{
  "file_path": "D:\\Python\\Quantum\\STATE.md",
  "new_string": "# STATE.md \u2014 NGUYEN-RH-1 RESUME POINT\n# \u2b50 READ THIS FIRST at the start of every session.\nLast updated: 2026-08-19 | Session 009 COMPLETE (Phase 4 done: 4.1\u20134.4;\nindependent review downgraded all three theorem tags; repair map\n4.5a\u20134.5d opened; E-027 detector stress test ALL CHECKS PASS).\n\n## CURRENT POSITION\n\u2714 PHASE 0, 1 COMPLETE. PHASE 3 (Track C): 3.1\u20133.8 CLOSED (R-004).\nPHASE 2 (Track B): 2.6\u20132.13 mapped to exhaustion at current technology.\n**PHASE 4 COMPLETE (session 009)** \u2014 synthesis D-052 (corrected),\nsanity gate D-053 (PASS), independent review D-054 (4.3b), stress\ntest E-027/D-055 (4.4).\n\n**Honesty state after 4.3b (D-054):** all three theorem STATEMENTS\nstand; TAGS downgraded:\n- T-1 (R-004): [PROVED-CONDITIONAL(RH-to-T\u2080; finite verification;\n  R-bound constants E-021-measured, tracing open \u2192 4.5a)]. D-039's\n  digamma proof broken (Binet tail |W| \u2273 35; recurrence Cauchy misuse);\n  D-036 \u00a71 far-sum gives O(|t| log t) \u2014 reference-point trick missing.\n  Margins (67 705 @ N\u2080, 2.46e26 @ N\u2081) absorb any plausible error.\n- T-2 (R-003 + D-041): factorization + (\u2605) \u27fa (1.12) [PROVED modulo\n  bookkeeping \u2192 4.5d]; (1.12) \u27f9 spec(D) \u2282 \u211d [PROVED via Suzuki + D1];\n  \u21d0 [SKELETON, L3/L4].\n- T-3 (valley theorem): upper bounds [PROVED-CONDITIONAL(RH)] (with\n  K(\u03b3,\u03b3) \u2264 1/m_\u03b3 fix); lower bounds + profile [SKETCH \u2014 D-049 proofs\n  broken (F9/F10), repair 4.5c; measured laws stand]; detector\n  restated by E-027 (three levels: bare immediate / pencil magnitude /\n  carrier \u03c0/(2a) exact).\n\n## ACTIVE PLAN (next stage, in order)\n1. **Repairs 4.5a\u20134.5d** (restore full tags):\n   - 4.5a: D-036 \u00a71 reference-point rewrite (Davenport trick \u2192 traced\n     A, B\u2080) + D-039 repair (Binet split at t ~ |V|/2; sharper\n     recurrence majorant) + I_I housekeeping (contour length 2(\u221an+1),\n     certified |\u03b6\u2032/\u03b6(2)|).\n   - 4.5b: symbolic concavity proof of B(n) components, n \u2265 20.\n   - 4.5c: valley lower bounds (node-killing test functions or\n     averaged-density lemma); B3 sinc\u00b2((z\u2212\u03b3)a) fix; m_\u03b3 fix; detector\n     restatement (bare-form indefiniteness for \u03b4 > 0 \u2014 likely provable\n     exactly via strict Schwarz; needs 4.5d's domain write-up).\n   - 4.5d: L1 write-up + (I1) form-domain identification + R-003\n     bookkeeping (punctured compacts for (1.12); R_\u03c4 distributional).\n2. Paper assembly update (draft.tex): fold D-054/D-055 corrections \u2014\n   the valley-theorem section and the detector remarks must be scoped\n   per the new tags; add E-027 to the numerics appendix.\n3. Phase 5 (Track D) \u2014 Lean formalization skeletons for the repaired\n   proofs.\nDeprioritized: a=13+ push; adversary-ceiling LP; T4b; E-021 n=10\u2074\nrerun (cheap via caches); E-023b acb box cover.\n\n## NEXT ACTION\nSession 010: repair 4.5a (R-004 hardening \u2014 the flagship theorem's\nfull tag). D-036 \u00a71 rewrite with the reference-point trick; D-039\nBinet/recurrence repairs; then 4.5b (symbolic concavity, routine).",
  "old_string": "# STATE.md \u2014 NGUYEN-RH-1 RESUME POINT\n# \u2b50 READ THIS FIRST at the start of every session.\nLast updated: 2026-08-19 | Session 008 COMPLETE (regroup + R7\nregistration + task 3.8/R-004 + task 2.10/E-022 done). Clean state.\n\n## CURRENT POSITION\n\u2714 PHASE 0, 1 COMPLETE. PHASE 3 (Track C): **3.1\u20133.8 ALL CLOSED** \u2014\nthe Li line has delivered its maximal deliverable (R-004). PHASE 2\n(Track B): 2.6\u20132.8 closed; (\u2605) design done; tasks 2.9/2.10/2.11 open.\n\n**R-004 (ledger #6), session 008:** \u03bb_n > 0 for all\n1 \u2264 n \u2264 N\u2081 = (T\u2080\u22121)\u00b2 = 8 999 999 999 994 000 000 000 001 \u2248 9.0\u00d710\u00b2\u2074,\nT\u2080 = 3\u00d710\u00b9\u00b2 (Platt\u2013Trudgian 2021). Tag [PROVED-CONDITIONAL(computational\nRH-to-T\u2080; finite verification; one numeric sub-bound)]. Conservative\nwindow N\u2081* = 2.73\u00d710\u00b2\u00b9. All constants traced (D-037); support\ncomputation E-023 ALL CHECKS PASS. Criterion-3 deliverable.\n\nTwo live fronts (unchanged):\n- Front A (Li, task 3.5): COMPLETE via R-004 (finite-height theorem is\n  the maximal content of the Li route \u2014 Voros wall, D-032/D-033).\n- Front B (Suzuki, task 2.9): (\u2605) \u27fa Cor 1.6 \u27fa \"Krein-space operator D\n  has real spectrum\" via strong resolvent convergence; walls W-1\n  (index), W-2 (persistence = RH), W-3 (circularity) (D-035).\n\n## ACTIVE PLAN (next stage, in order)\nFORK COMPLETE (session 008): Fork 1 (D-045 symbol identification) +\nFork 2 (D-046 \u03bb-channel sizing) both done. Outcomes: symbol = 1 +\nzero-comb (plateau \u2192 2a free PW); \u03bb-channel INFEASIBLE \u2192 G-004.\n**Awaiting user decision on the next direction** \u2014 candidates:\n1. Symbol-space formulation of the ground-state regime (D-046 revival\n   condition (ii): work with the comb-symbol directly, no\n   discretization) \u2014 the one live opening on the (\u2605) route.\n2. 2.9 remainder paper-work: L1 (unconditional embedding), L3/L4.\n3. Phase 4 synthesis / paper assembly (draft.tex) \u2014 the year's\n   results (R-003, R-004, spectral-measure theory) are paper-ready.\n4. Track A monitoring / new-domain sweep (\u00a74A.5) for the residual\n   equilibrium obstruction.\nOptional hardening of R-004 (not required): D-039 (prove digamma\nsub-bound \u2192 H2 to proved), E-023b (acb box cover for M\u2080).\nDeprioritized: a=13+ push; adversary-ceiling LP; T4b (see D-029/D-030/\nD-031 revival conditions); E-021 n=10\u2074 rerun (cheap via caches).",
  "replace_all": false
}