D-024 — E-016 analysis: Herglotz fine structure at θ=π (Q-S3c) — partial

Date: 2026-08-18 | Session: 005 | Task: Q-S3c Status: PARTIAL — machinery validated (V1, V2, V4 pass); V3 (zero-Poisson reconstruction) fails at rel ~0.3, root cause open. Diagnostic findings on the θ=π zero structure recorded.

1. Setup

For entire W with all-real zeros z_j and exponential factor e^{κz}, W′/W(z) = κ + Σ_j 1/(z − z_j) (symmetric sum; ±-pairs make Σ 1/z_j² absolutely convergent), so with ρ_W(y) := −(1/π) Im(W′/W)(iy), ρ_W(y) = −Im κ/π + (1/π) Σ_j y/(y² + z_j²) (Poisson-smoothed density). E-016 computes ρ_W two independent ways: (i) analytic Galerkin z-derivative of V(z) = ĥ(z) Σ c_j e^{izx_j} (exact, no finite differences); (ii) located zeros + Euler–Maclaurin lattice tail (spacing π/a, phase matched at z_last = 300). Target comparison: ρ_F(y) = (1/π)[2/y + ξ′/ξ(½+y) − ξ″/ξ′(½+y)], cross-checked vs finite differences of log|F(iy)| at 1.3e-8, ξ vs flint at 8.8e-17.

2. What passed [NUMERICAL-VALIDATED]

• V1: plateau of ρ_W on y∈[10,20] equals a/π to ≤0.15% at all a — the Herglotz density sees exactly the maximal lattice (consistent with E-014 H1).
• V2: ρ_F analytic vs finite-diff of log|F(iy)|: 1.33e-8 (floor set by mpmath-zeta ~16-digit noise in this env; documented caveat).
• V4: mirror symmetry v−(x)=v+(−x), rel < 1e-6 (machinery regression).
• Real-gauge identity: c = √(n₊/n₋) = 1 EXACTLY (mirror symmetry), so for real z, Vm = conj(Vp) and W(a,π;z) = 2i(z Im Vp − Re Vp): pure imaginary, zeros = sign changes of the real gauge g(z) = z Im Vp − Re Vp. This makes θ=π zero location EXACT (brentq), unlike the |W|-minima + polish used in E-014 P1/P3 at θ=0 (which was exact for a different reason: θ=0 gauge 2(z ReV + ImV) also real).

3. New diagnostic findings [NUMERICAL]

• F1. θ=π zeros are real. Sign changes of g confirmed at h=0.01; the all-real-zeros property (Thm 1.5) survives Galerkin at θ=π.
• F2. Stiff first-zero law: z₁(θ=π)·a = 4.504, 4.499, 4.497, 4.496 at a = 4, 6, 8, 10 — nearly constant ≈ 4.5 (≈ 1.43π), drifting down ~0.2% over the sweep (above the 4e-4 h-drift scale). The θ=π lattice is NOT the rigid half-shift (π/2a)(2j−1): the shift relative to the θ=0 lattice grows from ~0.34 toward the half-period 0.39 as j grows (a=4: shifts 0.339, 0.363, 0.372 for j=1,2,3).
• F3. Defect at the origin: no θ=π zero in (0, π/a) at any a (min |g| there is O(30) vs zero-crossing scale). With θ=0 zeros at 0, ±π/a, …, strict interlacing is locally broken near 0 — the first θ=π zero sits at ~4.5/a ≈ 1.43·(π/a), i.e. inside the SECOND θ=0 interval. Global density is unchanged (counts 381/572/763/954 in (0,300] vs aT/π = 382/573/764/955).
• F4. θ=π vs θ=0 on raw L2 against ρ_F: essentially tied (a=4: d_pi=1.2201 vs d_zero=1.2165; same to 0.3% at all a). The raw y∈[1,20] L2 metric is plateau-dominated and does NOT resolve the H4 (E-014) θ=π preference; a small-y-weighted metric is needed.

4. V3 failure — status and candidate causes

ρ_W^analytic − ρ_W^recon at a=4: +1.274 (y=0.5), +0.637 (1), +0.313 (2), +0.073 (5), −0.179 (10), −0.993 (20). Monotone-decreasing, one zero crossing near y≈7. Candidates (not yet discriminated): (a) Im κ ≠ 0 plus a y-dependent term. A pure constant Im κ gives a constant shift; the observed diff is NOT constant (spans 2.3), so κ alone cannot explain it. κ is measurable directly: Im κ = Im(W′/W)(0) = Im(g′(0)/g(0))-type evaluation at z=0. (b) Low-zero bookkeeping. Located zeros start at 1.1261 (a=4); count 381 vs lattice 382. A single missing zero at small z₀ adds (2/π)y/(y²+z₀²): at y=0.5, z₀≈1.1 that is +0.43 — same sign, right magnitude for part of the +1.27 residual, not all. (c) Large-z zero-finder contamination. Even with brentq on the real gauge, spurious/missed brackets at z ~ 100–300 (where Vp is a near-cancelling sum of ~10³ oscillatory terms — float64 noise floor) distort recon increasingly with y (each zero at z₀ contributes ~ (2/π)y/z₀² for y ≪ z₀; errors accumulate over ~10³ zeros). The −0.99 drift at y=20 has this signature. Discriminating test: recompute recon using only zeros ≤ 60 with EM tail from z_last=60 — if the y=20 mismatch changes materially, (c) is confirmed. Resolution = first task of session 006.

5. Honesty notes

• All E-016 numbers at h=0.01 (z₁ h-drift 4e-4 relative, E-010).
• F2's "4.5" is a 4-point fit; whether z₁·a → a limit or keeps drifting needs a > 10 (Q-S3b backend) and/or h-refinement.
• ρ_F trusted to ~1e-9 (mpmath zeta noise), ample for diagnostics; NOT a certified value.
• E-016 JSON records the V3 failure; CSV contains the full ρ_W/ρ_F grids for reuse.