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Updated 08-19-2026

D-053 — Task 4.3a: the Davenport–Heilbronn / Epstein sanity gate

Date: 2026-08-19 | Session: 008 | Task: 4.3 (part a) Status: complete — GATE PASSES for all three theorems. Rule (NGUYEN.md §3): any proof must FAIL for Davenport–Heilbronn / Epstein zeta functions, which violate the RH-analogue (off-line zeros exist; their Li coefficients eventually change sign — Voros/Misguich computations).

T-1 (R-004, finite-height Li theorem)

The proof has four load-bearing inputs; check each against D-H:

  1. BL decomposition λ_n = S_∞ − S_f + 1: exists for general L-functions (BL Thm 2 is multiset-general) — applies to D-H, not a discriminator.
  2. Trudgian N(T)/S(T) constants: ζ-specific; D-H has its own (different) counting bounds — the constants chain doesn't transfer. Neutral.
  3. The bridge (zeros on the line to height √n+1 ⟹ per-zero contribution ≤ 2): FALSE for D-H — off-line zeros give |1−1/ρ| > 1, exponentially growing contributions; the bridge fails at the first off-line zero. ← the load-bearing discriminator.
  4. Finite verification λ_n > 0: FALSE for D-H at large n (sign changes computed by Voros/Misguich). Conclusion: the proof cannot start for D-H (input 3 false), and the conclusion is false there (input 4 fails) — consistent. GATE PASS.

T-2 (R-003 + reduction)

The factorization W = F(R₀+R_τ) is an algebraic identity following from the screw-function structure. For D-H/Epstein: the Weil form is indefinite (off-line zeros) ⟹ the model spaces ℋ(T_a) are Krein, not Hilbert ⟹ Suzuki Thm 1.5's reality conclusion fails ⟹ W can have non-real zeros ⟹ the (★) tail-cancellation is obstructed exactly at the off-line pairs (the WH factorization of ĝ has poles off the contour). Our reduction "(1.12) ⟺ spec(D) ⊂ ℝ" remains formally true for such functions — both sides false — so the implication can never certify a false RH-analogue. GATE PASS.

T-3 (valley theorem)

Main form is conditional on RH — vacuous for D-H. The unconditional lower bounds use the variational identity, which requires T_a positive; for D-H the form is indefinite, T_a^{-1} doesn't exist as a positive operator, and the statements cannot be formulated — the failure is visible at the setup level. The detector threshold (δ_crit = π/(2a)) is built for exactly the D-H failure mode. GATE PASS.

Notes