Current task: —

Updated 08-17-2026

Session 001 — 2026-08-17 — Archive Installation

Goal

User requirement: multi-stage project with computer shutdowns between stages. All work must be persisted so no stage is ever redone. Also: a scientific paper will be written from this archive.

Decisions (user-approved)

Work done

Outcome

Archive ready. No research content produced yet.

Next session

Begin Phase 0.1 (Notes.txt quantum-equation index) — see STATE.md.

Addendum — Phase 0.1 completed (same day)

Task: Parse Notes.txt; index all quantum equations for Track B.

Output: research/derivations/D-001-quantum-formalism-index.md

Key findings while indexing (cross-pollination, per §4A.2):

  1. Q6 (Heisenberg uncertainty) ↔ paper's Lemma 2.1 critical-density Gabor frame AND the ≈0.682 bandwidth-one ceiling: Fourier-support limits ARE uncertainty constraints. Formalizing via Benedicks-type uncertainty principles is a concrete Track A.1.1 sub-attack. → logged as open thread.
  2. Q12 (harmonic oscillator / Hermite functions): self-dual Fourier eigenbasis — natural test functions for CGdL20-style SDP (Track A.1.2).
  3. Q14 (Ehrenfest → semiclassics): Gutzwiller trace formula with primes as periodic orbits = formal restatement of Weil explicit formula; the master heuristic of Track B (Berry–Keating program).
  4. Q2/Q13: unitary (Schrödinger) vs. dissipative (heat) evolution cleanly separates Track B (spectrum) from Track C (de Bruijn–Newman Λ=0).

Tooling: extended db.py with add-derivation and add-graveyard commands.

Outcome: 0.1 complete. D-001 is now the living master tool index.

Addendum 2 — Phase 0.2a completed (same day)

Task: Paper statements + notation map (sub-stage 0.2a of 0.2).

Setup: Read tool cannot parse PDFs in this environment → installed pypdf into .venv; extracted all 17 pages to research/paper/anthropic-paper-fulltext.txt (reusable by all future sub-stages — no re-extraction needed).

Output: research/derivations/D-002-paper-statements-notation.md

Outcome: 0.2a complete. Next: 0.2b (zero side, §4, Props 4.1–4.3) → D-003.

Addendum 3 — Phase 0.2b completed (same day)

Task: Reproduce the zero side (paper §4) independently → D-003.

Work: Re-derived and checked every step of Props 4.1–4.3, Remark 4.4, Cor 4.5 (not copied): block structure P/Q with the 2×2 signature-(1,1) pull-back for off-line pairs (verified the needed m_ρ = m_{1−ρ̄} pairing); trace identity tr G̃ = N(I′) + O(T^{1/2}L²) via Lemma 2.1 at complex z + Paley–Wiener tails; tail ‖Ẽ‖₁ ≪ T^{−1/2} (D₀=√T threshold confirmed necessary); C²-smoothing necessity confirmed as a genuine constraint on future window design. All claims [VERIFIED] — table in D-003.

Attack-relevant observations (logged in D-003 §Observations):

  1. Zero side is entirely frame-theoretic — Lemma 2.1 does all the work; a Track B operator model must reproduce the critical-density Gabor kernel (links D-001 Q6 → Phase 2.3).
  2. Off-line zeros are charged to Q at cost 2n₊(Q) — the "4b" of Lemma 3.2 is literally the price of not assuming RH.
  3. Slack identified: tr P ≤ N_0(I′) forfeits on-line multiple-zero multiplicity mass (m_ρ−1 per multiple zero) — a concrete refinement target for beating 2/3.

Outcome: 0.2b complete. Next: 0.2c (prime side §5) → D-004.

Addendum 4 — Phase 0.2c completed (same day)

Task: Reproduce the prime side (paper §5) independently → D-004.

Work: Re-derived and checked: auxiliary envelope estimates (5.1–5.7); Prop 5.2 reduction to the double integral (edge/tail errors; log log T source identified as ∫ϑ ≪ log L); Prop 5.3 archimedean term; Prop 5.4 prime term — diagonal n=m gives main term via (5.11), off-diagonal bounded (not evaluated) by Montgomery–Vaughan with δₙ⁻¹ ≤ 2n; Prop 5.5 cross-term decoupling; Lemma 5.6 window constant (the G″ = ψ″+2ψ = 0 stationarity trick re-derived by hand); Theorem 5.7 assembly. Also read §6 chain and Remark 6.1 (λ-parameter).

Hand-verified numerics: R(ψ₀) = 4/3; R(ψ_MT) = ½+(1/√2)cot(1/√2) = 1.327325 ⟹ 2−R = 0.672675, (3−R)/2 = 0.836338 — matches paper's 0.6725/0.83625. (Task 0.4 will re-confirm symbolically in sympy.)

Attack-relevant observations (in D-004):

  1. X ≤ T edge confirmed precisely: off-diagonal error O(L²X) must be o(TL³); X = T^{1+δ} destroys it — obstruction map entry confirmed.
  2. H(λ) = 2 − 1/λ − λ/3 is increasing on (0,1] ⟹ shortening the window never helps — negative knowledge, do NOT re-test.
  3. THE insertion point for task 1.4: the off-diagonal prime-pair sum O₁ is bounded away by MV, never evaluated; any unconditional support-(1+δ) form-factor fragment enters ‖G̃‖²_HS exactly there.

Outcome: 0.2c complete. Next: 0.2d (§3 + §6; includes task 0.3, independent Lemma 3.2 reproduction) → D-005.

Addendum 5 — Phase 0.2d + task 0.3 completed (same day)

Task: Linear algebra §3 + chain assembly §6 → D-005; fulfills task 0.3.

Work: Independent reproductions: Lemma 3.1 (inertia under pull-back, non-invertible case via injectivity on U); Lemma 3.2 (rank–trace) with every inequality step justified — Jordan decomposition, tr(PQ⁺)≥0, von Neumann, x²≥2x−1, (x−2)²≥0; equality case P=Π₁, Q=2Π₂ confirmed; Remark 3.3 variational view + Cauchy–Schwarz recovery; Lemma 3.4 (Weyl). Full chain (1.2) verified step by step for both (i) and (ii), plus Theorem B (χ-twist) sketch.

Key structural findings (in D-005):

  1. FOUR loss-points of Lemma 3.2 mapped: dropped 2tr(PQ⁺); von Neumann alignment; x²≥2x−1 tight only at p−n=1; x²≥4x−4 tight only at q=2. Certificate tight exactly at eigenvalue pattern {1}_P,{2}_Q — this explains WHY simple zeros and off-line pairs saturate the method, and that escaping 2/3 needs data breaking simultaneous tightness (= moments beyond second / support > 1, matching §7.2 mechanistically).
  2. Genealogy confirmed: (3.1) = matrix m²≥2m−1; flat charge 4 version = m²≥3m−2 (i.e. (m−1)(m−2)≥0).
  3. Multiple on-line zeros are the worst case for the certificate (cost 2 in N, return 0 rank) — cross-track hook to Levinson-style mollifier ideas for tightening tr P₁ + 2(s₂+p) ≤ N.

Outcome: 0.2d + 0.3 complete. Next: 0.2e (§7 sharpness/ceiling + App A) → D-006. Last sub-stage of 0.2.

Addendum 6 — Phase 0.2e completed; Phase 0.2 COMPLETE (same day)

Task: §7 (sharpness/ceiling/conditional) + Appendix A (Lean) → D-006.

Work: Full ceiling analysis captured: (a) dimension ceiling 100% + the "method cannot distinguish 2/3 from all" statement; (b) sharpness of all three counts (extremal = 2/3 simple orthogonal + 1/6 doubles); (c) two integrality levels; (d) CMS/Christoffel moment formulation; (e) Rudnick–Sarnak wall (Xᵏ ≤ T^{2−ε}, at X≍T only k≤2); (f) conditional ladder HL(k₀): m_k(1) = 1,4/3,2,13/4, Λ₂(0;1) = 5/36, HL(4) → 13/18, HL*(∞) → 100% simple but NOT RH; bandwidth-one ceiling ≈ 0.682 via Lean + interval arithmetic (p₀ = 1−a_N ≤ 0.6818287; audit caveat: EnclOK interval-certified, not kernel-checked). §7.3 conditional cubic-moment certificate (0.85082 on RH). Remark 7.1 ξ′ results (0.85838/0.86864 simple on-line unconditional). Appendix A: toolchain v4.33.0-rc2, Mathlib 51e6992efd06, tag v1.0, 3 standard axioms only; von Neumann + Sylvester contributed to Mathlib.

Synthesis: mechanistic obstruction map built (D-006 §6) — 5 walls, each with precise mechanism and escape requirement. Phase 1 attacks now have exact insertion points.

Outcome: Phase 0.2 COMPLETE (D-002..D-006). Paper fully mastered. Next: 0.4 (sympy R(ψ) verification), then 0.5 (needs git clone — network), 0.6 (numerical sandbox).

Addendum 7 — Phase 0.4 completed (same day)

Task: Symbolic verification of R(ψ₀) and R(ψ_MT) (task 0.4) → E-001.

Setup: installed sympy 1.14.0 / mpmath 1.3.0 / numpy 2.5.2 into .venv.

Output: research/numerics/scripts/E-001-window-constants.py + data/E-001-window-constants.{csv,json}. Symbolic R(ψ) quotient computed by direct sympy integration for both windows; ψ_MT identity checked to 60 digits (sympy simplify does not fully reduce the trig form — documented). Both PASS: R(ψ₀) = 4/3 exactly R(ψ_MT) = ½ + (1/√2)cot(1/√2) = 1.32749929632058835426562… certificates: 2−R = 0.67250070368… (paper: 0.6725); (3−R)/2 = 0.83625035…

ERROR CAUGHT (honesty log): E-001 flagged a mismatch vs D-004's hand-computed R(ψ_MT) = 1.327325. Diagnosis: hand-arithmetic slip in the decimal evaluation of cot(1/√2) (1.170078 written; correct 1.17026056). The symbolic identity was always correct. D-004 corrected with a dated CORRECTION note; summary table updated. Lesson: hand decimals are advisory only — machine verification decides. (The adversarial-review loop worked as designed.)

Outcome: 0.4 complete. Next: 0.5 (zeta-23-lean; needs network — ask user) and 0.6 (numerical sandbox).

Addendum 8 — Phase 0.5 completed (same day)

Task: Clone/inspect zeta-23-lean; confirm axiom audit → D-007.

Work: Cloned to zeta-23-lean/ (HEAD at tag v1.0 = 3635e74; user approved network use). Full structure mapped (Statement, Unconditional, Final/FinalMult, ThmD/E/DE, XiPrime, PairCeiling, LinAlg, MV, Poisson, WeilEF, FromPNTPlus, comparator/, AUDIT.md). Independent static audit: 0 real axiom declarations (2 grep hits are inside a tactic doc-comment, verified by reading the file); 0 real sorries outside the trusted challenge stubs (3 hits are comments); challenge stubs carry deliberate sorries as designed. Consistent with AUDIT.md: all #print axioms → [propext, Classical.choice, Quot.sound] only.

Deferred (recorded, not silently skipped): full kernel rebuild (lake build, ~8890 jobs incl. Mathlib) → Phase 5/Track D or a dedicated sub-stage when we contribute Lean lemmas. Detailed read of PairCeiling/LawN256.lean → scheduled with thread T3 (Phase 1.1).

Notable: repo proves Thm C (5/6) via rank–trace parameter c=3 (ZeroBlockData.mult_three) vs paper's c=2 regrouping — same constant, two formulations. c₁* = 0.75329… = 1/1.32749930 ✔ consistent with E-001.

Outcome: 0.5 complete. Next: 0.6 (numerical sandbox; split 0.6a/0.6b).

Addendum 9 — Phase 0.6 completed; PHASE 0 COMPLETE (same day)

Task: Numerical sandbox → E-002 (10⁴ zeros), E-003 (finite G̃), D-008.

E-002 saga (engineering lessons):

  1. Serial run: killed by 2-min default timeout at n=1000 (~8.7 z/s).
  2. Resume run: killed at 10 min, n=3000 — BUT the next window lost its work: Python's file buffer wasn't flushed on kill (computed 3001–4000, saved nothing). FIX: per-row flush + atomic part-files.
  3. Rewrote as 15-way multiprocess (zetazero(n) independent per n): 15 part-files, all completed inside one 10-min window; merge + checks in seconds. Two more bugs surfaced and fixed: cp1252 console can't print ≪/²/− (ASCII-only prints now; files UTF-8); float64 reload broke the γ₁ 1e-20 check (now parsed from CSV text at full precision).
  4. FINAL: ALL CHECKS PASS. 10,000 zeros @ dps 25; γ₁ verified; RvM err 0.32; min gap 0.0377; γ₁₀₀₀₀ = 9877.782654005501142774099.

E-003 (finite G̃, T ∈ {1000,2000,4000}, ψ₀ & ψ_MT):

PHASE 0 COMPLETE. All six tasks done, all artifacts committed. Next session: Phase 1.1 (re-derive 0.682 ceiling; start with PairCeiling/LawN256.lean per thread T3 + §4A.5 domain sweep).

Addendum 10 — Task 1.1 completed (same day)

Task: Re-derive the bandwidth-one ceiling; identify every non-tight inequality → D-009.

Work: Read the PairCeiling Lean suite directly (Defs/Ceiling/ Stability/Bridge/NearCUE/RowCert/LawN256/CeilingLaw256, ~2000 lines). Re-derived the ceiling mechanism: certificate value v = c₀ + ∫₀¹r(k)kdk; validity at the adversarial 256-periodic law (near-CUE rows to 3×10⁻⁴⁰, edge |D(1)| ≤ 0.82395317, simple fraction p₀ = 0.68182868746… exact rational, marks m∈{1,2}, LP-optimal) forces v ≤ p₀ + stability errors < 0.6819 for band-limited windows.

Key understanding gained: the ceiling is INFORMATION-THEORETIC, not analytic — two moments at bandwidth one cannot distinguish the sine process from a law with only 68.18% simple points. No inequality within this data class is "the" problem; the data itself is.

Inventory: 4 chain losses (L1–L4, from D-005), 4 ceiling losses (C1: edge 0.824; C2: triangle loss; C3: marks≤2 restriction; C4: N=256 grid), 3 structural gaps (S1 moments, S2 bandwidth, S3 o(N)-insensitivity).

Adversarial review of escape ideas:

Outcome: 1.1 complete; thread T3 closed. Next: 1.2 (SDP) or Idea C.

Addendum 11 — Idea C attacked and killed (G-001) (same day)

Decision: user chose (b) Idea C first (multiple-zero bounds in chain).

Quantification BEFORE literature (execution-loop discipline): the chain charges on-line doubles (s₂) and off-line pairs (p) identically (2 in N, flat charge 4 in Q′-budget). Substitution: replace each double by a depth-δ off-line pair; every bandwidth-one two-moment datum changes by O(δ) → 0. ⟹ any s₂ ≤ κN input is vacuous against the adversary. Idea C DEAD without spending literature effort — [FAILED-ATTEMPT] with a clean, quotable obstruction. Logged as G-001 (graveyard) + NGUYEN.md §6 (+ Idea A zero-density row from D-009).

Lesson recorded: the binding wall is shallow off-line-pair invisibility, not moments/windows/multiplicities. s₂-side inputs are provably useless in this framework. Salvage: sharper question for thread T4 — WHY does the ξ′-analogue escape this symmetry?

Outcome: G-001 committed. Next: task 1.2 (SDP numerics) per STATE.md, or 1.3 (third moment) if user prefers.

Addendum 12 — Task 1.3 warm-up (thread T1) completed (same day)

Decision: user approved 1.3 starting with T1 warm-up.

Results (D-010 + E-004):

  1. Λ₂(0) = 5/36 verified BY HAND from the moment matrix (det 5/108, (M⁻¹)₀₀ = 36/5). [VERIFIED]
  2. Scaling insight: in spacing-1 units the paper's construction is L-independent — the limit object is the sinc Gram matrix G̃ = Σ_j S(γ_j−k)S(γ_j−k') with S(x)=sin(πx)/(πx), integer grid; m₁=1 is the exact identity Σ_k S(x−k)²=1 (= sharp-window Lemma 2.1).
  3. E-004 Monte Carlo (60 GUE(400) bulk samples, W=600): m = 0.9939, 1.3271, 1.9945, 3.2489 vs targets 1, 4/3, 2, 13/4 — all within edge/ sampling tolerances; Λ₂(0) = 0.1453 vs 0.1389. CONFIRMED.
  4. Limiting density: near-uniform[0,2] perturbation (matches uniform through m₃; m₄ 3.25 vs 3.2). Not a named density.
  5. Erratum noted: paper's "d = N(T,2T) + O(L)" should be o(N) — N−d = (2log2−1)(T/2π) = O(T); inconsequential (E-003 confirms the arithmetic). Saved for errata appendix.
  6. Debugging path (logged for method): first E-004 run exposed a 1/L² normalization bug — diagnosed by matching the wrong values to the wrong formula (0.0988 = (2π/L)/L² at L=4). ✔ adversarial numerics working.

Outcome: T1 closed. Ready for task 1.3 proper (third-moment diagonal method failure analysis).

Addendum 13 — Task 1.3 analysis completed (same day)

Task: What fragment of the third moment survives unconditionally? → D-011.

Derivation chain:

  1. tr G̃³ reduces (via Lemma 2.1) to a triple zero-correlation with band-limited kernel Φ(γ₁−γ₂)Φ(γ₂−γ₃)Φ(γ₃−γ₁). [VERIFIED algebra]
  2. Prime side: diagonal method needs shift-correlations of THREE P_X's; off-diagonal control via MV costs ~X²/T ⟹ Rudnick–Sarnak X³ ≤ T^{2−ε} ⟺ λ ≤ 2/3. (k=2 works at λ=1 only because of the 1/(2π)² margin.)
  3. THE WALL, precisely: at bandwidth λ the grid has d = λN rows ⟹ certificate ≤ λ; third moment available only at λ ≤ 2/3. Availability and usefulness thresholds coincide EXACTLY at 2/3.
  4. THE HATCH (open): Lemma 2.1's no-aliasing needs only supp(ϕ∗ϕ) ⊂ (−2π/h, 2π/h) — an oversampled grid h = 2π/(qL′), q ≥ 1, keeps Poisson–Gabor EXACT (verified against the proof's H-support argument). At q=2, λ=2/3: d = 4N/3 — the dimension cap evaporates. Whether the extra rank is spectrally real or empty padding is numerically decidable on the sine-process model → E-005 designed (scan λ ∈ [1/2,1], q ∈ {1,2}, 3-moment Markov bound).

Outcome: 1.3 analysis complete. E-005 is the decider: positive → theorem path past 2/3 (unconditional!); negative → G-002 with exact obstruction. Awaiting user go-ahead for E-005.

Addendum 14 — E-005 run; moment ladder CLOSED (G-002) (same day)

E-005 results: calibrated correctly at (λ,q)=(1,1) (μ ≈ (1,4/3,2,13/4), CMS2 = 0.764 ≈ 3/4, CMS4 = 0.876 ≈ 31/36 — matches paper/theory). Then:

Verdict: G-002 — moment-ladder route closed from both sides (analytics D-011 + model numerics E-005). Three independent walls logged. Task 1.3 marked complete with answer "nothing survives usefully".

Salvage: E-005 is a calibrated CMS testbed; q-invariance of n₊ is a lemma-grade observation; cap table is obstruction-map exhibit material.

Strategic state after G-002: of the three escape directions from the 2-moment framework, only support > 1 (prime pairs, task 1.4) remains in Track A; otherwise Tracks B/C. Phase 1.5 decision gate now has all inputs except the 1.4 literature review.

Addendum 15 — Tasks 1.4 + 1.5 completed; PHASE 1 DONE (same day)

1.4 (D-012): web research per §4A.3 (3 arXiv API queries, all logged to D-001 Part III). Findings: (i) ALL post-paper pair-correlation work is conditional (narrow box / PCC / HL); (ii) best unconditional prime-pair results = Goldbach exceptional sets X^{0.72} (Pintz 2018), X^{7/10} (Zhao 2025, ineffective) — generic lags only. (iii) LAG-SCALE ANALYSIS (new): the off-diagonal O₁ terms with |log(n/m)| ≲ 1/T are COHERENT; their control requires prime-pair distribution at lags h ≲ X/T = T^δ — ultra-short lags where nothing unconditional exists. Answer to 1.4: no unconditional support-(1+δ) fragment available; missing input located precisely.

1.5 DECISION GATE (D-013): > 2/3 NOT achievable within the framework's current data class. Six-wall map assembled (W1 sharpness, W2 0.682 ceiling, W3 moment ladder, W4 multiplicity symmetry, W5 zero-density blindness, W6 short-lag prime pairs). Five necessary conditions for any fix stated. Open routes ranked: Track C > Track B > Track A-with-new-arithmetic > ξ′ sharpening > GL(2) extension. Track A → monitoring mode.

Phase 1 complete (1.2 deferred as calibration-only, documented). Session total: Phase 0 + Phase 1, 13 D-files, 5 experiments, 2 graveyard entries, ~17 commits.