D-014 — Track C landscape (2026 literature) + cross-pollination plan

Date: 2026-08-17 | Session: 002 | Task: 3.1 (setup) | Status: complete Method: web research per §4A.3 (arXiv, 2026-08-17); logged in D-001 Part III.

1. The 2026 Weil-form frontier (live literature)

Suzuki 2026 (arXiv:2606.09096) — "Weil's quadratic form via the screw function". One framework unifying Yoshida (1992), Bombieri (2001/2003), Connes–Consani (2023), Connes–Consani–Moscovici (2025+). Screw functions make the distributional Weil form continuous-function tractable. ALL UNCONDITIONAL. KEY CONJECTURE: a self-adjoint Hilbert–Pólya operator arises as the a→∞ limit of self-adjoint nonlocal first-order differential operators on [−a,a]. ⟹ THE Track C↔B bridge, in the literature, 2 months old.

Groskin 2026 (arXiv:2607.02828v3) — finite Guinand–Weil dictionary + archimedean tail order for the Connes–van Suijlekom/CCM truncation (prime cutoff c, band N):

• Thm 1: every Galerkin vector v yields an explicit band-limited test function g_v whose zero sum EQUALS ⟨v,Qv⟩ exactly (dimension 2N+1 source quotient; non-collapsing pole-neutral subfamily).
• Thm 2: omitted archimedean tail = totally positive Cauchy–Stieltjes increment ⟹ certification budget B_T ~ (2N+1)ρ log T/(π²T), ρ = 2π/log c. Rule: finite-cutoff positivity ⟹ cutoff-free positivity; eigenvalue < −B_T ⟹ certifies an off-line zero; [−B_T,0) = indeterminate.
• Verified over first 512 zeros; reproducibility package included.

Kim–Hong–Kim–Choi–Jang–Kim 2026 (arXiv:2607.24830v2) — numerical Suzuki operator: Archimedean spectral law to 30 digits; operator-form positivity criterion (bounded residual growth ⟺ zeros on line; injected off-line zero → exponential blow-up); lowest eigenvalue strictly positive, SUPEREXPONENTIALLY decaying. Explicitly not an RH proof.

Connes 2026 (arXiv:2602.04022) — survey; extremizing Weil form with primes < 13 approximates first 50 zeros (accuracy 2.6e-55–1e-3); approximants PROVED on-line (finite objects); convergence open.

2. Where NGUYEN-RH-1 sits in this landscape

Our inherited machinery is EXACTLY a Weil-form truncation with special properties: G̃ = Gabor-basis Galerkin compression at critical density (Lemma 2.1 = no-aliasing frame; Groskin's "dictionary" property holds for us: G̃ values are exact zero sums by construction (2.11)). The Anthropic paper adds what this literature does NOT have: unconditional control of rank/positive index of such truncations via the prime-side second moment (Thm 5.7). The Suzuki/Groskin axis has the truncation theory but no unconditional second-moment input.

3. The cross-pollination question (task 3.4), now precise

Q-3.4a (inertia input): The paper's chain gives unconditional lower bounds on rank/n₊ of G̃-type truncations (that's what 2/3 IS). Groskin's rule certifies full positivity from finite-cutoff eigenvalues vs budget B_T. QUESTION: does the unconditional n₊ lower bound + B_T budget sharpen the indeterminate zone [−B_T, 0) — i.e. can the second-moment input shrink the uncertifiable band?

Q-3.4b (positivity horizon): Kim et al. report the lowest eigenvalue of the Suzuki operator decays superexponentially. If truncated Weil forms at support a have min eigenvalue decaying superexponentially in a, then certifying positivity at support a needs precision ~ 1/λ_min(a) — a COMPLEXITY WALL for the positivity route. Quantify: λ_min vs support in our sandbox (E-006). This is an obstruction-map measurement.

Q-3.4c (operator candidates, Track B feed): Suzuki's nonlocal first-order operators on [−a,a] are concrete self-adjoint candidates. Compare their spectra against our E-002 zeros at a-scales we control. (Longer-term; Phase 2.4 adjacency: de Branges/canonical systems.)

4. Next sub-stage: E-006 design (task 3.1 numerics)

Compute the eigenvalue floor of our Gabor-truncated Weil form G̃ as a function of bandwidth λ (support a = λ·log T): λ ∈ {0.5, 1.0, 1.5, 2.0} at T = 2000 (zeros from E-002; λ > 1 = beyond provable positivity). Measure: λ_min(λ), log-decay rate, comparison with Groskin budget B_T. Interpretation: if λ_min stays comfortably positive at λ = 2, finite data sees no horizon yet; rate extraction → extrapolate where the indeterminate zone starts to matter. All [NUMERICAL-EVIDENCE].