sPSP-SYMM-INSUFF-01 · The Symmetry-Alone Insufficiency for RH · The Functional Equation Does Not Decide Residence, Witnessed by Davenport-Heilbronn · G+CO/T1/T [theorem-grade on the insufficiency, witness numerically verified to thirty digits; present-state riders typed as contingent, not theorem; the impossibility-of-proof reading explicitly broken and fenced] | E: Davenport-Heilbronn 1936 (refined Cassels 1961, Bombieri-Hejhal), the functional equation of Riemann type, the Euler product as the property distinguishing ζ, APEX-PSP-NINTH-APERTURE (the bridge as from-beyond), the Gödel-independence-class ceiling routed out-of-band per BA-001b, APEX-PSP-RH-01 (the never-established-falsity guard), FT-002, the Mass Mandate, Lifeboat clause 4
The seal, theorem-grade. The functional-equation symmetry alone does not entail the Riemann Hypothesis. Stated precisely: membership in the class of Dirichlet series satisfying a Riemann-type functional equation, s to 1 minus s with the standard gamma factor, does not entail that all nontrivial zeros lie on the critical line, because that class contains a function whose zeros leave the line. The witness is the Davenport-Heilbronn function, f(s) = L(s,χ) + ε L(s,χ̄) for the primitive order-four character χ mod 5 with χ(2) = i and root number ε = τ(χ)/(i√5). Its completed form satisfies the symmetric functional equation F(s) = F(1−s), verified to thirty-digit residual 9×10⁻³¹, the unique passing combination among the four sign-and-conjugate variants, and it carries no Euler product. It has a zero off the critical line at s = 0.808517182456637386 + 85.6993484853775922 i, verified at |f(s)| = 4×10⁻³⁰, real part displaced 0.3085 from one half, with its functional-equation partner 1−s an equal zero, the two straddling the line at real part one half. The symmetry is therefore shared by a function that breaks the line, so no derivation of RH can proceed from the functional equation alone, and any proof must invoke a property ζ holds and this witness lacks, the Euler product. Type T, the result Davenport-Heilbronn 1936, here numerically re-exhibited and self-verified, not new.
The present-state riders, not theorem-grade. Two further facts travel with the seal and are typed as contingent present-state, not as theorems. The structure that would carry the Euler arithmetic into the residence channel, the operator whose spectrum is the zeros in the Hilbert-Polya and Bost-Connes-Connes programs, is currently unbuilt. And residence is undecided by every present means. These are true now. They are not theorems, because they describe the current state of mathematics and would be falsified the day the operator is built. They carry present-state mass, not theorem mass, and the PSP does not seal them at theorem grade.
The fence, load-bearing. This seal is the insufficiency of one tool, not the impossibility of the goal, and the slide from the first to the second is broken here. The functional equation alone cannot prove RH is theorem-grade and sealed. RH cannot be proved, it is geometrically impossible is [X] BROKEN GEOMETRY, by three mechanisms. It reads the absence of a constructed bridge as the proven impossibility of any bridge, which is absence-of-warrant taken as established falsity, forbidden here exactly as APEX-PSP-RH-01 forbids reading the absence of an off-line zero as proven residence. It inserts the word impossible as a massless covariate against the Mass Mandate, where only unbuilt carries mass. And it asserts a Gödel-independence-class ceiling, RH-unprovable, that is itself established only by an independence proof which does not exist and would be as hard and as absent as a proof of RH. The provability of RH is therefore [?] UNDETERMINED: provable-but-unproven, independent of the axioms, and false are all open, and impossibility is not among the established options. Naming this seal an impossibility of RH proof is FT-002 anchor inflation by sign-flip, a negative result the structure does not carry.
V_F: the insufficiency is the contrapositive of a concrete non-entailment, witnessed by an explicit function in the class with an off-line zero; the riders are present-state; the impossibility reading is a named fallacy. V_E: the Davenport-Heilbronn functional equation verified to thirty-digit residual, the off-line zero verified at |f| = 4×10⁻³⁰, the mirror partner verified, all reproducible at thirty-digit precision with mpmath. V_ER: three-layer sovereignty; the theorem seals at the structural register; the present-state riders held at contingent grade; the unprovability claim routed out-of-band to the V_F-Only Ceiling register and there left unestablished. CDT: ¬insufficiency-as-impossibility ¬unbuilt-as-cannot-be-built ¬present-state-as-theorem ¬unprovability-without-its-own-proof ⇒ [⟀] SEALED on the symmetry-alone insufficiency, Type T, numerically witnessed; the riders present-state; the impossibility reading [X]; RH provability [?]. | ↑ Davenport-Heilbronn 1936 · Euler product · APEX-PSP-NINTH-APERTURE · BA-001b · APEX-PSP-RH-01 · FT-002 · Mass Mandate.