{
  "todos": [
    {
      "content": "Verify E-010 extended re-run data validity + re-register in db",
      "status": "completed",
      "activeForm": "Verifying E-010 re-run data"
    },
    {
      "content": "Commit housekeeping (papers, Notes.txt) + E-010 extension",
      "status": "completed",
      "activeForm": "Committing housekeeping and E-010"
    },
    {
      "content": "Debug E-011 precision plumbing (53-bit signature at digit ~14)",
      "status": "in_progress",
      "activeForm": "Debugging E-011 precision plumbing"
    },
    {
      "content": "Re-run E-011 and resolve checks (or document obstruction)",
      "status": "pending",
      "activeForm": "Re-running E-011"
    },
    {
      "content": "Session-end protocol: journal, db, checklist, commit",
      "status": "pending",
      "activeForm": "Running session-end protocol"
    }
  ]
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    "filePath": "D:\\Python\\Quantum\\NGUYEN.md",
    "content": "        (zeta halving 2^{\u2212(\u03c3\u22122)}\u00b7dirichletC2),\n        one_add_div_pow_norm_le_exp_four_ninths (kernel envelope\n        e^{(4/9)\u03c3} \u2014 single-exponential, no geometric series),\n        Jlag_dogleg_norm_le_envelope (dirichletC2\u00b7(4e^{\u2212c\u03c3}+4\u00b72^{\u2212\u03c3})),\n        integral_exp_neg_le (FTC machine), cdog_gt (c \u2265 0.2487),\n        exp_four_thirds_le (e^{4/3} \u2264 3.8073),\n        **hIntegral_Jlag_dogleg_norm_le \u2264 8.38** (charged with\n        dirichletC2 \u2264 1; tight R(1.5) = 5.94 = documented tightening\n        path). E-034 re-validated at Lean constants: ALL CHECKS PASS\n        (margins 13 160/88 911/2.458e26).\n      \u2192 DL-4 COMPLETE (session 040, LiDogLeg.lean +~160 lines, build\n        8773, audit clean): kFun_right_vertical_norm_le_exp_sqrt_half_add_one\n        (w = 2\u221an kernel variant, e^{\u221an/2}+1),\n        **vIntegral_Jlag_right_norm_le** \u2264 (U\u2212T)(e^{\u221an/2}+1)\n        2^{\u2212(2\u221an\u22122)}\u00b7dirichletC2 \u2014 exponentially tiny; VRightBound\n        charged IIMaj/2 (provable with dirichletC2 \u2264 1; IIMaj/4 =\n        tightening path).\n      \u2192 DL-5 COMPLETE (session 040, LiDogLeg.lean +~210 lines, build\n        8773, audit clean): **hIntegral_Jlag_low_le_dogLegUpper** +\n        **hIntegral_Jlag_low_neg_norm_le_dogLegUpper** (both \u2264\n        ((\u221an+1)/2)(e^{5/2}+1)/2 + 8.38 + IIMaj/2 for T \u2208 [\u221an, \u221an+1],\n        U = (3/2)T) + dogLegUpper_two_sides \u2014 the full dog-leg PROVED\n        in Lean, no perimeter, nothing above T_cut.\n      \u2192 DL-6 COMPLETE (session 040, LiDogLeg.lean +~350 lines, build\n        8773, audit clean): the direct Dirichlet bound at re = 2\u221an \u2014\n        **norm_logDeriv_zeta_le_iimZeta \u2264 0.57\u00b72^{\u2212(2\u221an\u22122)} (n \u2265 5),\n        NO dirichletC2 \u2264 0.57 needed** (log2-leading + 3^{\u2212(2\u221an\u22122)}\n        tail with dirichletC2 \u2264 1); **V_right \u2264 IIMaj/4 EXACTLY**;\n        DL-5 upgraded to the quarter charge (dogLegUpper = VLeft +\n        8.38 + IIMaj/4 per side \u27f9 1.5\u00b7IIMaj total in Bn, the\n        convexity-safe coefficient). E-034 re-validated at IIMaj/4:\n        ALL CHECKS PASS.\n      \u2192 ii\u2032-6-a2 COMPLETE (session 040): DogLegRight def + Bn\n        redefined (horizontal share 2(e\u2074+1)(7H\u0302) + 2\u00b7DogLegRight,\n        T = \u221an+1 UNCHANGED \u2014 N\u2081 = (T\u2080\u22121)\u00b2 \u2248 9\u00d710\u00b2\u2074 PRESERVED);\n        margin lockstep at coefficient level; 4/3-bracket strengthening\n        cherry-picked onto master (e11ad95); marginTilde_convex\n        re-proved (1.5\u00b7IIMaj, strict); base points re-certified\n        (Bn(8e4) \u2208 [273k, 332k], Bn(10\u2075) \u2264 352k \u27f9 margin(8e4) \u2265\n        4 001, hbase2 66 001 < 80 501); endpoints UNCHANGED; build\n        GREEN (8773 jobs); audit: 41 decls, 3 standard axioms, 0 sorryAx.\n      THEN THE ORIGINAL PLAN RESUMES (D-074 \u00a78 return map): ii\u2032-6-c\n      (four side bounds with the repaired horizontal) \u2192 ii\u2032-6-d (fold\n      \u27f9 sfBoundProp) \u2192 ii\u2032-7 (closure at the ORIGINAL N\u2081 \u2248 9\u00d710\u00b2\u2074).\n      Fallbacks: (a) c = 1.5 paused branch (N\u2081 = 4\u00d710\u00b2\u2074); (b) c = 2\n      (N\u2081 = 2.25\u00d710\u00b2\u2074); (c) T = \u221an hidden-cancellation open question\n      (D-073 \u00a78 D4).\n      \u2192 **ii\u2032-6-c STARTED (session 040 block B)**: FINDING F-E (D-074\n      \u00a79) \u2014 the I_I kernel charge e^{\u221an/2}+1 is off by one at\n      re w = 2\u221an near t = 0 (true value e^{\u221an/2+1/8}; latent in the\n      original D-036 charge); REPAIR ADOPTED: I_I charged honestly as\n      e^{1/3}\u00b7IIMaj (uniform kernel \u2264 e^{\u221an/2+1/3}+1, n \u2265 4), the\n      overrun absorbed by the fold's free dropped 2\u03c0 (e^{1/3} \u2264 1.4 \u2264\n      2\u03c0) \u2014 NO change to Bn/IIMaj/convexity/pinch/base points\n      (fatten-IIMaj rejected: iimaj_bracket_bound's 4.56 is linear).\n      **6-c-i COMPLETE (a8c5b1e)**: LiSideBounds.lean scaffolding\n      (Z2hat_mono, nearZerosSub_mult_sum_le, logDeriv continuity,\n      Jlag_norm_eq, both zero-freeness lemmas, rectangle side\n      decomposition, VIntegral/HIntegral machinery, even symmetry).\n      **ii\u2032-6-c..6-d COMPLETE (session 041)**: 6-c-ii I_III\n      (vIntegral_Jlag_left_norm_le, 6e5a98f), 6-c-iii I_I F-E form\n      (vIntegral_Jlag_right_le_exp_third_IIMaj, 9bd8ba9), 6-c-iv the\n      mid-horizontal segment machine (arcsinh-FTC from scratch \u2014\n      integral_one_div_sqrt_sq_add_sq + endpoints + per-zero \u2264\n      2\u00b7log(16.5/\u03b4) by pure log-mono, 1fed1df), 6-c-v the horizontal\n      assembly (mid + DL-5 tails both sides, 5d3dc0a), and **6-d THE\n      FOLD (774fec5): `sfBoundProp : SfBoundProp` PROVED** \u2014 |Sf n| \u2264\n      Bn FAlt n on [10\u2075, N\u2081] from EXACTLY the six declared perimeter\n      hypotheses (RHUpTo, ZeroCountProp, ZeroCountProp2,\n      ZeroCountGlobalProp, NoLowZeros, MZeroProp); the pinch NI + 5.5R\n      \u2264 7\u00b7H\u0302 discharged by a fully elementary log-mono chain (no\n      certified log numerics); the bridge cloned at T \u2264 3\u00b710\u00b9\u00b2 (the\n      range actually used \u2014 T \u2264 \u221an+1 holds exactly incl. n = N\u2081).\n      **ii\u2032-6 COMPLETE.**\n- [x] ii\u2032-7 Closure: theorem sfBoundProp : SfBoundProp; final\n      lambda_pos_finite_height_closed (3 perimeter hypotheses only);\n      full axiom audit; paper Appendix B + tag upgrade (D-062 \u00a74\n      endpoint); archive closure + D-037 line-63 erratum pointer.\n      \u2192 COMPLETE (session 041): **lambda_pos_finite_height_closed\n      PROVED** (a0943a6, LiClosed.lean) \u2014 \u03bb_n > 0 for ALL 1 \u2264 n \u2264\n      N\u2081 = (3\u00d710\u00b9\u00b2\u22121)\u00b2 from EXACTLY SEVEN declared hypotheses (the two\n      computational certificates + the five analytic perimeter Props);\n      full axiom audit (AuditSession041: 29 decls incl. the closed\n      chain \u2014 3 standard axioms only, 0 sorryAx); Paper A closure\n      (b64a839 + the session-042 review revision, compiles clean\n      11 pp \u2014 Appendix B rewritten: second tier COMPLETE + the\n      verbatim signature + the instantiation list + the dependency\n      chain; the B(n) display corrected + the F-A/F-B/F-E\n      correction-note footnote + the F-D dog-leg promoted to \u00a73.1;\n      abstract/intro to the seven-hypotheses headline; hedged novelty\n      (Oesterl\u00e9's unpublished precursor noted); AI Contribution\n      Statement factual refresh); archive closure (D-037 erratum\n      block + addendum, R-004 Lean-closure status, NGUYEN.md,\n      STATE.md, journals 041\u2013042). **SESSION-042 REVIEW REVISION\n      (user review)**: the finite-verification certificate\n      Misguich\u2013Voros (deformed sequence, private communication) was\n      RETRACTED and replaced by Johansson 2015 \u00a7 4.2 (rigorous\n      ordinary-\u03bb_n to 10\u2075); LiVerifiedUpTo reduced (5\u00b710\u2075) \u2192 (10\u2075)\n      (79c96f2, matches the checkable evidence \u2014 the assembly's\n      hfinite branch uses exactly n < 10\u2075).\n\n## 6. GRAVEYARD (documented dead ends \u2014 REQUIRED logging)\n| # | Attempt | Track | Obstruction found | Date |\n|---|---------|-------|-------------------|------|\n| G-001 | Multiple-zero bounds as chain constraint (Idea C) | A | s\u2082/p substitution symmetry (depth\u21920 pairs = doubles in all 2-moment data) | 2026-08-17 |\n| (A) | Zero-density input into chain (D-009 Idea A) | A | depth\u21920 off-line pairs invisible to N(\u03c3,T) bounds | 2026-08-17 |\n| G-002 | Moment-ladder certificates (tr G\u0303\u1d4f, k\u22653; oversampled grids) | A | RS availability \u03bb\u22642/k bites before usefulness; caps 0.607N (k=3, \u03bb=2/3) and 0.502N (k=4, \u03bb=1/2) both < 2/3; n\u208a q-invariant (E-005) | 2026-08-17 |\n| G-003 | Density certificates \u2192 de Bruijn\u2013Newman \u039b (task 3.3) | C | one-sided information: density theorems lower-bound the on-line fraction, \u039b needs an upper bound on the complex fraction; extremal vs averaged (E-012); asymptotic vs explicit | 2026-08-17 |\n| G-004 | \u03bb \u2192 \u03bb_a\u207b ground-state channel (task 2.13 follow-up) | B | disjoint feasibility windows: ground-state resolution needs a \u2272 1.3 (\u03bb_a superexponential), zero resolution needs a \u2273 4\u20136; discretization floor out-ranks \u03bb_a; D-046 sizing | 2026-08-19 |\n\n## 7. RESULT LEDGER\n| # | Claim | Status tag | Location |\n|---|-------|-----------|----------|\n| 0 | \u22652/3 simple on-line (inherited) | [PROVED] (R2) | Paper Thm A |\n| 1 | Galerkin realization of Suzuki's Weil-form operator A_a validated to 4.5e-6 against the zero-side explicit formula; Weil formula conventions machine-verified (Gaussian test, 9e-16) | [NUMERICAL-VALIDATED] | E-009a/b, D-019 \u00a71 |\n| 2 | \u03bb_a (bottom of Weil-form spectrum) positive at a = 0.25..5 with steep cliff (0.414 \u2192 \u22641.7e-12), consistent with Yoshida \u03bb_a>0 \u2200a \u27fa RH; independently matches Kim et al. superexponential decay report | [NUMERICAL] (upper bounds; conforming Galerkin) | E-009b, D-019 F1 |\n| 3 | Li coefficients \u03bb_n > 0 for all n \u2264 3000 computed from the arithmetic side (BL Thm 2, zero-free; flint Stieltjes + Arb \u03b6, dps 1000\u20131450), cross-validated vs E-008 zero side; \u03bb_1 to 102 digits; refined asymptotic ratio 1.00216 at n=2000, 0.999845 at n=3000 | [NUMERICAL-VALIDATED] | E-011, E-015, R-001 |\n| 4 | \u03be\u2032-certificate strength mechanism: perfect-square node in the diagonal law D\u2081 \u2248 s(1\u22122s)\u00b2 (prime-scale cancellation p ~ \u221a(T/2\u03c0)); Remark 7.1 constants reproduced to 5e-6; s\u2082/p symmetry does NOT break; gain not portable to \u03be | [NUMERICAL-VALIDATED] | E-013, R-002, D-022 |\n| 5 | Exact factorization W(a,\u03b8;z) = F\u00b7(R\u2080+R_\u03c4), F = z\u00b2\u03be/\u03be\u2032, with R\u2080 = 2e^{a+i\u03b8/2}sin(az\u2212\u03b8/2)+o(e\u1d43) closed form; Suzuki Cor 1.6 \u27fa (\u2605) tail-cancellation (Wiener\u2013Hopf remainder, RH-gated); W(a,\u03b8;0) = 0 double \u2200a,\u03b8; explains full E-010\u2013E-016 phenomenology | [PROVED (modulo std tempered-FT bookkeeping)] + [NUMERICAL-VALIDATED] (E-017: skeleton 3.7%\u21920.009%, off-axis cancellation 99.8%, \u03b8=\u03c0 preferred) | R-003, D-026, D-027, E-017 |\n| 6 | Explicit finite-height Li theorem: \u03bb_n > 0 for all n \u2264 N\u2081 = (T\u2080\u22121)\u00b2 \u2248 9.0\u00d710\u00b2\u2074 (T\u2080 = 3\u00d710\u00b9\u00b2 Platt\u2013Trudgian); conservative window N\u2081* = 2.73\u00d710\u00b2\u00b9; explicit assembled chain (S_\u221e elementary lower bound + traced contour constants + margin convexity + finite verification; statement-level precursor Oesterl\u00e9 2000 unpublished, published weaker window Lagarias 2007). **FULLY MACHINE-PROVED 2026-08-26 (tier ii\u2032 complete): `LiSideBounds.sfBoundProp` + `LiClosed.lambda_pos_finite_height_closed` from exactly 7 declared hypotheses (2 computational certificates + 5 analytic perimeter Props); axiom audit on the full chain: 3 standard axioms only, 0 sorryAx; B(n) corrected in 4 places during the closure (F-A/F-B/F-D/F-E, D-037 erratum block + Paper A \u00a73 Step-2 footnote), statement unchanged; finite-verification certificate corrected session 042 (Misguich\u2013Voros RETRACTED \u2014 deformed sequence \u2014 \u2192 Johansson 2015 \u00a7 4.2 rigorous ordinary \u03bb_n to 10\u2075; LiVerifiedUpTo (5\u00b710\u2075) \u2192 (10\u2075))** | [PROVED-CONDITIONAL(computational RH-to-T\u2080; finite verification)] \u2014 R-bound fully traced D-056 (A = 10.5, B\u2080 = 48, honest C\u2080; E-028 verified, 26\u00d7 slack); N\u2080 = 80 000; margin convexity PROVED D-057 (E-029) | R-004, D-037, D-056, D-057, E-023, E-028; Lean: LiSideBounds.sfBoundProp, LiClosed.lambda_pos_finite_height_closed |\n\n## 8. SESSION PROTOCOL & RESEARCH ARCHIVE (MANDATORY)\n\nMulti-stage project: computer is shut down between stages. NOTHING may live\nonly in conversation. The archive below is the single source of truth.\n\n### 8.1 Archive map\n```\nSTATE.md                    \u2b50 resume point \u2014 READ FIRST every session\nresearch/\n\u251c\u2500\u2500 journal/session-NNN.md  per-session narrative log\n\u251c\u2500\u2500 results/R-NNN-*.md      one file per result (+ index.md ledger)\n\u251c\u2500\u2500 graveyard/G-NNN-*.md    one file per dead end (+ index.md)\n\u251c\u2500\u2500 derivations/D-NNN-*.md  worked derivations (+ index.md)\n\u251c\u2500\u2500 numerics/scripts/       deterministic experiment scripts\n\u251c\u2500\u2500 numerics/data/          E-NNN-*.csv + E-NNN-*.json metadata sidecars\n\u251c\u2500\u2500 research.db             SQLite catalog (sessions, results, graveyard,\n\u2502                           derivations, experiments) \u2014 schema in\n\u2502                           numerics/scripts/schema.sql, CLI: db.py\n\u2514\u2500\u2500 paper/                  paperA.tex (Paper A, split 2026-08-20),\n                            paperB.tex (research-note skeleton),\n                            draft.tex (FROZEN pre-split combined copy),\n                            references.bib\n```\n\n### 8.2 Storage conventions\n- Results/derivations/graveyard: plain Markdown, honesty tag on every claim.\n- Numerics: DUAL storage \u2014 CSV+JSON sidecar (human/paper-readable) AND\n  SQLite registration (queryable), written together via db.py.\n- Data is cached; scripts never recompute unless --force.\n- Paper assembled only from tagged files in results/.\n\n### 8.3 Session START protocol\n1. Read STATE.md \u2192 resume point, next action, open threads, blockers.\n2. Skim latest journal entry + results/index.md.\n\n### 8.4 Session END protocol (before any shutdown \u2014 no exceptions)\n1. Write journal/session-NNN.md (work, decisions, outcomes).\n2. File every result/dead end/derivation in its own file; update index.md\n   files; register in research.db.\n3. Update NGUYEN.md checklist (\u00a75), Graveyard (\u00a76), Result Ledger (\u00a77).\n4. Update STATE.md LAST with an explicit \"NEXT ACTION:\" line.\n5. git add + commit with session number in the message.\nNOTE: save artifacts IMMEDIATELY when produced (crash-safe), not batched.\n\n### 8.4a FORK RULE (user directive, 2026-08-17)\nWhen the path FORKS (multiple candidate directions), do ALL of them \u2014\none by one \u2014 then pause to gather thoughts and evaluate the next step\ntogether. No path is dropped at a fork by choice alone; paths are only\nclosed by documented obstruction (Graveyard) or deprioritized by\nevidence after all forks are explored.\n\n### 8.5 PACING & SUB-STAGES (user directive, 2026-08-17)\n- NO RUSH. This is long-term research; correctness and rigor outrank speed.\n  Take whatever time a stage needs to be done right.\n- SUB-STAGE RULE: if a stage will take a few hours, break it into smaller\n  sub-stages, each ending in a clean, committed, resumable state (per \u00a78.4).\n  Record sub-stage breakdowns in STATE.md.\n- BREAK PROMPT: if work is running long \u2014 ESPECIALLY approaching 5:00 PM\n  local time \u2014 pause and ASK the user whether to continue, take a break,\n  or log off (run \u00a78.4 end protocol before stopping).\n\n### 8.6 SESSION LOG (summary; details in research/journal/)\n- Session 001 (2026-08-17): Blueprint + mission prompt ingested; research\n  archive installed (git + SQLite + LaTeX skeleton). \u2192 Begin Phase 0.\n- Session 002 (2026-08-17): Phase 0 + Phase 1 complete (Track A \u2192\n  monitoring mode, D-013); Track C begun (3.1 D-014, E-006\u2013E-008, 3.4a/b,\n  D-018 Suzuki scoping).\n- Session 003 (2026-08-17): Q-3.4c complete \u2014 E-009a (Suzuki g validated),\n  E-009b (Galerkin A_a, validated vs zero side 4.5e-6; \u03bb_a cliff),\n  D-019 (analysis; Q-S1 verdict; E-010 proposed). Weil explicit-formula\n  conventions machine-verified (D-001 NT-1).\n- Session 004 (2026-08-17): recovered killed session (E-011 orphaned\n  mid-edit); paper archive reorg; E-010 extended (a\u22648, \u03bb=\u22121); ALL 4 FORKS\n  COMPLETE \u2014 E-010 (fork 1, session 003); E-011/R-001 (fork 2: Li \u03bb_n > 0,\n  n \u2264 2000, arithmetic side); 3.3 answered NO (fork 3: D-021 \u2192 G-003,\n  E-012); T4 CLOSED (fork 4: D-022/R-002 \u2014 \u03be\u2032 strength = perfect-square\n  node in D\u2081, not portable to \u03be; T4b opened). Q-S3 attacked (D-023,\n  E-014): Cor 1.6 density wall mapped to a=10 (maximal lattice,\n  interlacing exact, no thinning \u2014 scale limitation, not a wall);\n  \u03b8=\u03c0 candidate found; Q-S3a/b/c queued. Detached-run machinery\n  established.\n- Session 005 (2026-08-18): user approved do-all-one-by-one plan\n  (A3\u2192A1\u2192A2\u2192C\u2192B, D in background). D launched: E-015 Li n\u22643000\n  detached (in progress at stop; sandbox blocks multiprocessing spawn \u2014\n  detached launches need sandbox disabled). A3 done: E-016 + D-024 \u2014\n  Herglotz \u03c1_W at \u03b8=\u03c0 vs \u03c1_F; V1/V2/V4 pass, V3 (zero-Poisson\n  reconstruction) fails rel ~0.3, root cause open; findings: \u03b8=\u03c0 zeros\n  real, z\u2081(\u03c0)\u00b7a\u22484.5 stiff law, no \u03b8=\u03c0 zero in (0,\u03c0/a), raw-L2 \u03b8=\u03c0\u2248\u03b8=0\n  (plateau-dominated metric). Stopped mid-V3-root-cause (user break;\n  3 network dropouts, no state loss).\n- Session 006 (2026-08-18): REGROUP (user-requested). Full archive audit:\n  all 16 experiments have CSV+JSON sidecars and db registrations; fixed\n  db hygiene (sessions 3\u20135 registered; row_counts corrected for E-010\u2013\n  E-016). Checklist revised: Phase 2 restructured \u2014 Suzuki/Cor 1.6 line\n  recognized as the live spectral program (2.1 partial, 2.4 active; new\n  tasks 2.6 V3-root-cause, 2.7 Q-S3a analytic, 2.8 Q-S3b backend);\n  3.4 checked off (answered YES); NEW tasks 3.5 (Li asymptotic attack \u2014\n  most direct criterion-#1 path), 1.6 (\u03be\u2032-world ceiling), 1.7 (T4b\n  scoping). Active order confirmed: 2.6 \u2192 2.7 \u2192 2.8 \u2192 1.6 \u2192 1.7, with\n  3.5 as standing high-value background thread.\n- Session 006 cont. (2026-08-18): task 2.6 COMPLETE \u2014 E-016 V3\n  root-caused (E-016b: winding-excess diagnostic finds double zero at\n  origin, \u00b1i\u03b5 pair; EM-tail factor-y bug found + confirmed quantitatively)\n  and REPAIRED (E-016c: V5 1.75e-16, V6 \u22644.4e-15, V3R 1.5\u20132.0e-5, ALL\n  CHECKS PASS); D-025 filed; E-016 F1\u2013F4 stand.\n- Session 006 cont.2 (2026-08-18): task 2.7 COMPLETE \u2014 **breakthrough\n  reduction**: W(a,\u03b8;z) = F\u00b7(R\u2080+R_\u03c4) EXACT (F = z\u00b2\u03be/\u03be\u2032, R\u2080 closed-form\n  sine skeleton); Cor 1.6 \u27fa (\u2605) tail-cancellation (Wiener\u2013Hopf remainder,\n  RH-gated). R-003 promoted to ledger (#5). E-017: off-axis cancellation\n  already 99.8% at a=4; \u03b8=\u03c0 preferred; open content isolated to \u211d.\n- Session 006 cont.3 (2026-08-18): task 2.8(i) complete \u2014 E-018 descent\n  profile: (\u2605) cancellation strip-uniform (median |M|/|R\u2080| ~5\u20137e-3,\n  y\u2080\u2208[0.1,1]); open content = zero set of M only; \u03b8-preference found\n  metric-dependent. D-028.\n- Session 006 cont.4 (2026-08-18): task 2.8(ii) complete \u2014 backend v2\n  (blockwise \u039b to e\u00b2\u2074, validated 2.2e-7/4.4e-8) + E-019 at a=12: NO\n  transition (maximal lattice, z\u2081\u00b7a=4.495, gap CV \u2193, origin pair +2.0000\n  winding excess, \u03b3-proximity chance-level, envelope stationary). 2.8\n  CLOSED; a=13+ deprioritized by evidence (D-029). \u2192 Next: 1.6\n  (\u03be\u2032-world ceiling).\n- Session 006 cont.5 (2026-08-18): task 1.6 COMPLETE \u2014 E-020b window\n  optimum for kernel D\u2081 = 0.86864 to 1e-6 (quartic window IS the\n  optimum; control recovers MT); \u03be\u2032-world saturated, nothing transfers\n  to \u03be; E-020 adversary-pool limitation documented (D-030). User\n  decision: do 1.6 then 1.7 as queued (option #1). \u2192 Next: 1.7 (T4b\n  scoping).\n- Session 006 cont.6 (2026-08-18): task 1.7 COMPLETE \u2014 T4b CLOSED\n  (D-031): node mechanism iterates formally (\u03be\u2033 \u2248 0.94 leading-order)\n  but G-001 symmetry caps every \u03be\u207d\u1d4f\u207e data class below 100%; Speiser/\n  Levinson\u2013Montgomery route to RH closed at every order; revival\n  condition noted. Session-005 queue (A3\u2192A1\u2192A2\u2192C\u2192B) NOW FULLY DONE.\n  \u2192 Next: 3.5 (Li asymptotic attack) or (\u2605) analytic attack.\n- Session 006 cont.7 (2026-08-18): task 3.5 DESIGN COMPLETE (user\n  directive: do 3.5 then (\u2605), then analyze for next stage). D-032\n  literature map: Voros \u2014 \"\u03bb_n>0 \u2200n\u2265N\u2080\" \u27fa RH at ANY N\u2080; Lagarias\n  Thm 6.1 finite-height bridge is the exploitable skeleton. D-033:\n  achievable deliverable = EXPLICIT finite-height theorem (\u03bb_n>0 \u2200n\u2264N\u2081\n  \u2248 10\u00b2\u2070\u201310\u00b2\u2074 from Platt T\u2080 + traced constants + finite verification);\n  wall = three-way arithmetic cancellation (triangle bounds explode).\n  New tasks 3.6/3.7/3.8 (D-036 constants, E-021 S_\u221e, D-037 assembly).\n- Session 006 cont.8 (2026-08-18): task (\u2605) DESIGN COMPLETE. D-034:\n  WH factorization + RH is unexplored territory (26 arXiv hits all\n  Riemann\u2013Hilbert); Krein/Gohberg\u2013Feldman/de Branges\u2013Krein machinery\n  exists. D-035: resolvent-convergence route \u2014 D_{a,\u03b8(a)}\u2192D in Krein\n  space; Cor 1.6 \u27fa D has real spectrum (deliverable D1); three-layer\n  wall (index/persistence=RH/circularity); off-axis 99.8% cancellation\n  is TAUTOLOGICAL (entirety of W forces it), the \u211d-envelope is the\n  whole problem. New tasks 2.9/2.10/2.11.\n- Session 007 (2026-08-18/19): task 3.6 marked done; task 3.7 (E-021)\n  COMPLETE after 6-bug root-cause saga (mpmath zeta ~58-digit ceiling,\n  GAMMA_E float literal, 3\u00d7 parse-before-dps, \u03c3-omission boundary\n  (15/14)\u207f, float64 zeros). E-021 ALL CHECKS PASS: V1 4.8e-25 vs\n  E-015; A = 0.9, B\u2080 = 1.0 measured; c\u2080 = \u22121/4 (+1/(24n) correction);\n  |S_f| max 13.52 at n = 3298 vs traced bound ~2280 (~170\u00d7 slack).\n  Resources folder (29 RH-equivalence notes) read in full. Next: 3.8\n  (D-037 \u2192 R-004 finite-height theorem assembly).\n- Session 008 (2026-08-19): REGROUP + R7 registration (resources atlas\n  indexed: resources/index.md, manifest R7, D-001 Part III pointer).\n  Task 3.8 COMPLETE \u2014 **R-004 finite-height Li theorem**: \u03bb_n > 0 for\n  all n \u2264 N\u2081 = (T\u2080\u22121)\u00b2 \u2248 9.0\u00d710\u00b2\u2074 (T\u2080 = 3\u00d710\u00b9\u00b2 Platt\u2013Trudgian);\n  conservative window N\u2081* = 2.73\u00d710\u00b2\u00b9. E-023 support computation ALL\n  CHECKS PASS: D-036 |t|<10 gap found + closed (M\u2080 = 1.3352), S_\u221e\n  elementary lower bound validated vs exact (min slack 9.52), N\u2080 =\n  5\u00d710\u2074 adopted (crossover 32 871 traced / 30 217 measured), margin\n  convex, margin(N\u2081) = 2.46\u00d710\u00b2\u2076. Ledger #6. Criterion-3 deliverable \u2014\n  the maximal content of the Li route short of RH. Task 2.10 COMPLETE\n  same session: E-022/E-022b + D-040 \u2014 spectral measures \u03bc_a (Clark\n  weights 1/K_a(z_k,z_k)) for a \u2264 12; **valley observable**: K_a(z,z)\n  dips exactly at the true \u03b3's, \u0394(a) \u2248 3.33 \u2212 4.76/a saturating at\n  fixed \u03b8=\u03c0; \u03b3-information flows through weights, not positions; mass\n  within \u00b10.1 of \u03b3's 63\u219272.5%; new constraint on 2.9-D1 (\u03b8(a) must\n  drive \u0394 \u2192 \u221e). Next: 2.9-D1, 2.11 (D-038); optional D-039 digamma\n  upgrade.\n- Session 009 (2026-08-19): **PHASE 4 COMPLETE.** Session-008 leftovers\n  (D-052 4.1/4.2, D-053 4.3a gate PASS) committed. 4.3b (D-054): four\n  independent-instance reviews + hand-verification \u2014 ALL THREE TAGS\n  downgraded (D-039 proof broken/bound true; D-036 \u00a71 far-sum\n  O(|t| log t) hole \u2014 reference-point trick missing; D-049 B1/B3 lower\n  bounds broken; \u03b4_crit one-direction; T-2 \u21d0 skeleton; D-052 accounting",
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