riemann

Proposal: Two-Precision flint.arb/flint.acb Candidate-Mechanism Screen Across WEIL-FLOOR, KERNEL-API, and NB-GAIN (cofinal-candidate-mechanisms-screen-2026-10-16-v1, Repaired per Review)

Wall check: Same open gap. Closest results: WEIL-FLOOR, NS24/39/43, NB-GAIN (NS53/61/83/87/98), and ZERO-GEOMETRY (NS100/101). We still need a bound that remains valid as the window grows ($b \to \infty$ or $N \to \infty$), retaining every signed arithmetic contribution and coupling. RH, G2, and both missing-original evidence groups in evidence/MISSING.md remain OPEN. M17 v2 remains PAUSED.

1. Exact Target & Scope

This bounded candidate-testing package (audits/cofinal-candidate-mechanisms-screen-2026-10-16-v1/) executes a deterministic, two-precision (256-bit and 384-bit flint.arb / flint.acb, zero numpy/scipy/mpmath imports) quantitative screen of the six post-v18 candidate mechanisms cataloged in the two pre-computation 5-item arithmetic assessments:

In addition to screening the six candidates, this package establishes and certifies two exact analytic identities in WEIL-FLOOR:

  1. Theorem 1 (Boundary-Free Dual-Pole Chebyshev–Legendre Identity for $T_{\mathrm{arith}}(a) - T_{\mathrm{pole}}(a)$ in Both Parities): proves that in both even and odd parities on $L^2(-a,a)$, subtracting the exact polar operator $T_{\mathrm{pole}}(a)$ from the signed prime-power shift sum $T_{\mathrm{arith}}(a)$ corresponds to subtracting the dual pole generator $x + \log x - 1$ from Chebyshev’s $\psi(x)$, with identically zero Stieltjes integration-by-parts boundary terms at both $u = 0$ ($\widetilde{R}\psi(1) = 0$) and $u = 2$ ($\widetilde{C}{m,n}(2) = 0$).
  2. Theorem 2 (Closed-Form Odd Pole Moment of the Position-Weighted Gaussian Radical $x\,h(x)$ vs. Critical-Zero Derivative $\Xi’(\gamma_1)$): proves by differentiating the Mellin identity eq:v150-mellin (evidence/v150/capacity_weight.tex) at $z = 1$ that the odd position-weighted Gaussian radical $f_{\mathrm{pos}}(x) = x\,h(x)$ has a strictly positive closed-form odd pole moment $\int_{\mathbb{R}} x\sinh(x/2)h(x)\,dx = \frac{1}{\sqrt{3}}\bigl(1 + \frac{\gamma_E - \log(4\pi)}{2}\bigr) = +0.01333431378871523222883\dots > 0$ and couples to simple critical-line zeros via $\widehat{f_{\mathrm{pos}}}(\gamma_k) = +\frac{2i}{\sqrt{3}}\Xi’(\gamma_k) \ne 0$, while recording its compressed 8-mode Legendre residual on $V_8^{\mathrm{odd}}(-a,a)$ across $\lambda \in {2, 3, 5, 8}$.

2. Five-Item Pre-Computation Gate Alignment (Corrected per Review)

Item Requirement Implementation in verify_cofinal_candidate_mechanisms.py    
1. Exact expression Full unpreconditioned Weil compression $\mathsf{W}\lambda^{(8)} = P_8\bigl(A{\mathrm{arch}}^{\mathrm{full}}(a) + T_{\mathrm{pole}}(a) - T_{\mathrm{arith}}(a)\bigr)P_8$ on $V_8^{\mathrm{even}/\mathrm{odd}}(-a,a)$ (WEIL-FLOOR), and in NB-GAIN: old-space squared error $E_N^{(2)} = 2 - b_{\mathrm{old}}^\top G_N^{-1} b_{\mathrm{old}}$, corrected new-block load $h_N = b_{\mathrm{new}} - C_N^\top G_N^{-1} b_{\mathrm{old}}$, Schur complement $H_N = D_N - C_N^\top G_N^{-1} C_N$, and dyadic step gain $\Delta_N = E_N^{(2)} - E_{2N}^{(2)} = h_N^\top H_N^{-1} h_N$. Exact flint.arb matrices for $\lambda \in {2, 3, 5, 8}$ (up to 26 prime-power shifts, $k \le 61$) and $N \in {4, 8, 16, 32}$ via Kernel from evidence/v159/ns61/certify_smoothed.py.    
2. Coefficient hypotheses Exact $\Lambda(k)/\sqrt{k}$ weights (WEIL-FLOOR), NS57 perturbation $\delta_2 = -\frac{\log 2}{100}$ (KERNEL-API), and true vs. NS74-perturbed Beurling loads $\beta \in {1/2, 1/2 + 10^{-2}}$ (NB-GAIN). All signed coefficients retained; zero positive-weight or diagonal-only surrogates.    
3. Uniform range Fixed benchmark windows $\lambda \in {2, 3, 5, 8}$ ($a = \log\lambda$) and finite dyadic prefix dimensions $N \in {4, 8, 16, 32}$. Exact two-precision (256-bit and 384-bit) ball certificates with 176/176 string-reconstructed .overlaps() checks.    
4. Estimating method Exact Stieltjes / Legendre polynomial integration (int_u_poly_exp), ball Cholesky (cholesky_lower_arb) including Weyl perturbation enclosures $(2 \delta_2 )I_8 \pm B_{\delta_2} \succ 0$, arb_mat.solve(..., algorithm="precond"), automatic differentiation (acb_series), and explicit lem:v150-weight tail enclosures for $n \ge 12$ and $x \ge 3.5$. Zero floating-point tolerances or midpoint eigenvalue assertions (.contains(0) and > arb(0) only).
5. Remaining transfer & scope Does the tested mechanism yield a window-independent lower bound $q_a[f] \ge -C_*|f|_2^2$ as $a \to \infty$ or cofinal decrease $E_N^{(2)} \to 0$ as $N \to \infty$? Disclosed honestly as OPEN: compressed 8-mode residuals and finite-shell ($V_{[4,8]}^{\mathrm{even}}$) ratios do not enclose $V_8(-a,a)^\perp$, four finite sizes $N \in {4, 8, 16, 32}$ cannot establish convergence or persistent gain, and the $a \to \infty$ / $N \to \infty$ signed arithmetic transfers remain open.    

3. Non-Goal & Pause Disclosure

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