pureVectorCDoublingData_requires_extra_input
Pure completed-ξ symmetry together with the functional-equation doubling recurrence fails to force a zero-composition witness at every zero. Researchers working on the critical-line problem in Recognition Science would cite this no-go result to justify the need for additional analytic input from the Euler or Hadamard side. The proof proceeds by contradiction: it applies the assumed universal statement to an explicit off-critical toy example and invokes the forcing property of any zero-composition witness to reach a contradiction.
claimIt is not the case that for every completed-ξ surface and every complex number ρ, if ρ is a zero obeying reflection symmetry, conjugation symmetry, and the doubling recurrence, then there exists a zero-composition witness at ρ. Formally, ¬∀ Ξ, ρ (obeys pure symmetry and doubling at ρ ⇒ admits zero-composition witness at ρ).
background
This module formalizes the main stage gate for Vector C: functional-equation reflection symmetry plus conjugation symmetry give pairing data on zeros, but do not by themselves force the critical line. We certify this with an explicit toy completed-ξ surface whose zeros occur on the off-line pair Re(s) = 1/4, 3/4. A completed-ξ surface is a function xi : ℂ → ℂ equipped with the reflection property xi(1 − s) = xi(s) and the conjugation property xi(conj s) = conj(xi s). The pure doubling data at a zero ρ further requires that the functional reflection, conjugation, and critical reflection of ρ are also zeros, together with the recurrence doubledZeroDefect(ρ) = 2 (zeroDefect(ρ))² + 4 zeroDefect(ρ). A zero-composition witness at ρ consists of a zero-composition law such that the law evaluated at the deviation of ρ equals 1; any such witness forces ρ to lie on the critical line. Upstream, the minimal completed-ξ surface is described as: 'Minimal completed-ξ symmetry data for Vector C. reflection is the completed functional equation, and conjugation is the standard reality symmetry. Any stronger zero-location constraint must be added on top of this surface; it is not present here by default.'
proof idea
The proof is by contradiction. Assume the universal quantification holds. Instantiate it on the pair furnished by the offline example theorem, which satisfies the pure doubling data but lies off the critical line. This produces a zero-composition witness. The lemma that any zero-composition witness forces its point onto the critical line then yields the desired contradiction.
why it matters in Recognition Science
This no-go theorem shows that the pure functional-equation package is insufficient to force the critical line, even after incorporating the doubling recurrence. It therefore directs attention to the need for additional structure from the Euler or Hadamard side, as noted in the module comment: 'any successful upgrade from honest zeta-derived phase data to ZeroCompositionWitness must use genuinely extra analytic input not present in the pure FE package; in this repository the natural candidates live on the Euler/Hadamard side'. The result sits at the Vector C stage gate and closes one possible route in the forcing chain.
scope and limits
- Does not assert that stronger Vector C data cannot force the critical line.
- Does not supply the extra analytic input required from Euler or Hadamard factors.
- Does not apply when the completed-ξ surface already encodes quantitative factorization or phase bounds.
- Does not address the location of all zeros, only whether the pure symmetry package suffices.
formal statement (Lean)
139theorem pureVectorCDoublingData_requires_extra_input :
140 ¬ (∀ (Ξ : CompletedXiSurface) (ρ : ℂ),
141 PureVectorCDoublingData Ξ ρ → Nonempty (ZeroCompositionWitness ρ)) := by
proof body
Term-mode proof.
142 intro h
143 obtain ⟨Ξ, ρ, hpure, hline⟩ := pureVectorCDoublingData_offline_example
144 rcases h Ξ ρ hpure with ⟨w⟩
145 exact hline (zeroCompositionWitness_forces_on_critical_line w)
146
147/-- In particular, pure FE symmetry plus the current doubling law do not by
148themselves force the critical line. -/