← Session 21 — Tier ii′ planning + launch: close SfBoundProp in Lean (D-064)
thinking (134 chars)
D-064 registered. Now write the journal entry, update STATE.md, update NGUYEN.md (session log + tier ii′ activation note), and commit.
Registered. Now the journal, STATE.md, and NGUYEN.md updates.
tool: Write
input
{
"file_path": "D:\\Python\\Quantum\\research\\journal\\session-014.md",
"content": "# Session 014 \u2014 2026-08-21 \u2014 Tier ii\u2032 launched: design + two discoveries\n\n## Directive\nUser chose option (a): do tier ii\u2032 (close SfBoundProp in Lean) BEFORE\nsubmission, with Paper A frozen and submittable at all times. Freeze\nprotocol in force: no edits to paperA.tex/paperB.tex/draft.tex/\nreferences.bib until ii\u2032-7; no edits to existing Lean proofs except the\nliCoeff correctness fix (below, flagged); commit at every sub-stage.\n\n## Work done (sub-stage ii\u2032-0: design \u2192 D-064)\n1. Baseline verified clean (git: only untracked LaTeX artifacts).\n2. Two parallel inventory agents surveyed: (i) vendored residue/\n meromorphic API; (ii) Zeta23 zero infrastructure + NguyenRH1 defs +\n Mathlib zeta/digamma/Jensen/Borel\u2013Carath\u00e9odory surface.\n3. Read liCoeff/Sf/perimeter defs, D-036, D-056, D-037 (full), and the\n primary source (Lagarias math/0404394 \u00a74\u20136, archived PDF).\n4. Axiom-checked all planned Zeta23 dependencies LIVE: zero_sum_inv_sq,\n zeta_local_zero_count, zeta_reflect_zero, zeta_mult_reflect,\n RectangleIntegral'_eq_sumResiduesIn, ResidueMult, logDerivResidue,\n riemannZetaLogDerivResidue \u2014 ALL 3 standard axioms only.\n5. Settled the exact identity chain NUMERICALLY (dps 40\u201350, mpmath +\n E-021 zero cache; see D-064 \u00a73 table).\n6. Wrote D-064 (design doc: discoveries, routes R1\u2013R5, lemma graph,\n revised sub-stage map ii\u2032-1a..7). Registered in index.md + db.\n\n## DISCOVERY 1 \u2014 liCoeff encoding defect (correctness, not cosmetic)\n`liCoeff` (LiSkeleton.lean:39) is an unordered tsum whose family is NOT\nSummable for any n \u2265 1 (summand ~ n/|\u03b3|, \u03a3 mult/|\u03c1| = \u221e by RvM).\nTherefore liCoeff n = 0 (junk), making `LiVerifiedUpTo` FALSE and\n`SfBoundProp` FALSE as encoded (at n = 10\u2075: |Sf| \u2265 432 501 > 347 000\n\u2265 Bn by the certified LiBasePoints bounds). The session-012 theorems\nare vacuously true; the session-013 audit could not see this (it checks\naxioms, not hypothesis truth). Paper A's mathematics is unaffected \u2014\npure encoding defect. FIX DESIGNED (D-064 \u00a71): paired symmetric form\n(manifestly real, absolutely summable via pair bound |pair| \u2264\nC(n)\u00b7mult/|\u03c1|\u00b2 + Zeta23's proved `zero_sum_inv_sq`). Scheduled as\nsub-stage ii\u2032-1a (first executable item).\n\n## DISCOVERY 2 \u2014 the crux identity is Sf = Res\u2081 EXACTLY\nThe D-036/D-037 bookkeeping, settled against Lagarias's paper and\nverified numerically:\n- **Sf(n) := S_\u221e(n) + 1 \u2212 \u03bb_n = Res\u2081(n) = \u03a3_{j=1}^n C(n,j)\u00b7c_{j\u22121}**\n (c_k = Laurent coefficients of \u2212\u03b6\u2032/\u03b6 at 1). Follows from Lagarias\n (4.10) [\u03c4_j = \u03b7_j + (\u22121)\u02b2\u03c3_{j+1} \u2212 (\u22121)\u02b2\u03b4] summed binomially.\n Verified: (4.10) at j=0 exact to 15 digits (\u03bb\u2081 = 0.02309570896612103);\n crux consequence \u03bb\u2082 = 0.09234573523 matches E-015's certified\n 0.09234573522804667 exactly.\n- **Residue theorem**: sides(T) = \u03bb_n(T) + triv + Res\u2081(n), verified to\n 2e-21 (n=2) by direct contour integration.\n- **Assembly is TAIL-FREE**: |Sf| \u2264 |\u03bb_n(T)| + |sides| + |triv| \u2264\n 2N(T) + [side pieces] + 1 = B(n) \u2014 no tail term anywhere (the crux\n absorbs it). This is why D-037's B(n) has none.\n- The w=1 pole (order n+1) residue computation reduces to a Leibniz\n rule: (w\u22121)^{n+1}J = [w\u207f \u2212 (w\u22121)\u207f]h(w), h = (w\u22121)(\u2212\u03b6\u2032/\u03b6) analytic \u2014\n Res = \u03a3_j C(n,j)c_{j\u22121} confirmed algebraically.\n- SIDE EFFECT: D-037's line-63 compressed identity needs an erratum\n pointer to D-064 (correct reading: Sf = Res\u2081 = sides \u2212 \u03bb_n(T) \u2212 triv).\n\n## Inventory highlights (why tier ii\u2032 is feasible)\n- Rectangle residue theorem EXISTS (vendored FromPNTPlus\n `RectangleIntegral'_eq_sumResiduesIn`, axiom-clean) \u2014 needs the\n simple-pole scaffold extended to finite order (bounded, R1).\n- `logDerivResidue`, `ResidueMult`, `riemannZetaLogDerivResidue`\n vendored and clean.\n- Mathlib HAS Borel\u2013Carath\u00e9odory, Jensen's formula, digamma,\n HadamardThreeLines, MeromorphicAt/Order/NormalForm/FactorizedRational.\n- Zeta23 HAS proved \u03a3 mult/(1+\u2016\u03b3_\u03c1\u2016\u00b2) < \u221e (zero_sum_inv_sq), local\n zero count, RvM, reflect/conj symmetry, window finiteness.\n- Mathlib has NO Weierstrass/Hadamard factorization; FromPNTPlus\n StrongPNTPrefix has the local Blaschke/Borel toolkit.\n\n## Route decisions (D-064 \u00a74)\nR1 build higher-order residue machinery (1\u20132 sess); R2 crux via (4.10)\n\u2014 investigate zeta-23 EF reuse (\u03b22) first, Hadamard fallback (\u03b21)\n(2\u20134 sess, the long pole); R3 R-bound via LOCAL route\n(Borel\u2013Carath\u00e9odory/Jensen/Blaschke \u2014 machinery exists) (2\u20133 sess);\nR4 the old \"w=1 \u2194 S_\u221e\" monster is KILLED by Discovery 2 (factors into\nR1's Leibniz computation + R2's (4.10)); R5 perimeter unchanged.\n\n## State at stop\nD-064 complete; sub-stage map ii\u2032-1a..7 (total ~9\u201316 sessions).\nNEXT: ii\u2032-1a (liCoeff pairing fix + summability). Paper A untouched\n(compiles at session-013 state; submission-ready fallback intact \u2014\nD-064 \u00a78).\n"
}
raw content_json (database row)
[{"type":"thinking","thinking":"D-064 registered. Now write the journal entry, update STATE.md, update NGUYEN.md (session log + tier ii′ activation note), and commit.","signature":""},{"type":"text","text":"Registered. Now the journal, STATE.md, and NGUYEN.md updates."},{"type":"tool_use","id":"Write_61","name":"Write","input":{"file_path":"D:\\Python\\Quantum\\research\\journal\\session-014.md","content":"# Session 014 — 2026-08-21 — Tier ii′ launched: design + two discoveries\n\n## Directive\nUser chose option (a): do tier ii′ (close SfBoundProp in Lean) BEFORE\nsubmission, with Paper A frozen and submittable at all times. Freeze\nprotocol in force: no edits to paperA.tex/paperB.tex/draft.tex/\nreferences.bib until ii′-7; no edits to existing Lean proofs except the\nliCoeff correctness fix (below, flagged); commit at every sub-stage.\n\n## Work done (sub-stage ii′-0: design → D-064)\n1. Baseline verified clean (git: only untracked LaTeX artifacts).\n2. Two parallel inventory agents surveyed: (i) vendored residue/\n meromorphic API; (ii) Zeta23 zero infrastructure + NguyenRH1 defs +\n Mathlib zeta/digamma/Jensen/Borel–Carathéodory surface.\n3. Read liCoeff/Sf/perimeter defs, D-036, D-056, D-037 (full), and the\n primary source (Lagarias math/0404394 §4–6, archived PDF).\n4. Axiom-checked all planned Zeta23 dependencies LIVE: zero_sum_inv_sq,\n zeta_local_zero_count, zeta_reflect_zero, zeta_mult_reflect,\n RectangleIntegral'_eq_sumResiduesIn, ResidueMult, logDerivResidue,\n riemannZetaLogDerivResidue — ALL 3 standard axioms only.\n5. Settled the exact identity chain NUMERICALLY (dps 40–50, mpmath +\n E-021 zero cache; see D-064 §3 table).\n6. Wrote D-064 (design doc: discoveries, routes R1–R5, lemma graph,\n revised sub-stage map ii′-1a..7). Registered in index.md + db.\n\n## DISCOVERY 1 — liCoeff encoding defect (correctness, not cosmetic)\n`liCoeff` (LiSkeleton.lean:39) is an unordered tsum whose family is NOT\nSummable for any n ≥ 1 (summand ~ n/|γ|, Σ mult/|ρ| = ∞ by RvM).\nTherefore liCoeff n = 0 (junk), making `LiVerifiedUpTo` FALSE and\n`SfBoundProp` FALSE as encoded (at n = 10⁵: |Sf| ≥ 432 501 > 347 000\n≥ Bn by the certified LiBasePoints bounds). The session-012 theorems\nare vacuously true; the session-013 audit could not see this (it checks\naxioms, not hypothesis truth). Paper A's mathematics is unaffected —\npure encoding defect. FIX DESIGNED (D-064 §1): paired symmetric form\n(manifestly real, absolutely summable via pair bound |pair| ≤\nC(n)·mult/|ρ|² + Zeta23's proved `zero_sum_inv_sq`). Scheduled as\nsub-stage ii′-1a (first executable item).\n\n## DISCOVERY 2 — the crux identity is Sf = Res₁ EXACTLY\nThe D-036/D-037 bookkeeping, settled against Lagarias's paper and\nverified numerically:\n- **Sf(n) := S_∞(n) + 1 − λ_n = Res₁(n) = Σ_{j=1}^n C(n,j)·c_{j−1}**\n (c_k = Laurent coefficients of −ζ′/ζ at 1). Follows from Lagarias\n (4.10) [τ_j = η_j + (−1)ʲσ_{j+1} − (−1)ʲδ] summed binomially.\n Verified: (4.10) at j=0 exact to 15 digits (λ₁ = 0.02309570896612103);\n crux consequence λ₂ = 0.09234573523 matches E-015's certified\n 0.09234573522804667 exactly.\n- **Residue theorem**: sides(T) = λ_n(T) + triv + Res₁(n), verified to\n 2e-21 (n=2) by direct contour integration.\n- **Assembly is TAIL-FREE**: |Sf| ≤ |λ_n(T)| + |sides| + |triv| ≤\n 2N(T) + [side pieces] + 1 = B(n) — no tail term anywhere (the crux\n absorbs it). This is why D-037's B(n) has none.\n- The w=1 pole (order n+1) residue computation reduces to a Leibniz\n rule: (w−1)^{n+1}J = [wⁿ − (w−1)ⁿ]h(w), h = (w−1)(−ζ′/ζ) analytic —\n Res = Σ_j C(n,j)c_{j−1} confirmed algebraically.\n- SIDE EFFECT: D-037's line-63 compressed identity needs an erratum\n pointer to D-064 (correct reading: Sf = Res₁ = sides − λ_n(T) − triv).\n\n## Inventory highlights (why tier ii′ is feasible)\n- Rectangle residue theorem EXISTS (vendored FromPNTPlus\n `RectangleIntegral'_eq_sumResiduesIn`, axiom-clean) — needs the\n simple-pole scaffold extended to finite order (bounded, R1).\n- `logDerivResidue`, `ResidueMult`, `riemannZetaLogDerivResidue`\n vendored and clean.\n- Mathlib HAS Borel–Carathéodory, Jensen's formula, digamma,\n HadamardThreeLines, MeromorphicAt/Order/NormalForm/FactorizedRational.\n- Zeta23 HAS proved Σ mult/(1+‖γ_ρ‖²) < ∞ (zero_sum_inv_sq), local\n zero count, RvM, reflect/conj symmetry, window finiteness.\n- Mathlib has NO Weierstrass/Hadamard factorization; FromPNTPlus\n StrongPNTPrefix has the local Blaschke/Borel toolkit.\n\n## Route decisions (D-064 §4)\nR1 build higher-order residue machinery (1–2 sess); R2 crux via (4.10)\n— investigate zeta-23 EF reuse (β2) first, Hadamard fallback (β1)\n(2–4 sess, the long pole); R3 R-bound via LOCAL route\n(Borel–Carathéodory/Jensen/Blaschke — machinery exists) (2–3 sess);\nR4 the old \"w=1 ↔ S_∞\" monster is KILLED by Discovery 2 (factors into\nR1's Leibniz computation + R2's (4.10)); R5 perimeter unchanged.\n\n## State at stop\nD-064 complete; sub-stage map ii′-1a..7 (total ~9–16 sessions).\nNEXT: ii′-1a (liCoeff pairing fix + summability). Paper A untouched\n(compiles at session-013 state; submission-ready fallback intact —\nD-064 §8).\n"},"extra_content":null}]