Session 006 — Regroup & plan revision (user-requested)

Date: 2026-08-18 | Type: organizational (no new experiments)

Purpose

User asked for a full regroup before continuing: review goal, data, scripts, notes, checklist progress; revise the checklist; check off what qualifies; add new tasks the work so far implies; confirm everything is documented and saved; then state the next step.

Archive audit (findings + fixes)

All experiments E-001…E-016 have BOTH CSV and JSON sidecars; every script has data and vice versa. E-007b/E-009b are registered secondary runs sharing scripts with E-007/E-009 (documented pattern, not orphans).
research.db hygiene fixed this session:
Sessions 003–005 were unregistered → registered (now 1–5 present).
row_count was 0 for E-010/E-011/E-012/E-013/E-014/E-016 → recounted from CSVs and corrected (48 / 2000 / 9999 / 3 / 80 / 3519); E-015 also corrected to 3000.
Paper: draft.tex still a 56-line skeleton (5-entry references.bib) — acceptable at this stage; population begins in Phase 5 or when the first [PROVED] result lands.
Git was clean at session start; no data at risk.

Strategic assessment

Goal unchanged: RH proof or maximal rigorous progress (criteria 1–4).
Two live leads, everything else is monitoring/background: 1. Suzuki / Cor 1.6 reduction line (Track B): an explicit RH-equivalent convergence target. Status: maximal lattice + exact interlacing to a=10 (E-014), θ=π candidate (H4 read-off), θ=π zeros real with stiff z₁·a≈4.5 law (E-016 F1/F2), but V3 Herglotz reconstruction mismatch (rel ~0.3) unresolved and the required density-collapse transition not yet visible (scale limitation — needs a≫10 and/or the analytic v± limit). 2. Li coefficients (Track C): λ_n > 0 certified to n=3000 from the arithmetic side; ratio → 1. The asymptotic attack (prove λ_n>0 for all n≥N₀) is now its own task — it is the most direct criterion-#1 path open to us.
Recognition recorded: the Suzuki line IS Phase 2 realized (T_a is a canonical-system/de Branges construction; Galerkin A_a is a validated Weil-form operator). Phase 2 restructured accordingly.

Checklist changes (committed to NGUYEN.md)

Phase 2 rewritten: 2.1 → [~] (partial via Suzuki Galerkin), 2.4 → [~] (active via Suzuki line), new tasks 2.6 (E-016 V3 root cause), 2.7 (Q-S3a analytic v± limit), 2.8 (Q-S3b backend to a~12–14).
3.4 → [x] (question answered YES; continuation lives in Phase 2).
NEW 3.5: Li-coefficient asymptotic attack.
NEW 1.6: ξ′-world bandwidth-one ceiling (LawN256 analogue).
NEW 1.7: T4b scoping (node mechanism on higher derivatives ξ⁽ᵏ⁾).
3.2 note updated: further n-pushes only in support of 3.5.
Nothing pruned as unnecessary; 1.2 stays DEFERRED (D-013 rationale).

Active order (carries forward the user-approved A3→A1→A2→C→B queue,

renumbered to the revised checklist)

2.6 — root-cause E-016 V3 (three candidates + discriminating tests in D-024 §4). THIS is the next action.
2.7 — Q-S3a analytic v± → A_∞^{−1}e^{±·}.
2.8 — Q-S3b prime-sum backend → a ~ 12–14.
1.6 — ξ′-world ceiling. 5. 1.7 — T4b scoping. Background/standing: 3.5 (Li asymptotics — start with §4A.3 literature sweep when a slot opens); E-015-class runs launch detached when useful.

Decisions

No new ledger result promoted from E-016: F1–F4 stay in D-024 until V3 is resolved (honesty rule — a failed check's diagnostics are [NUMERICAL] hints, not results).
Break-prompt rule and detached-run rule (sandbox disabled) reaffirmed.

Session 006 cont. — Task 2.6: E-016 V3 root cause (COMPLETE)

What was done

Wrote E-016b (T1–T5 discriminating tests). Key new diagnostic: ARGUMENT PRINCIPLE on circles |z|=r — the real-gauge brentq scan can only see real zeros; winding counts all.
Breakthrough observation: the D-024 residual's low-y part matches 2/(πy) to 4 digits — the fingerprint of a zero pair at |z|≈0 missing from the recon.
E-016b results: winding excess EXACTLY 2 at all radii (r=1e-6…0.39), a=4 AND a=10. Pair located at ±iε, ε=5.8e-9 (a=4), 2.1e-9 (a=10) — Galerkin W has a DOUBLE ZERO AT THE ORIGIN (artifact vs Thm 1.5; driven by Σc≈0; this is D-024 F3's "defect at the origin" identified). recon+pair kills the y=0.5 residual: +1.274 → +0.0007.
Second defect found: EM tail in E-016 recon has a SPURIOUS FACTOR y (∫y/(y²+z²)dz = arctan, no y). Predicts the y=20 deficit exactly at both a (a=10: −2.53 predicted vs −2.53 observed). The notail variant matching analytic to +0.09 was the smoking gun: analytic side had the correct tail all along.
Refuted: candidate (a) Im κ (κ≈0 by symmetry; T1 = noise ratio); candidate (c) float64 zero-finder noise (mp60 dps-50 relocation identical); float64 analytic cancellation (E-016c V6: direct vs dps-50 ≤ 4.4e-15 for all y≤40, both a).
E-016c repair: rescaled stable evaluation (V5 1.75e-16 vs dps-50) + corrected recon (tail fix + pair): V3R 1.99e-5 (a=4), 1.46e-5 (a=10) vs 0.02 threshold — ALL CHECKS PASS.
Filed D-025; registered E-016b/E-016c/D-025 in research.db; updated derivations/index.md, NGUYEN.md (2.6 → [x]), STATE.md.

Lessons logged (D-025 §8)

Cross-checks need an independent ABSOLUTE anchor (plateau law a/π caught a shared missing-parentheses bug that implementation-vs- implementation agreement missed).
The winding-excess diagnostic is now standard equipment for the Suzuki line (use in 2.8 at a~12–14).
E-016/E-016b scripts frozen as historical record; corrected recon formula lives in E-016c.

Task 2.7 (Q-S3a): analytic v± → A_∞^{−1}e^{±·} — start with §4A.3 literature sweep (Suzuki §7, canonical systems inverse problems).

Session 006 cont.2 — Task 2.7: Q-S3a (COMPLETE — R-003 breakthrough)

What was done

Read Suzuki 2606.09096 in full. §8.3 contains the key: the first-kind equation g∗v± = h± with h± = (e^{±x} ± i)·1_{[−a,a]} and the formal FT step the paper only uses heuristically.
Made it EXACT (D-026): with the out-of-interval tail τ±, W(a,θ;z) = F(z)·(R₀ + R_τ), F = z²ξ/ξ′ — Cor 1.6's target itself. R₀ closed form = 2e^{a+iθ/2}sin(az−θ/2) + o(eᵃ) ((1±iz) = i(z∓i) kills the prefactors exactly). Cor 1.6 ⟺ (★): R_τ cancels the sine skeleton up to a zero-free remainder — finite-interval Wiener–Hopf remainder on ĝ = z⁻²ξ′/ξ; RH enters via symbol factorization (D-H/Epstein gate consistent).
Instant reconciliation with ALL prior numerics: θ=0 lattice πk/a (E-014 H3); θ-independence + interlacing (H1); z₁·a=4.5 vs 4.712 deformation (F2); growth-dominated read-off + phase agreement (H4); and the E-016b origin pair — v̂±(0)=0 ⟹ W(a,θ;0)=0 double EXACTLY (the ±iε pair = numerical splitting of the exact degeneracy).
E-017 (measurement): R_τ = W_gal/F − R₀ computed directly at a=4..10. Findings (D-027): off-axis cancellation already 99.8% (y=16.5) at a=4; arg M = ±90° exactly; cancellation weakens toward ℝ (0.2%→4.7% at y=0.5); θ=π preferred (4.7% vs 6.8%) — first quantitative reproduction of the H4 preference. Open content of Cor 1.6 isolated: the strip around ℝ.
R-003 promoted to result ledger (#5): [PROVED (modulo std tempered-FT bookkeeping)] + [NUMERICAL-VALIDATED]. All artifacts registered (E-017, D-026, D-027, R-003).

Lessons logged

Suzuki's paper hides the exact identity in a "suggests" paragraph (§8.3 end): read the appendix sections, not just the main theorems.
The Galerkin ±iε "artifact" (D-025) turned out to be the numerical shadow of an EXACT double zero — a failed check that became a confirmation of the factorization. Cross-domain consistency pays.

Task 2.8 (Q-S3b): E-018 descent profile (|R_τ/R₀| on z = x+iy₀, y₀ ∈ {1, 0.3, 0.1}) first — cheap, informs the backend push to a ~ 12–14 with winding-based zero-flow tracking.

Session 006 cont.3 — Task 2.8(i): descent profile (COMPLETE)

What was done

E-018: |M|/|R₀| on z = x+iy₀, y₀ ∈ {1, 0.3, 0.1}, a=4..10, θ=π/0. ALL CHECKS PASS (C1 off-axis W < 1e-10 vs dps-50 after fixing a sign bug in the mpmath reference; C2 complex-F vs flint acb 5.3e-10).
Findings (D-028): median |M|/|R₀| ~ 5–7e-3 UNIFORM across the strip — the (★) cancellation does not break down toward the real axis; it is the limiting off-axis behavior already. Slowly increasing with a. θ-ordering REVERSED vs E-017's imaginary-axis point — the θ=π preference is metric-dependent (recorded honestly; any θ(a) theory must explain both metrics).
Structural conclusion: the entire remaining content of Cor 1.6 is the a-evolution of M's REAL zero/pole configuration (γ-locking + outflow); no off-axis envelope measurement can see it. Task 2.8(ii) refocused accordingly: zero-flow, γ-locking metric, strip winding counts at a=12 (needs blockwise e²⁴ sieve backend).

Task 2.8(ii): backend v2 (blockwise Λ, checkpointed prefix sums, e²⁴)

E-019 at a=12 (zero-flow: inflow bounce? γ-locking? strip winding).

Session 006 cont.4 — Task 2.8(ii): backend v2 + a=12 (COMPLETE — no transition)

What was done

Built prime_backend_v2.py: exact segmented Λ sieve (survivors = primes; prime powers by exact index), blockwise to e²⁴ with running cumsums, query-offline design, npz cache. Two bugs caught by own validation gates (multiples-vs-index Λ; off-by-one cumsum limit — O(0.2) at small n because t−log n ~ 1/n is not small there). Validated: V-A1 2.2e-7 (float64 reordering floor), V-A2 pipeline at a=10 (Q dev 4.4e-8, z₁ dev 6.9e-9, count exact).
E-019 at a=12: maximal lattice (1145/1146, window edge only); z₁·a = 4.4953 (stiff law continues, slow drift down); gap CV 0.186 vs 0.198 at a=10 — lattice STIFFENS; origin pair persists (+2.0000 winding excess, dense-edge rerun after an undersampling artifact at 400/edge); NO other non-real zeros (|Re|≤40, |Im|≤0.1); γ-proximity 5e-5–2e-4 is chance-level (minimizing γ changes with a; h-noise ~5e-4); envelope |M|/|R₀| stationary 6.5e-3.
D-029: task 2.8 CLOSED ("anything change by a=12?" — NO). a=13+ deprioritized by evidence (§8.4a), revivable in ~30 min via backend cache. γ-locking claim protocol established (fixed-γ tracking over ≥3 a-values + beat h-noise + beat 1e-4 chance level).

Task 1.6: ξ′-world bandwidth-one ceiling (LawN256 analogue).

Session 006 cont.5 — Task 1.6: ξ′-world ceiling (COMPLETE — saturated)

Context

User asked to double-check the plan: original Phase 1 (1.1–1.5) IS complete; 1.6/1.7 were NEW regroup additions (session-005 queue items C and B). User chose option #1: do 1.6 then 1.7 as queued.

What was done

E-020 (adversary-side, LawN256-mirror pool LP): control pool matches rows exactly (ε=0) but min-simple mix = 0.9959 (paper 0.6818) — finite pool can't realize the position-OPTIMIZED extremal adversary; D₁-law rows unrealizable by the pool (ε=155.6). Documented as machinery limitation (needs exact-rational continuous-position LP).
E-020b (attack-side, the decisive computation): Rayleigh-quotient window optimization for kernel D₁. CONTROL recovers the MT window (R=1.32749947 vs 1.32749930, 1.7e-7) — machinery validated. TARGET: R_opt = 1.13135921 vs R_quartic = 1.13136020 — the paper's quartic window is the window-class optimum to 1e-6; certificate 0.86864 is SATURATED.
D-030 filed; 1.6 CLOSED: the perfect-square node buys ~0 headroom beyond the paper's own window; nothing transfers to ξ (R-002 stands).

Task 1.7 (T4b scoping): higher-derivative ξ⁽ᵏ⁾ node mechanism — scope only; verdict pursue/close.

Session 006 cont.6 — Task 1.7: T4b scoping (COMPLETE — closed)

What was done

Read Zeta23/XiPrime/Coeff/Basic.lean: the family C(N;Λ⋆) at primes is Λ(p)·(log p/Λ⋆ − 1) — degree-1 in log p; the D₁ node = its root at log p = Re Λ⋆. Mechanism iteration is formal: degree-k prime polynomial → k nodes.
Leading-order pair integrals: s(1−2s)² = 1/15 (fill ×2.124 → 0.14162 ✓ E-013); s(1−2s)⁴ = 0.0286 → ξ″ flat ≈ 0.94 [HEURISTIC estimate]. Split nodes worse than co-located.
Verdict (D-031): CLOSE T4b. (i) ξ⁽ᵏ⁾ certificates don't touch the ξ/ξ′ records; (ii) RH via Speiser/Levinson–Montgomery needs 100% ξ⁽ᵏ⁾-on-line, unreachable in the 2-moment data class at EVERY order (G-001 symmetry); (iii) cost high. Revival condition noted.
Session-005 queue (A3→A1→A2→C→B) now FULLY DONE.

Two standing fronts (user's call): 3.5 Li asymptotic attack (criterion-#1 path) or the (★) analytic attack (Suzuki line).

Session 006 cont.7+8 — both fronts' designs (COMPLETE) + next-stage analysis

User directive

Do 3.5 then (★), one after another, then analyze data for next stage.

3.5 (Li asymptotic attack) — design complete

D-032 literature map (arXiv sweeps + Lagarias math/0404394 PDF read directly, Thms 5.1/6.1 + §6 contour argument).
KEY structural fact (Voros math/0404213, math/0506326): "λ_n > 0 ∀n≥N₀" ⟺ RH at ANY N₀ — no partial-strength version exists.
ACHIEVABLE deliverable (D-033): EXPLICIT finite-height theorem — λ_n > 0 ∀ n ≤ N₁ (est. 10²⁰–10²⁴) from Lagarias Thm 6.1's bridge (RH to height T ⟹ S_f = O(√n log n) for n ≤ T²/4log²T) + Platt's T₀ ≈ 3e12 + traced constants + finite verification. Nobody has assembled this chain. Wall: arithmetic-side triangle bounds explode ((15/14)ⁿ); the three-way cancellation IS RH.
Work plan: 3.6 (D-036 constants), 3.7 (E-021 S_∞ exact), 3.8 (D-037 assembly → R-004).

(★) (tail-cancellation) — design complete

D-034: WH factorization + RH is unexplored (26 arXiv hits all Riemann–Hilbert); Krein/Gohberg–Feldman/de Branges–Krein machinery exists and fits.
D-035: route = strong resolvent convergence D_{a,θ(a)} → D in the (indefinite) Krein-space limit — Cor 1.6 ⟺ "D has real spectrum" (deliverable D1, criterion-3-shaped). Three-layer wall: W-1 index unknown (off-line pairs), W-2 real-spectrum persistence = RH, W-3 factorization circular for proofs.
Honest reading: the 99.8% off-axis cancellation is TAUTOLOGICAL (entirety of W forces it); the ℝ-envelope (~0.5%) carries the whole zero-consolidation problem. E-017 F5's slow increase of the leftover with a is the honest signal: nothing converging yet.

Next-stage analysis (the data's verdict)

Both fronts are RH-equivalent at their cores (Voros for Li; W-2 for (★)) — no surprise; but BOTH yielded concrete achievable packages: 3.6–3.8 (an explicit theorem) and 2.9-D1/2.10/2.11 (a reduction + new observables).
Recommended order (STATE.md ACTIVE PLAN): 3.6 → 3.7 → 2.10 → 3.8 → 2.9-D1 → 2.11. The finite-height Li theorem is the most concrete new result within reach; E-022 (spectral measures) is cheap and opens the Krein-frame analysis.
Machine/time notes: 3.7/2.10 are short CPU; 3.6/3.8/2.9-D1/2.11 are paper-work.

Session 006 cont.9 — task 3.6 done; 3.7 at ~90% (MID-TASK STOP, 5 PM)

What was done

3.6 (D-036, COMPLETE, committed): explicit contour bounds for ζ traced through Lagarias §6 — N(T) via Trudgian, partial-fraction remainder, segment bounds; dominant term I_III ≈ 4.7√n log n.
3.7 (E-021, ~90%, interrupted): exact S_∞(n) + |S_f(n)| + tightened constants. Debugging saga (all fixed in current script): (i) 30k zeros via mp.zetazero too slow → reduced to 4.6k (T=5000) + partial-summation tails + npy cache; (ii) flint acb zero-division in the ξ fd → zeta_logder switched to mpmath (E-018 C2-validated); (iii) η strings held "(x + 0.0j)" → mp.re before nstr; (iv) 5e7 string parses bottleneck → parse once at full dps; (v) n=10⁴ rows cost ~2 h (cost ∝ n²·dps) → N_MAX=5000 (1e4 rerun is cheap via caches); (vi) KEY FIX: S_f convention — Σ(−1)^{j−1}C(n,j)η_{j−1} sums +C|η| and explodes ~(4/3)^n (V1 rel dev 1e143, caught by E-015 cross-check); the validated E-011/BL convention is S_f = Σ C(n,j)η_{j−1} PLAIN (η alternates internally). Failed CSV/JSON deleted; caches kept.
Partial results already in log: A_use=0.9, B0_use=1.0 (|R| max 4.75); η₀ = −0.576975 = −γ ✓ (σ₁-tail 1.2e-3); S_∞ residual → c₀ ≈ −0.25000 (Lagarias Thm 5.1 O(1) constant for ζ).

RESUME (STATE.md IN PROGRESS)

Rerun E-021 --force (caches → ~20 min), then register, D-037, commit.