgs in #print axioms LOCUS.factors_iff_not_sep_list
/-- info: 'LOCUS.Channel' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.Channel
/-- info: 'LOCUS.Token.toEconomy' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.Token.toEconomy
/-- info: 'LOCUS.Reading.orbits' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.Reading.orbits
/-- info: 'LOCUS.Reading.owed' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.Reading.owed
/-- info: 'LOCUS.Reading.complete' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.Reading.complete
/-- info: 'LOCUS.branchOf' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.branchOf
/-- info: 'LOCUS.emit' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.emit
/-- info: 'LOCUS.Reading.state' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.Reading.state
/-- info: 'LOCUS.three_states' depends on axioms: [propext] -/
#guard_msgs in #print axioms LOCUS.three_states
/-- info: 'LOCUS.no_seat_no_locus' depends on axioms: [propext] -/
#guard_msgs in #print axioms LOCUS.no_seat_no_locus
/-- info: 'LOCUS.unlocated_is_plain_open' depends on axioms: [propext] -/
#guard_msgs in #print axioms LOCUS.unlocated_is_plain_open
/-- info: 'LOCUS.located_never_silent' depends on axioms: [propext] -/
#guard_msgs in #print axioms LOCUS.located_never_silent
/-- info: 'LOCUS.crossed_requires_complete' depends on axioms: [propext] -/
#guard_msgs in #print axioms LOCUS.crossed_requires_complete
/-- info: 'LOCUS.crossed_requires_channel' depends on axioms: [propext] -/
#guard_msgs in #print axioms LOCUS.crossed_requires_channel
/-- info: 'LOCUS.halt_carries_owed' depends on axioms: [propext] -/
#guard_msgs in #print axioms LOCUS.halt_carries_owed
/-- info: 'LOCUS.unchannelled_is_halt' depends on axioms: [propext, Quot.sound] -/
#guard_msgs in #print axioms LOCUS.unchannelled_is_halt
/-- info: 'LOCUS.halt_reports_owed' depends on axioms: [propext] -/
#guard_msgs in #print axioms LOCUS.halt_reports_owed
/-- info: 'LOCUS.owed_zero_is_unchannelled' depends on axioms: [propext, Quot.sound] -/
#guard_msgs in #print axioms LOCUS.owed_zero_is_unchannelled
/-- info: 'LOCUS.one_bit_apart' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.one_bit_apart
/-- info: 'LOCUS.legacy_xi' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.legacy_xi
/-- info: 'LOCUS.legacy_o' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.legacy_o
/-- info: 'LOCUS.legacy_bare' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.legacy_bare
/-- info: 'LOCUS.k4_locus' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.k4_locus
/-- info: 'LOCUS.slotLists' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.slotLists
/-- info: 'LOCUS.slotLists_twelve' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.slotLists_twelve
/-- info: 'LOCUS.slotLists_pos' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.slotLists_pos
/-- info: 'LOCUS.allReadings' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.allReadings
/-- info: 'LOCUS.reading_space' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.reading_space
/-- info: 'LOCUS.census_no_seat_no_locus' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.census_no_seat_no_locus
/-- info: 'LOCUS.census_located_never_silent' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.census_located_never_silent
/-- info: 'LOCUS.census_crossed_iff_complete_channelled' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.census_crossed_iff_complete_channelled
/-- info: 'LOCUS.census_halt_reports_owed' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.census_halt_reports_owed
/-- info: 'LOCUS.census_owed_zero_is_unchannelled' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.census_owed_zero_is_unchannelled
/-- info: 'LOCUS.census_three_states' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.census_three_states
/-- info: 'LOCUS.Token.render' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.Token.render
/-- info: 'LOCUS.Token.ascii' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.Token.ascii
/-- info: 'LOCUS.sealAscii' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.sealAscii
/-- info: 'LOCUS.render_receipted' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.render_receipted
/-- info: 'LOCUS.render_injective_in_owed' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.render_injective_in_owed
/-- info: 'LOCUS.voidImage' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.voidImage
/-- info: 'LOCUS.void_precedes_economy' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.void_precedes_economy
/-- info: 'LOCUS.legacy_records_not_promoted' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.legacy_records_not_promoted
/-- info: 'LOCUS.no_rh_locus' does not depend on any axioms -/
#guard_msgs in #print axioms LOCUS.no_rh_locus
/-- info: 'PSP.ORIENT.V3' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.ORIENT.V3
/-- info: 'PSP.ORIENT.M3' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.ORIENT.M3
/-- info: 'PSP.ORIENT.det3' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.ORIENT.det3
/-- info: 'PSP.ORIENT.neg3' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.ORIENT.neg3
/-- info: 'PSP.ORIENT.signMatrices' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.ORIENT.signMatrices
/-- info: 'PSP.ORIENT.lock_scalar_sign_blind' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.ORIENT.lock_scalar_sign_blind
/-- info: 'PSP.ORIENT.det_sign_flips_in_odd_dimension' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.ORIENT.det_sign_flips_in_odd_dimension
/-- info: 'PSP.QUAT.Q' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.QUAT.Q
/-- info: 'PSP.QUAT.qmul' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.QUAT.qmul
/-- info: 'PSP.QUAT.qi' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.QUAT.qi
/-- info: 'PSP.QUAT.qj' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.QUAT.qj
/-- info: 'PSP.QUAT.qk' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.QUAT.qk
/-- info: 'PSP.QUAT.qneg1' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.QUAT.qneg1
/-- info: 'PSP.QUAT.quat_basis_laws' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.QUAT.quat_basis_laws
/-- info: 'PSP.QUAT.qnorm' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.QUAT.qnorm
/-- info: 'PSP.QUAT.isInt' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.QUAT.isInt
/-- info: 'PSP.QUAT.isHalf' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.QUAT.isHalf
/-- info: 'PSP.QUAT.hurwitzOK' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.QUAT.hurwitzOK
/-- info: 'PSP.QUAT.grid' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.QUAT.grid
/-- info: 'PSP.QUAT.tuples4' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.QUAT.tuples4
/-- info: 'PSP.QUAT.hurwitz_units_24' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.QUAT.hurwitz_units_24
/-- info: 'PSP.QUAT.grid2' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.QUAT.grid2
/-- info: 'PSP.QUAT.tuples4i' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.QUAT.tuples4i
/-- info: 'PSP.QUAT.norm2_shell_split' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.QUAT.norm2_shell_split
/-- info: 'PSP.CD.cdconj' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.CD.cdconj
/-- info: 'PSP.CD.padd' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.CD.padd
/-- info: 'PSP.CD.psub' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.CD.psub
/-- info: 'PSP.CD.cdmul' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.CD.cdmul
/-- info: 'PSP.CD.e' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.CD.e
/-- info: 'PSP.CD.smul' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.CD.smul
/-- info: 'PSP.CD.octonion_associator' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.CD.octonion_associator
/-- info: 'PSP.CD.sx' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.CD.sx
/-- info: 'PSP.CD.sy' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.CD.sy
/-- info: 'PSP.CD.zero16' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.CD.zero16
/-- info: 'PSP.CD.sqnorm' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.CD.sqnorm
/-- info: 'PSP.CD.sedenion_zero_divisor' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.CD.sedenion_zero_divisor
/-- info: 'PSP.CASCADE.cascade' does not depend on any axioms -/
#guard_msgs in #print axioms PSP.CASCADE.cascade
/-- info: 'PSP.CASCADE.patterns8' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.CASCADE.patterns8
/-- info: 'PSP.CASCADE.spec' does not depend on any axioms -/
#guard_msgs in #print axioms PSP.CASCADE.spec
/-- info: 'PSP.CASCADE.first_failure_terminal_8' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.CASCADE.first_failure_terminal_8
/-- info: 'PSP.CASCADE.halt_free_iff_all_pass_8' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.CASCADE.halt_free_iff_all_pass_8
/-- info: 'PSP.GROUNDLESS.o0Neighbors' does not depend on any axioms -/
#guard_msgs in #print axioms PSP.GROUNDLESS.o0Neighbors
/-- info: 'PSP.GROUNDLESS.fiveCubeCoords' does not depend on any axioms -/
#guard_msgs in #print axioms PSP.GROUNDLESS.fiveCubeCoords
/-- info: 'PSP.GROUNDLESS.deg5' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.GROUNDLESS.deg5
/-- info: 'PSP.GROUNDLESS.groundless_cascade_degree' depends on axioms: [propext] -/
#guard_msgs in #print axioms PSP.GROUNDLESS.groundless_cascade_degree
/-- info: 'PSP.TOKEN.level' does not depend on any axioms -/
#guard_msgs in #print axioms PSP.TOKEN.level
/-- info: 'PSP.TOKEN.terminal_tokens_level_and_distinct' does not depend on any axioms -/
#guard_msgs in #print axioms PSP.TOKEN.terminal_tokens_level_and_distinct
/-- info: 'PSP.TWOGROUP.two_group_census' does not depend on any axioms -/
#guard_msgs in #print axioms PSP.TWOGROUP.two_group_census
theorem codexCone : True :=
let _ := @Nat.factorial
let _ := @CoreRat.div_pos_of_le
let _ := @CoreRat.div_gcd_pos
let _ := @CoreRat.coprime_div_gcd
let _ := @Ratio.natAbs_signed
let _ := @Ratio.«mk'»
let _ := @Ratio.instOfNatRat
let _ := @Ratio.instAddRat
let _ := @Ratio.instMulRat
let _ := @Ratio.instNegRat
let _ := @Ratio.instSubRat
let _ := @Ratio.inv
let _ := @Ratio.instDivRat
let _ := @Ratio.instLTRat
let _ := @Ratio.instLERat
let _ := @Ratio.instDecidableLtRat
let _ := @Ratio.instDecidableLeRat
let _ := @Ratio.pow
let _ := @Ratio.instHPowRatNat
let _ := @Ratio.instToStringRat
let _ := @Ratio.num_zero
let _ := @Ratio.den_zero
let _ := @Ratio.«mk'_zero_num»
let _ := @Ratio.zero_mul
let _ := @Ratio.mul_zero
let _ := @Ratio.canonical
let _ := @Ratio.ext
let _ := @Ratio.inv_zero
let _ := @Grade.rank
let _ := @Grade.weakest
let _ := @Grade.weakest_is_a_link
let _ := @Grade.weakest_idempotent
let _ := @Claim.join
let _ := @Claim.join_grade
let _ := @deltaM
let _ := @socialWeight
let _ := @FTOE.Even
let _ := @FTOE.OddAt
let _ := @FTOE.WhollyOdd
let _ := @FTOE.T1_wall
let _ := @FTOE.T2_coalition
let _ := @FTOE.T3_no_fixed_point
let _ := @FTOE.T4_one_bit
let _ := @FTOE.T5_crossing
let _ := @FTOE.T6a_deed_anchor
let _ := @FTOE.T6b_deed_supply
let _ := @FTOE.fTOE_port
let _ := @FTOE.Involution
let _ := @FTOE.FixedPointFree
let _ := @FTOE.T7_global_wall
let _ := @FTOE.T8_torsor_forward
let _ := @FTOE.T8_torsor_backward
let _ := @FTOE.T9_torsor_inverse_left
let _ := @FTOE.T9_torsor_inverse_right
let _ := @FTOE.T10_factorization
let _ := @FTOE.T11_separation
let _ := @FTOE.T12_encoder_injective
let _ := @FTOE.T14_wall_factorization
let _ := @FTOE.T13_wholly_odd_is_odd_everywhere
let _ := @FTOE.flipF
let _ := @FTOE.T15_price_bijection_forward
let _ := @FTOE.T15_price_bijection_backward
let _ := @FTOE.fTOE_core
let _ := @FTOE.OrbitRel
let _ := @FTOE.OrbitRel_symmetric
let _ := @FTOE.Dtau
let _ := @FTOE.calib
let _ := @FTOE.calib_even
let _ := @FTOE.descend
let _ := @FTOE.priceForward
let _ := @FTOE.priceBackward
let _ := @FTOE.T16_left_inverse
let _ := @FTOE.T16_right_inverse
let _ := @FTOE.T16_price_bijection
let _ := @FTOE.canonD0
let _ := @FTOE.flipF_involution
let _ := @FTOE.T16_canonical
let _ := @FTOE.T17_odd_everywhere_iff_wholly_odd
let _ := @K4.val
let _ := @K4.tau
let _ := @K4.target
let _ := @K4.rho
let _ := @K4.tau_involution
let _ := @K4.tau_fixed_point_free
let _ := @K4.target_wholly_odd
let _ := @K4.register_even
let _ := @K4.val_lifts_mod_210
let _ := @K4.wall
let _ := @K4.pair_distinct
let _ := @K4.price_one_bit
let _ := @K4.price_full_fibre
let _ := @K4.wallClaim
let _ := @Audit.Ans.toBool
let _ := @Audit.ClaimInput.anyUnknown
let _ := @Audit.sealL
let _ := @Audit.DrillInput.anyUnknown
let _ := @Audit.drillScreen
let _ := @Audit.refusalGates
let _ := @Audit.compartmentGates
let _ := @Audit.auditClaim
let _ := @Audit.compartmentGrade
let _ := @Audit.unknown_routes_open
let _ := @Audit.contentless_refused
let _ := @Audit.det3
let _ := @Audit.det3_two_id
let _ := @INVERSION.ItFromBit
let _ := @INVERSION.it_from_bit_refuted
let _ := @INVERSION.it_from_it
let _ := @K4.it_from_it_canonical
let _ := @K4.it_from_bit_refuted_canonical
let _ := @K4.inversionClaim
let _ := @GEO.qmul
let _ := @GEO.qconj
let _ := @GEO.qadd
let _ := @GEO.qsub
let _ := @GEO.qnorm2
let _ := @GEO.qhalve
let _ := @GEO.qquarter
let _ := @GEO.qcanon
let _ := @GEO.q1
let _ := @GEO.qi
let _ := @GEO.qj
let _ := @GEO.qk
let _ := @GEO.chirality
let _ := @GEO.noncommutative
let _ := @GEO.hurwitz
let _ := @GEO.hurwitz_norm
let _ := @GEO.hurwitz_products_even
let _ := @GEO.hurwitz_closed
let _ := @GEO.hurwitzReps
let _ := @GEO.units_mod_sign_twelve
let _ := @GEO.conjClass
let _ := @GEO.classSizesAux
let _ := @GEO.insNat
let _ := @GEO.insSort
let _ := @GEO.class_equation
let _ := @GEO.shell2
let _ := @GEO.shell_twentyfour
let _ := @GEO.no_half_integer_norm2
let _ := @GEO.allGates
let _ := @GEO.gates_twelve
let _ := @GEO.gates_valid
let _ := @GEO.gates_complete
let _ := @GEO.invCount
let _ := @GEO.perms24
let _ := @GEO.a4
let _ := @GEO.a4_order
let _ := @GEO.actG
let _ := @GEO.a4_simply_transitive
let _ := @GEO.tetraVerts
let _ := @GEO.tetraEdges
let _ := @GEO.tetraFaces
let _ := @GEO.euler_closure
let _ := @GEO.basisU
let _ := @GEO.perms6
let _ := @GEO.relabel_parity
let _ := @GEO.O8
let _ := @GEO.omul
let _ := @GEO.osub
let _ := @GEO.onorm2
let _ := @GEO.e8
let _ := @GEO.octonion_associator
let _ := @GEO.hPairs
let _ := @GEO.norm_composition_H
let _ := @GEO.oPairs
let _ := @GEO.norm_composition_O
let _ := @GEO.det2
let _ := @GEO.gf2Mul
let _ := @GEO.gf2Vecs
let _ := @GEO.gf2Sols
let _ := @GEO.triaxial_lock_gf2
let _ := @GEO.triaxial_lock_gf2_count
let _ := @GEO.triaxial_open_gf2_count
let _ := @GEO.V3
let _ := @GEO.M3
let _ := @GEO.det3
let _ := @GEO.adj3
let _ := @GEO.mm3
let _ := @GEO.diag3
let _ := @GEO.mv3
let _ := @GEO.vscale
let _ := @GEO.V3Z
let _ := @GEO.M3Z
let _ := @GEO.det3Z
let _ := @GEO.adj3Z
let _ := @GEO.mm3Z
let _ := @GEO.diag3Z
let _ := @GEO.adjugate_identity_int
let _ := @GEO.AdjugateIdentity3
let _ := @GEO.adj3Samples
let _ := @GEO.adj3_executed
let _ := @GEO.cramerSolve
let _ := @GEO.cramer_executed
let _ := @GEO.golAdmit
let _ := @GEO.gateNames
let _ := @GEO.gateNames_twelve
let _ := @GEO.gateScreenGo
let _ := @GEO.gateScreen
let _ := @GEO.gateScreen_clean
let _ := @GEO.gateScreen_first_failure
let _ := @GEO.gateScreen_names_failed_gate
let _ := @GEO.gateScreen_arity
let _ := @GEO.ClaimInputRA.anyUnknown
let _ := @GEO.rowCascadeRACore
let _ := @GEO.rowCascadeRA
let _ := @GEO.unknown_routes_open_ra
let _ := @GEO.raDecode
let _ := @GEO.raBattery
let _ := @GEO.ra_battery_f2
let _ := @GEO.ra_ledger_refused
let _ := @GEO.ra_ledger_iii
let _ := @GEO.ra_ledger_ii
let _ := @GEO.ra_ledger_void
let _ := @GEO.ra_ledger_aGivenRA
let _ := @GEO.RAVerdict.toToken
let _ := @GEO.GolToken.toToken
let _ := @GEO.gateScreenToken
let _ := @GEO.demoInputA
let _ := @GEO.demoInputVoid
let _ := @GEO.demoInputOpen
let _ := @ROOT.Actuates
let _ := @ROOT.actuation_universal
let _ := @ROOT.raClaim
let _ := @ROOT.boolCut
let _ := @ROOT.nest_31
let _ := @ROOT.nestClaim
let _ := @ROOT.throneEmpty
let _ := @ROOT.throne_is_RA
let _ := @SEALS.sealL
let _ := @SEALS.sealG
let _ := @SEALS.sealM
let _ := @SEALS.seals_distinct
let _ := @SEALS.seal_names_match
let _ := @SEALS.sealG_walk
let _ := @POSTULATE.OneCut
let _ := @POSTULATE.F2
let _ := @POSTULATE.F2_kills
let _ := @POSTULATE.BSDContract
let _ := @POSTULATE.F6
let _ := @TONGUE.enlarge
let _ := @TONGUE.unclosed
let _ := @TONGUE.earned_freedom
let _ := @TONGUE.colocClaim
let _ := @TONGUE.coloc_grade
let _ := @TONGUE.ra_grade_pinned
let _ := @TONGUE.wall_grade_pinned
let _ := @TONGUE.inversion_grade_pinned
let _ := @TONGUE.nest_grade_pinned
let _ := @terminalWall
let _ := @terminalPrice
let _ := @TERMINAL
let _ := @terminalClaim
let _ := @terminal_grade_pinned
let _ := @IAM.iamToken
let _ := @IAM.iam_unwitnessed_withheld
let _ := @IAM.iam_witnessed_speaks
let _ := @IAM.narcissus_truth_table
let _ := @IAM.one_bit_freedom
let _ := @IAM.IamToken.toToken
let _ := @IAM.codexSelfRead
let _ := @IAM.codex_self_read_interior
let _ := @IAM.terminalWord
let _ := @IAM.terminal_word
let _ := @GEO.SignPattern
let _ := @GEO.allSigns
let _ := @GEO.signComp
let _ := @GEO.reflection_eight
let _ := @GEO.reflection_eight_is_two_cubed
let _ := @GEO.reflections_abelian
let _ := @GEO.reflections_involutive
let _ := @GEO.signVal
let _ := @GEO.reflMat
let _ := @GEO.reflection_reverses_orientation
let _ := @GEO.sandwich
let _ := @GEO.reflection_blind_executed
let _ := @GEO.V5
let _ := @GEO.cube5
let _ := @GEO.neighbors5
let _ := @GEO.fivecube_thirtytwo
let _ := @GEO.fivecube_vertices_distinct
let _ := @GEO.fivecube_degree_five
let _ := @GEO.counts_forced
let _ := @TONGUE.deletionOp
let _ := @TONGUE.e3
let _ := @TONGUE.deletion_idempotent
let _ := @TONGUE.deletion_singular
let _ := @TONGUE.deletion_minor_nonsingular
let _ := @TONGUE.deletion_annihilates
let _ := @TONGUE.linear_fixes_zero
let _ := @TONGUE.deletion_no_left_inverse
let _ := @TONGUE.deletion_monoid_not_group
let _ := @TONGUE.tongue_obedience
let _ := @OFFICE.discriminator
let _ := @OFFICE.discriminatorQ2First
let _ := @OFFICE.order_loadbearing
let _ := @OFFICE.citation_cannot_promote
let _ := @UC.routeHalt
let _ := @UC.routeHalt_fidelity
let _ := @UC.universalCascade
let _ := @UC.uc_seal_failure_terminal
let _ := @UC.uc_routes_eight
let _ := @UC.uc_routes_five
let _ := @UC.uc_unmeasured
let _ := @UC.circularityScreen
let _ := @UC.circularity_refused_at_target
let _ := @UC.road_requires_conjunction
let _ := @UC.circRows
let _ := @UC.circ_space
let _ := @OMEGA.kbRat
let _ := @OMEGA.ln2Rat
let _ := @OMEGA.landauerRat
let _ := @OMEGA.omegaBoundary
let _ := @OMEGA.omega_zero_bits
let _ := @OMEGA.omega_paid_floor
let _ := @OMEGA.one_bit_freedom_priced
let _ := @OMEGA.omega_reversible_branch
let _ := @OMEGA.omega_refuses_nonpositive_temp
let _ := @OMEGA.omega_monotone
let _ := @OMEGA.omega_linear
let _ := @OMEGA.landauer_zero_bits
let _ := @OMEGA.omega_verdict_fidelity
let _ := @AEGIS.refusalConst
let _ := @AEGIS.aegisReset
let _ := @AEGIS.aegisGuard
let _ := @AEGIS.aegis_constant
let _ := @AEGIS.aegis_deed_increments
let _ := @AEGIS.aegis_reads_parameter
let _ := @AEGIS.aegisFourLogicRun
let _ := @AEGIS.aegis_battery
let _ := @NOMOS.allStrings
let _ := @NOMOS.allStrings_length
let _ := @NOMOS.twoPowPos
let _ := @NOMOS.descs
let _ := @NOMOS.descs_length
let _ := @NOMOS.incompressibility_cap
let _ := @NOMOS.nomosRows
let _ := @NOMOS.incompressibility_executed
let _ := @NOMOS.decodeId
let _ := @NOMOS.decodePad
let _ := @NOMOS.compressible_bound_universal
let _ := @NOMOS.cbUniversal
let _ := @NOMOS.cb_id_8_2
let _ := @NOMOS.cb_pad3_8_2
let _ := @NOMOS.cb_pad2_6_1
let _ := @NOMOS.cb_pad4_12_3
let _ := @NOMOS.recoverable
let _ := @NOMOS.nomos_fuel_is_omega_floor
let _ := @NOMOS.nomosScreen
let _ := @NOMOS.nomosScreen_battery
let _ := @BRIDGE.ceiling
let _ := @BRIDGE.census
let _ := @BRIDGE.census_count
let _ := @BRIDGE.post_seal_disposition
let _ := @BRIDGE.weakest_never_above_left
let _ := @BRIDGE.bridge_ceiling_caps
let _ := @BRIDGE.successorOf
let _ := @BRIDGE.supersession_routes
let _ := @BRIDGE.bridgeScreen
let _ := @BRIDGE.bridgeScreen_total
let _ := @FOUND.sigmaP
let _ := @FOUND.pluralist_alternative_executed
let _ := @FOUND.posits
let _ := @FOUND.posits_loadbearing_on_nothing
let _ := @FOUND.gradeOfRank
let _ := @FOUND.posit_never_raises
let _ := @FOUND.bandOf
let _ := @FOUND.register_routing_fidelity
let _ := @FOUND.verdictInventory
let _ := @FOUND.verdicts_posit_free
let _ := @FOUND.engine_never_authority
let _ := @HALT.cascade
let _ := @HALT.patterns
let _ := @HALT.fftSpec
let _ := @HALT.first_failure_terminal_5
let _ := @HALT.first_failure_terminal_7
let _ := @HALT.first_failure_terminal_8
let _ := @HALT.first_failure_terminal_12
let _ := @HALT.halt_free_iff_all_pass_8
let _ := @HALT.gate_counts_recorded
let _ := @HALT.signPatterns
let _ := @HALT.ninth_gate_barred
let _ := @HALT.route
let _ := @HALT.ofUC
let _ := @HALT.route_is_routeHalt
let _ := @HALT.route_samples
let _ := @HALT.tokenBranch
let _ := @HALT.router_exclusive
let _ := @HALT.readingRoadsGate
let _ := @HALT.one_door_open
let _ := @HALT.no_walk_passes_the_aperture
let _ := @HALT.xiGates
let _ := @HALT.oGates
let _ := @HALT.gate_censuses
let _ := @HALT.rhHalt
let _ := @HALT.pnpHalt
let _ := @HALT.halts_routed
let _ := @HALT.legacy_halts_are_instances
let _ := @HALT.terminal_one_bit
let _ := @HALT.species_exhausted
let _ := @HALT.haltGrades
let _ := @HALT.ftoe_ceiling
let _ := @CROSSING.ExecFrame
let _ := @CROSSING.execFlip
let _ := @CROSSING.formalRead
let _ := @CROSSING.ran
let _ := @CROSSING.formalRead_even
let _ := @CROSSING.ran_wholly_odd
let _ := @CROSSING.execFlip_involution
let _ := @CROSSING.ran_odd_at_true
let _ := @CROSSING.wall
let _ := @CROSSING.deed
let _ := @CROSSING.price
let _ := @CROSSING.crossing
let _ := @CROSSING.identity_one_bit
let _ := @CROSSING.fibre4
let _ := @CROSSING.fibre4_sixteen
let _ := @CROSSING.fibre4_distinct
let _ := @CROSSING.price_on_this_file
let _ := @CROSSING.formal_never_odd
let _ := @CROSSING.constant_no_crossing
let _ := @CROSSING.harmony
let _ := @LOCUS.Separates
let _ := @LOCUS.Factors
let _ := @LOCUS.wall_of_sep
let _ := @LOCUS.swap
let _ := @LOCUS.swap_involution
let _ := @LOCUS.swap_record_even
let _ := @LOCUS.swap_flips_pair
let _ := @LOCUS.seat_of_sep
let _ := @LOCUS.k4_separated_pair
let _ := @LOCUS.k4_wall_of_pair
let _ := @LOCUS.readout
let _ := @LOCUS.factors_of_not_sep_list
let _ := @LOCUS.factors_iff_not_sep_list
let _ := @LOCUS.Channel
let _ := @LOCUS.Token.toEconomy
let _ := @LOCUS.Reading.orbits
let _ := @LOCUS.Reading.owed
let _ := @LOCUS.Reading.complete
let _ := @LOCUS.branchOf
let _ := @LOCUS.emit
let _ := @LOCUS.Reading.state
let _ := @LOCUS.three_states
let _ := @LOCUS.no_seat_no_locus
let _ := @LOCUS.unlocated_is_plain_open
let _ := @LOCUS.located_never_silent
let _ := @LOCUS.crossed_requires_complete
let _ := @LOCUS.crossed_requires_channel
let _ := @LOCUS.halt_carries_owed
let _ := @LOCUS.unchannelled_is_halt
let _ := @LOCUS.halt_reports_owed
let _ := @LOCUS.owed_zero_is_unchannelled
let _ := @LOCUS.one_bit_apart
let _ := @LOCUS.legacy_xi
let _ := @LOCUS.legacy_o
let _ := @LOCUS.legacy_bare
let _ := @LOCUS.k4_locus
let _ := @LOCUS.slotLists
let _ := @LOCUS.slotLists_twelve
let _ := @LOCUS.slotLists_pos
let _ := @LOCUS.allReadings
let _ := @LOCUS.reading_space
let _ := @LOCUS.census_no_seat_no_locus
let _ := @LOCUS.census_located_never_silent
let _ := @LOCUS.census_crossed_iff_complete_channelled
let _ := @LOCUS.census_halt_reports_owed
let _ := @LOCUS.census_owed_zero_is_unchannelled
let _ := @LOCUS.census_three_states
let _ := @LOCUS.Token.render
let _ := @LOCUS.Token.ascii
let _ := @LOCUS.sealAscii
let _ := @LOCUS.render_receipted
let _ := @LOCUS.render_injective_in_owed
let _ := @LOCUS.voidImage
let _ := @LOCUS.void_precedes_economy
let _ := @LOCUS.legacy_records_not_promoted
let _ := @LOCUS.no_rh_locus
let _ := @PSP.ORIENT.V3
let _ := @PSP.ORIENT.M3
let _ := @PSP.ORIENT.det3
let _ := @PSP.ORIENT.neg3
let _ := @PSP.ORIENT.signMatrices
let _ := @PSP.ORIENT.lock_scalar_sign_blind
let _ := @PSP.ORIENT.det_sign_flips_in_odd_dimension
let _ := @PSP.QUAT.Q
let _ := @PSP.QUAT.qmul
let _ := @PSP.QUAT.qi
let _ := @PSP.QUAT.qj
let _ := @PSP.QUAT.qk
let _ := @PSP.QUAT.qneg1
let _ := @PSP.QUAT.quat_basis_laws
let _ := @PSP.QUAT.qnorm
let _ := @PSP.QUAT.isInt
let _ := @PSP.QUAT.isHalf
let _ := @PSP.QUAT.hurwitzOK
let _ := @PSP.QUAT.grid
let _ := @PSP.QUAT.tuples4
let _ := @PSP.QUAT.hurwitz_units_24
let _ := @PSP.QUAT.grid2
let _ := @PSP.QUAT.tuples4i
let _ := @PSP.QUAT.norm2_shell_split
let _ := @PSP.CD.cdconj
let _ := @PSP.CD.padd
let _ := @PSP.CD.psub
let _ := @PSP.CD.cdmul
let _ := @PSP.CD.e
let _ := @PSP.CD.smul
let _ := @PSP.CD.octonion_associator
let _ := @PSP.CD.sx
let _ := @PSP.CD.sy
let _ := @PSP.CD.zero16
let _ := @PSP.CD.sqnorm
let _ := @PSP.CD.sedenion_zero_divisor
let _ := @PSP.CASCADE.cascade
let _ := @PSP.CASCADE.patterns8
let _ := @PSP.CASCADE.spec
let _ := @PSP.CASCADE.first_failure_terminal_8
let _ := @PSP.CASCADE.halt_free_iff_all_pass_8
let _ := @PSP.GROUNDLESS.o0Neighbors
let _ := @PSP.GROUNDLESS.fiveCubeCoords
let _ := @PSP.GROUNDLESS.deg5
let _ := @PSP.GROUNDLESS.groundless_cascade_degree
let _ := @PSP.TOKEN.level
let _ := @PSP.TOKEN.terminal_tokens_level_and_distinct
let _ := @PSP.TWOGROUP.two_group_census
trivial
/-- info: 'codexCone' depends on axioms: [propext,
POSTULATE.Algorithm,
POSTULATE.FrozenData,
POSTULATE.Row,
POSTULATE.SupplyClean,
POSTULATE.certifiedPairing,
Quot.sound,
ROOT.FormalDomain,
ROOT.L1m,
ROOT.L3m,
ROOT.RA,
ROOT.U,
ROOT.nest_21,
ROOT.nest_32,
ROOT.ΔE] -/
#guard_msgs in #print axioms codexCone