D-025 — E-016 V3 failure ROOT-CAUSED and REPAIRED (task 2.6 complete)

Date: 2026-08-18 | Session: 006 | Task: 2.6 (Q-S3c remainder) Status: COMPLETE. V3 failure = two recon-side defects (missing origin pair + EM-tail factor-y bug); analytic side was never broken. Repaired V3 passes at 1.5–2.0e-5 (threshold 0.02) — E-016c, ALL CHECKS PASS.

1. The failure (recap)

E-016 V3 compared ρ_W^analytic (Galerkin z-derivative) vs ρ_W^recon (located real zeros + Euler–Maclaurin lattice tail): residual at a=4 +1.274 (y=0.5), +0.637 (1), +0.313 (2), +0.073 (5), −0.179 (10), −0.993 (20); rel ~0.3. D-024 §4 listed candidates (a) Im κ ≠ 0, (b) low-zero bookkeeping, (c) large-z float64 zero-finder noise.

2. Key observation that cracked it

2/(πy) at y = 0.5, 1, 2 equals 1.2732, 0.6366, 0.3183 — matching the residual's low-y part to 4 digits. 2/(πy) is exactly the Poisson contribution of a zero pair at |z₀| ≈ 0 MISSING from the recon.

3. E-016b findings (argument-principle diagnostic)

New reusable diagnostic: the real-gauge brentq scan (E-016) can only see REAL zeros; winding numbers of W on |z| = r count ALL zeros.

• Winding excess = 2 at every radius (r = 1e-6 … 0.39 at a=4; same at a=10): Galerkin W(a,π;·) has exactly ONE extra zero pair near the origin, invisible to the real-axis scan.
• Pair located: zeros at ±iε, ε = 5.8e-9 (a=4), 2.1e-9 (a=10) — effectively a DOUBLE ZERO AT THE ORIGIN. Genuine: W(iy) = i·(real) by mirror symmetry, and Im W(iy) crosses 0 at y = ε with |W| at the dps-50 noise floor (2.8e-49).
• Origin of the pair: Σc ≈ 0 makes W(0) ≈ 0; the mirror symmetries (z→z̄, z→−z̄) force any near-origin non-real zero to be a purely imaginary pair. This is a Galerkin artifact: Thm 1.5 (all-real zeros) holds for the true W; the discretization collides the first real pair at 0 and splits it onto the imaginary axis. This IS the "defect at the origin" of D-024 F3, now identified.
• Confirmation: recon + exact pair contribution (2/π)·y/(y²−ε²) kills the y=0.5 residual: +1.274 → +0.0007 (a=4), +1.275 → +0.0017 (a=10).

4. The second defect: EM-tail factor-y bug (E-016 script)

The E-016/E-016b recon tail was coded as tail = (y/d)(π/2 − arctan(z₀/y)) ← SPURIOUS FACTOR y The integral is ∫_{z₀}^∞ y/(y²+z²) dz = π/2 − arctan(z₀/y), so the correct tail is (1/d)(π/2 − arctan(z₀/y)). The bug overshoots the recon by (y−1)·(correct tail): predicts −1.08 at y=20 (a=4) vs observed −0.99 ✓, and −2.53 at y=20 (a=10) vs observed −2.53 ✓✓ (exact). Smoking gun in E-016b: the notail variant (located zeros only) matched the analytic side to +0.09/+0.17 at y=20 — the analytic side had the correct tail all along.

5. Verdicts on the D-024 candidates + refutations

• (a) Im κ ≠ 0: MOOT. κ = W′/W(0) ≈ 0 by symmetry (x symmetric, c real ⟹ W(ℝ) = 2i·(real gauge)); the naive T1 measurement is a noise-level ratio of two tiny quantities (W(0)≈0), not a κ detection.
• (b) low-zero bookkeeping: CONFIRMED, but the missing object is the NON-REAL origin pair (§3), not a missing real zero. The 381-vs-382 count gap is just window-edge bookkeeping (382nd lattice point at ≈300.4 > Z_MAX).
• (c) large-z float64 zero-finder noise: REFUTED. mpmath dps-50 relocation of all zeros ≤ 60 changed nothing (mp60 ≡ trunc60 variant).
• (c′) float64 cancellation in the ANALYTIC ρ_W at large y (hypothesized mid-investigation, e^{±ay} dynamic range): REFUTED. E-016c V6: direct float64 vs mpmath dps-50 reference agrees to ≤ 4.4e-15 for all y ≤ 40 at both a=4 and a=10. The E-016 analytic machinery was exact.

6. E-016c repair (ALL CHECKS PASS)

• Rescaled stable evaluation V(z) = e^{i x_min z}Ṽ(z) (all exponents ≤ 0 for z = iy, y > 0), W′/W = i·x_min + B′/B, cross-validated vs mpmath dps-50: max rel dev 1.75e-16 (V5). [Note: the rescaled path is validated but not needed — the direct evaluation was already exact; kept as machinery for larger (a, y) where e^{±ay} cancellation will eventually bite: estimated y·a ≳ 300 in float64.]
• Recon with corrected tail + origin pair: V3R shape rms rel = 1.99e-5 (a=4), 1.46e-5 (a=10) vs threshold 0.02 — V3 REPAIRED with three orders of margin.

7. Consequences

• E-016 findings F1–F4 STAND: they were computed from the analytic side and the real-zero set, both validated (V6, mp60). Only the V3 check itself was affected, and it was a recon artifact.
• E-016/E-016b scripts are FROZEN as the historical record (their CSV/ JSON correspond to the buggy tail); the corrected recon lives in E-016c. Any future Poisson-recon work must use the E-016c formula.
• The winding-number diagnostic (§3) is now standard equipment for the Suzuki line: it detects non-real zeros that the real-gauge scan cannot see. Use in Q-S3b (2.8) when tracking zeros at a ~ 12–14.
• Follow-up (not blocking): h-refinement of ε (does the origin pair realign onto ℝ as h→0?), and the small-y-weighted θ-metric (D-024 F4) can now be built on the validated ρ_W.

8. Honesty notes

• V5 initially "passed" with a shared missing-parentheses bug in both the rescaled and mpmath paths (−xm outside the /π) — caught because the plateau value 3.99 ≠ a/π was obviously wrong. Lesson logged: cross-checks must include an INDEPENDENT absolute value (the plateau law a/π), not only agreement between two implementations.
• ε is indistinguishable from 0 at the h=0.01 resolution; treated as a double zero at the origin for the recon (difference < 1e-16 at y ≥ 0.5).
• eps fallback in E-016c reports ε = 1e-12 (scan floor): the pair term is 2/y to 1e-16 regardless; E-016b's findroot values (5.8e-9/2.1e-9) are the better estimates.