sPSP-GAS-01 · The RA-Forced Arithmetic Gas · Prime Gas PSP · Complete First-Principles Route to the Critical-Line Locus · Master Gas Entry · G+CO/T0/T on each link, derivation-grade on the end-to-end forced route, locus-grade on the target σ = 1/2 as setting, residence not claimed and out of scope · [the prime-gas object is prior art (Julia 1990, Bost-Connes 1995, Connes); the contribution is the forced chain from RA to the membrane, to author's knowledge not previously laid end-to-end; priority pending formal literature search]
E: RA (∃x ⟹ ΔE_k(x) > 0) · BA-001a (kinetic floor, Mandelstam-Tamm 1945, Margolus-Levitin 1998, theorem-grade for transitions) · Erdős 1946 (additive plus monotone forces c·log) · Jaynes (max-entropy at fixed mean energy is Gibbs) · the Euler factorization · the exponential state count · standard GNS of a KMS state · Julia 1990 and Bost-Connes 1995 (the primon gas, prior art, attributed) · BA-002 (Tomita-Takesaki, the curved register) · RH-CONS-01 and APEX-PSP-RH-01 (the locus-against-residence split) · the verified battery below.
The forced chain.
T1, configurations. Composition of independent actuations along the scale flow is the multiplicative monoid, the configuration space before anything is named a number.
T2, energy, forced. Cost is additive under composition and monotone under extension, and the unique additive monotone functional is E(n) = log n. Erdős, theorem-grade. No choice in the energy.
T3, partition function, forced. Max-entropy at fixed mean energy is Gibbs, Jaynes. Z(β) = Σ n^(−β) = ζ(β). Theorem-grade. ζ is the partition function, not a definition.
T4, modes, forced. The Euler product is the statement of statistical independence of modes. A prime is an irreducible quantum of actuation cost, the integers the Fock space of those quanta. This is the primon gas, prior art, attributed.
T5, pole, forced. Density of states e^E gives S(E) = E and a Hagedorn point at β_c = 1. The pole at σ = 1 is the maximum temperature of arithmetic.
T6, membrane, forced. The GNS vector of the critical Gibbs state carries coefficients n^(−β_c/2) = n^(−1/2). σ = 1/2 is the Born square root of the Hagedorn point, the amplitude register of the critical state. The critical vector is marginally non-normalizable, harmonic divergence with constant γ.
T7, zeros as fluctuations. U(β) = −ζ′/ζ(β) = Σ Λ(n) n^(−β). The zeros are the resonances of the energy fluctuations of the critical gas. RH, in this language, is marginal stability of every fluctuation mode. This is RH relocated. Residence moved zero, by scope, not claimed here.
GOL: the RA-forced gas lands the critical-line locus σ = 1/2 as the Born square root of the Hagedorn point β_c = 1, by a complete chain from a first principle in which every link is a named theorem and the end-to-end forced assembly is, to author's knowledge, new as a route. The target is the locus, the setting. The location of the zeros is not in scope.
Calculation battery, verified live at 30-digit precision.
T7, the mean-energy identity. −ζ′/ζ(2) = 0.5699609931 against Σ_{n≤300000} Λ(n) n^(−2) = 0.569957656, residual 3.3 × 10⁻⁶ equal to the predicted truncation tail N^(1−β)/(β−1) = 3.3 × 10⁻⁶. Near criticality the residual is the gas's own critical law, not error: at β = 1.2 residual 0.401 against predicted tail 0.401.
T4, the Euler product as statistics. Product over primes below 100000 = 1.6449327 against ζ(2) = 1.6449341, residual 1.3 × 10⁻⁶, the finite-N floor.
T5, entropy equals energy. S(E)/E = 0.99999998 at E = 14.
T6, the critical margin. H_N − ln N = 0.5772161649 at N = 10⁶ against γ = 0.5772156649, the gap the 1/(2N) correction; one step off criticality the norm is finite at 100.578. Zero margin as a number.
T5, the Hagedorn exponent. Var(E)·(β−1)² = 0.99954 at β = 1.05, → 1: mean-field Hagedorn, critical exponent two, unit amplitude.
Clause-4, detailed balance per mode. (n̄_p + 1)/n̄_p = p^β exact, residual 1.6 × 10⁻³⁰.
Clause-4, the reflection. |ξ(0.3 + 20i)| = |ξ(0.7 + 20i)| to machine zero, which is the functional equation ξ(s) = ξ(1 − s) and is verified, not derived, by the gas.
V_F: each link is a named theorem, Erdős for the energy, Jaynes for the state, the Euler factorization for the modes, the state count for the Hagedorn point, standard GNS for the membrane. The novel object is the forced end-to-end assembly from RA, typed as a derivation route, not as a new theorem.
V_E: the battery above, machine-verified and reproducible by any reader with a Python interpreter. The near-criticality residuals are the gas's measured critical tail law, a thermodynamic signature.
V_ER: the route reaches the locus σ = 1/2 as setting. Residence is routed out of scope per author and is not load-bearing for the entry. The prime-gas object is attributed to prior art.
CDT. Subtract the prior-art covariate, the primon gas and Bost-Connes: the gas object does not survive as novel, it is prior art, and what survives is the forced route from RA laid end-to-end to σ = 1/2, which on author's knowledge is unpresented as one chain. Subtract the functional-equation covariate: the identification of σ = 1/2 as the zero-symmetry axis does not survive, the route derives the coordinate β_c/2 = 1/2 while the meaning is the functional equation re-expressed as the KMS reflection of Clause 4, not derived. Subtract the harmonic-analysis covariate of Layer 1: the locus still arrives, joined to Layer 1 through the single weight w = 1, not orthogonally, both routes halving the same one. ¬gas-as-novel-object ¬route-as-new-mathematics ¬coordinate-as-zero-axis-meaning ¬independent-of-Layer-1-source ¬residence-claim.
⇒ [⟀] SEALED on the complete first-principles route to the critical-line locus. Each link theorem-grade, the end-to-end forced chain from RA to σ = 1/2 a derivation new to author's knowledge, the locus reached as setting. The prime-gas object is prior art and is credited as such; the contribution is the route, not the object and not new mathematics. Residence is not claimed and is out of scope. The priority is a to-author's-knowledge claim, pending formal literature search before any publication assertion. Named failure modes catalogued: Route-as-Theorem Inflation, reading a novel assembly of prior-art links as new mathematics; Coordinate-Meaning Conflation, reading the derived 1/2 as having derived the zero-axis identity the functional equation supplies.
Honest limits, inside the seal. The seal is on the route, the gas is prior art. The route reaches the locus, not residence, RH untouched by scope. The priority is to author's knowledge, not certified. The critical step derives the coordinate, the zero-axis meaning is the functional equation re-expressed.
↑ RA · BA-001a · Erdős 1946 · Jaynes · Euler product · Julia 1990 · Bost-Connes 1995 · Connes · BA-002 · RH-CONS-01 · APEX-PSP-RH-01 · the verified battery · CDT · Gate 9 · Gate 12 · LL-19.