← Session 34 — F-D repair scoping: 6-R side thread setup
thinking (2824 chars)
So D-071 and D-072 are NOT registered in research.db either, and not in index.md. Session 038's end-protocol was partially executed for those (they exist as files, are referenced in STATE.md/NGUYEN.md, but index.md + db registrations were missed — consistent with the session ending at the F-D discovery with the user pausing).
While I'm here fixing archival hygiene: I should register D-071 and D-072 in the db + index, AND create the D-073 seed file + index row + db registration. That keeps everything consistent. This is in-scope: the user asked about persistence/tracking, and I'm ensuring the new thread's anchor (plus fixing the two gaps from session 038 that would otherwise confuse the resume).
Let me:
1. Create research/derivations/D-073-fd-horizontal-repair.md — the mission seed: header (title, date, session, status IN PROGRESS, evidence planned), the mission prompt verbatim (as "Mission (agreed 2026-08-25)" section), and a progress log section stub for the thread to append to.
2. Append rows for D-071, D-072, D-073 to research/derivations/index.md.
3. Register all three in research.db via db.py add-derivation.
4. Update STATE.md NEXT ACTION to point at the D-073 mission file (replace "the thread prompt (agreed 2026-08-25)" with the concrete path).
5. Commit.
For the D-073 file content — the mission prompt verbatim from my earlier message, lightly adapted into file form. Keep it as the exact agreed scope. Also add a "Progress log" section at the end (empty, for the thread to fill — that's what makes stop-and-go precise within the thread).
db.py add-derivation signature: positional id, --title --task --status --file (NO --date flag per handoff notes).
D-071 file: research/derivations/D-071-ii-prime-6-design.md (exists per earlier ls).
D-072 file: research/derivations/D-072-finding-F-D-right-tail-stepping.md (exists).
D-073 file: research/derivations/D-073-fd-horizontal-repair.md (to create).
Titles for index/db:
- D-071: "ii′-6-0 design: honest-Bn redefinition (4F, (2/3)Z2hat) + 8-step protocol (user-approved); findings F-A/F-B/F-C; perimeter additions ZeroCountGlobalProp + MZeroProp; E-030 ALL CHECKS PASS" | task 5.1-ii′-6 | complete (design)
- D-072: "Finding F-D: the (6.12) stepping in Lagarias Thm 6.1 broken at T = √n (ratio e/2 > 1; right-tail ~e^{0.066√n}; E-031); inheritance into R-004's B(n); what survives; repair (R) T = c√n verified; literature status; open questions" | task 5.1-ii′-6 | documented
- D-073: "F-D repair side thread ii′-6-R: mission + plan — focused re-proof of the (6.12) horizontal right-tail (6-R-0 E-032 diagnostic → 6-R-1 repaired proof with explicit constants → 6-R-2 drafts)" | task 5.1-ii′-6-R | in-progress
Now write the D-073 file. Content: header + mission (the agreed prompt, verbatim-ish, formatted as the mission) + progress log stub.
Session 038's archive closure missed D-071/D-072 registrations (the session ended at the F-D discovery). I'll fix that while creating the D-073 mission anchor.
tool: Write ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\derivations\\D-073-fd-horizontal-repair.md",
"content": "# D-073 \u2014 F-D repair side thread ii\u2032-6-R: mission, plan, and repaired proof of the Thm 6.1 horizontal right-tail\nDate: 2026-08-25 | Session: 039 (mission agreed; thread START) | Status: IN PROGRESS \u2014 sub-stage 6-R-0 (E-032 diagnostic) is next\nEvidence: E-031 (finding), E-032 (planned diagnostic), E-033 (planned cross-validation)\nChecklist anchor: NGUYEN.md \u00a75 Phase 5B, item **ii\u2032-6-R**\n\n## 0. Mission (agreed 2026-08-25, user directive \u2014 this section IS the thread prompt, persisted so nothing lives only in conversation)\n\n**MISSION \u2014 focused re-proof of the right part of the horizontal\nsegments in Lagarias's Theorem 6.1 (Finding F-D follow-up).** This is\na SEPARATE, SELF-CONTAINED side thread. The main line (Phase 5B, Lean\ntier ii\u2032) is parked at the Finding F-D decision point awaiting exactly\nthis thread's deliverables. Do NOT resume main-line work; do NOT open\nLean formalization.\n\n**READ FIRST (in order):**\n1. STATE.md \u2014 resume point.\n2. research/derivations/D-072-finding-F-D-right-tail-stepping.md \u2014\n the finding: mechanism, E-031 evidence, inheritance, what survives.\n3. research/paper/lagarias-li-coefficients-automorphic-0404394.pdf,\n pages 27\u201329 \u2014 the actual Thm 6.1 proof passage ((6.9)\u2013(6.14)).\n4. research/derivations/D-071-ii-prime-6-design.md \u2014 the honest-Bn\n constant machinery this repair feeds (kernel charge, bridge, FAlt).\n5. research/numerics/scripts/E-031 script + research/numerics/data\n E-031 sidecars \u2014 the envelope evidence.\n\n**BACKGROUND (established; do not re-derive):**\n- The printed stepping (6.12) bounds the kernel ratio by e and (6.10)\n gains a factor 2 per unit step right from \u03c3 = 3, so the product\n ratio is \u2264 e/2 > 1: the product GROWS to a peak ~e^{0.066\u221an} at\n \u03c3 \u2248 0.36\u221an (E-031, measured). Both bounds are SHARP (zeta halving =\n Dirichlet m=2 term; kernel ratio |1\u22121/(s+1)\u00b2|\u207f \u2248 e^{n/T\u00b2} exact), so\n no side-wise/triangle refinement at T = \u221an can work.\n- Oscillation cannot rescue T = \u221an by standard tools: the phase\n \u03a6(\u03c3) \u2248 \u2212n^{3/2}/(\u03c3\u00b2+n) has |\u03a6\u2032| = O(1), so IBP/van der\n Corput/stationary phase gain O(1)\u2013polynomial factors against an\n e^{0.066\u221an} amplitude range. The only known exact evaluation is the\n residue identity itself, which is circular for this purpose.\n- Conjugation gives I_II + I_IV = 2i\u00b7Im(I_II): the closed identity\n constrains ONLY Im(I_II). Re(I_II) is unconstrained by the contour\n argument and may be of order ~envelope \u2014 i.e. the printed per-side\n claim |I_II| = O(\u221an) may be FALSE as a statement at T = \u221an.\n- The repair is one line inside Lagarias's own framework: he proves\n the horizontal bound for general T with T\u00b2 \u2265 n and only specializes\n T = \u221an + \u03b5 in the final line. Taking T = c\u221an + \u03b5 with\n c > 1/\u221a(log 2) \u2248 1.2012 makes the kernel ratio \u2264 e^{1/c\u00b2} < 2, the\n stepping works verbatim, and every other ingredient is T-generic\n (verified numerically, E-031 V3, c = 1.5 and 2).\n\n**DELIVERABLES:**\n- (D1) **E-032 DIAGNOSTIC** (cheap, decisive): compute the TRUE\n complex horizontal integral I_II at T = \u221an (zero-avoiding) for\n n = 10\u00b3, 10\u2074 (10\u2075 if cheap) by direct quadrature (mpmath, dps \u2265 40;\n integrand smooth, phase O(1) rad/unit; validate \u03b6\u2032/\u03b6 against the\n Dirichlet series where re > 2). Compare against the E-031 envelope.\n Verdict: Re(I_II) ~ envelope \u27f9 the per-side claim is FALSE at\n T = \u221an (only the pair survives); |I_II| \u226a envelope \u27f9 hidden\n cancellation exists. Record the numbers either way.\n- (D2) **THE REPAIRED PROOF** (this file, sections to be written):\n (i) dependency audit \u2014 every ingredient of Thm 6.1's proof listed\n and marked T-generic / T-dependent;\n (ii) the corrected stepping lemma at T = c\u221an, c > 1/\u221a(log 2):\n per-step product ratio \u2264 q := e^{1/c\u00b2}/2 < 1, geometric series from\n \u03c3 = 3, explicit right-tail bound \u222b\u2083^{\u03c3\u2081}, total horizontal bound\n |I_II| + |I_IV| with EXPLICIT constants (no untraced O's \u2014 use\n D-071's machinery: kernel charge e^{4/c\u00b2}+1, |\u03b6\u2032/\u03b6(2)|, the\n zero-avoiding (log T)\u00b2 bound);\n (iii) recommended c for the main line, quantified (2 margin /\n 1.5 range);\n (iv) consequence audit (Thm 6.1 conclusion + Thm 1.1 height);\n (v) numerical cross-validation of all new constants (E-033).\n- (D3) A one-paragraph \"repair summary\" suitable for the Lagarias\n letter (DRAFT only \u2014 do NOT edit any correspondence file).\n- (D4) A one-paragraph verdict on the T = \u221an per-side question,\n combining E-032 with the circularity analysis (DRAFT only).\n\n**HARD CONSTRAINTS:**\n- OUT OF SCOPE: re-proving any other part of Thm 6.1 (audit only);\n the T = \u221an salvage as a proof goal (D1's verdict settles whether\n anything there remains to pursue \u2014 if |I_II| comes out SMALL, STOP\n and report to the user before going further); Lean work; contacting\n anyone.\n- FREEZE PROTOCOL (user directive 2026-08-21): no edits to\n research/paper/paperA.tex, paperB.tex, draft.tex, references.bib;\n no edits to existing Lean files; no edits to\n research/correspondence/**. New files only: this file,\n research/numerics/scripts/E-032*.py / E-033*.py + data sidecars.\n- Honesty tag on every claim ([PROVED] / [NUMERICAL-VALIDATED] /\n [HEURISTIC]); numerics deterministic + cached, CSV/JSON sidecars,\n registered in research.db via db.py.\n- Session protocol per NGUYEN.md \u00a78; sub-stage rule (\u00a78.5): land D1\n first as its own committed sub-stage.\n\n**SUCCESS CRITERIA:**\n1. This file contains the complete repaired proof of the horizontal\n bound with explicit constants, plus the dependency and consequence\n audits, all honesty-tagged.\n2. E-032's verdict recorded with numbers.\n3. The main line can execute repair (R) from this file + the\n recommended c without any further analysis.\n\n## 1. Progress log (append one entry per sub-stage; this is the in-thread stop-and-go record)\n\n- 2026-08-25: mission agreed (user directive), thread registered in\n NGUYEN.md \u00a75 as ii\u2032-6-R; this file created as the resume anchor.\n NEXT: sub-stage 6-R-0 (E-032 diagnostic).\n\n<!-- Sections to be written by the thread:\n## 2. E-032 diagnostic (D1) \u2014 verdict + numbers\n## 3. Dependency audit (D2-i)\n## 4. The corrected stepping lemma (D2-ii) \u2014 repaired proof, explicit constants\n## 5. The choice of c (D2-iii)\n## 6. Consequence audit (D2-iv)\n## 7. Cross-validation (D2-v, E-033)\n## 8. Drafts (D3 letter paragraph; D4 per-side verdict paragraph)\n-->\n"
}
result 2026-08-25 22:40:16Z → 2026-08-25 22:40:16Z
{
"type": "create",
"filePath": "D:\\Python\\Quantum\\research\\derivations\\D-073-fd-horizontal-repair.md",
"content": "# D-073 \u2014 F-D repair side thread ii\u2032-6-R: mission, plan, and repaired proof of the Thm 6.1 horizontal right-tail\nDate: 2026-08-25 | Session: 039 (mission agreed; thread START) | Status: IN PROGRESS \u2014 sub-stage 6-R-0 (E-032 diagnostic) is next\nEvidence: E-031 (finding), E-032 (planned diagnostic), E-033 (planned cross-validation)\nChecklist anchor: NGUYEN.md \u00a75 Phase 5B, item **ii\u2032-6-R**\n\n## 0. Mission (agreed 2026-08-25, user directive \u2014 this section IS the thread prompt, persisted so nothing lives only in conversation)\n\n**MISSION \u2014 focused re-proof of the right part of the horizontal\nsegments in Lagarias's Theorem 6.1 (Finding F-D follow-up).** This is\na SEPARATE, SELF-CONTAINED side thread. The main line (Phase 5B, Lean\ntier ii\u2032) is parked at the Finding F-D decision point awaiting exactly\nthis thread's deliverables. Do NOT resume main-line work; do NOT open\nLean formalization.\n\n**READ FIRST (in order):**\n1. STATE.md \u2014 resume point.\n2. research/derivations/D-072-finding-F-D-right-tail-stepping.md \u2014\n the finding: mechanism, E-031 evidence, inheritance, what survives.\n3. research/paper/lagarias-li-coefficients-automorphic-0404394.pdf,\n pages 27\u201329 \u2014 the actual Thm 6.1 proof passage ((6.9)\u2013(6.14)).\n4. research/derivations/D-071-ii-prime-6-design.md \u2014 the honest-Bn\n constant machinery this repair feeds (kernel charge, bridge, FAlt).\n5. research/numerics/scripts/E-031 script + research/numerics/data\n E-031 sidecars \u2014 the envelope evidence.\n\n**BACKGROUND (established; do not re-derive):**\n- The printed stepping (6.12) bounds the kernel ratio by e and (6.10)\n gains a factor 2 per unit step right from \u03c3 = 3, so the product\n ratio is \u2264 e/2 > 1: the product GROWS to a peak ~e^{0.066\u221an} at\n \u03c3 \u2248 0.36\u221an (E-031, measured). Both bounds are SHARP (zeta halving =\n Dirichlet m=2 term; kernel ratio |1\u22121/(s+1)\u00b2|\u207f \u2248 e^{n/T\u00b2} exact), so\n no side-wise/triangle refinement at T = \u221an can work.\n- Oscillation cannot rescue T = \u221an by standard tools: the phase\n \u03a6(\u03c3) \u2248 \u2212n^{3/2}/(\u03c3\u00b2+n) has |\u03a6\u2032| = O(1), so IBP/van der\n Corput/stationary phase gain O(1)\u2013polynomial factors against an\n e^{0.066\u221an} amplitude range. The only known exact evaluation is the\n residue identity itself, which is circular for this purpose.\n- Conjugation gives I_II + I_IV = 2i\u00b7Im(I_II): the closed identity\n constrains ONLY Im(I_II). Re(I_II) is unconstrained by the contour\n argument and may be of order ~envelope \u2014 i.e. the printed per-side\n claim |I_II| = O(\u221an) may be FALSE as a statement at T = \u221an.\n- The repair is one line inside Lagarias's own framework: he proves\n the horizontal bound for general T with T\u00b2 \u2265 n and only specializes\n T = \u221an + \u03b5 in the final line. Taking T = c\u221an + \u03b5 with\n c > 1/\u221a(log 2) \u2248 1.2012 makes the kernel ratio \u2264 e^{1/c\u00b2} < 2, the\n stepping works verbatim, and every other ingredient is T-generic\n (verified numerically, E-031 V3, c = 1.5 and 2).\n\n**DELIVERABLES:**\n- (D1) **E-032 DIAGNOSTIC** (cheap, decisive): compute the TRUE\n complex horizontal integral I_II at T = \u221an (zero-avoiding) for\n n = 10\u00b3, 10\u2074 (10\u2075 if cheap) by direct quadrature (mpmath, dps \u2265 40;\n integrand smooth, phase O(1) rad/unit; validate \u03b6\u2032/\u03b6 against the\n Dirichlet series where re > 2). Compare against the E-031 envelope.\n Verdict: Re(I_II) ~ envelope \u27f9 the per-side claim is FALSE at\n T = \u221an (only the pair survives); |I_II| \u226a envelope \u27f9 hidden\n cancellation exists. Record the numbers either way.\n- (D2) **THE REPAIRED PROOF** (this file, sections to be written):\n (i) dependency audit \u2014 every ingredient of Thm 6.1's proof listed\n and marked T-generic / T-dependent;\n (ii) the corrected stepping lemma at T = c\u221an, c > 1/\u221a(log 2):\n per-step product ratio \u2264 q := e^{1/c\u00b2}/2 < 1, geometric series from\n \u03c3 = 3, explicit right-tail bound \u222b\u2083^{\u03c3\u2081}, total horizontal bound\n |I_II| + |I_IV| with EXPLICIT constants (no untraced O's \u2014 use\n D-071's machinery: kernel charge e^{4/c\u00b2}+1, |\u03b6\u2032/\u03b6(2)|, the\n zero-avoiding (log T)\u00b2 bound);\n (iii) recommended c for the main line, quantified (2 margin /\n 1.5 range);\n (iv) consequence audit (Thm 6.1 conclusion + Thm 1.1 height);\n (v) numerical cross-validation of all new constants (E-033).\n- (D3) A one-paragraph \"repair summary\" suitable for the Lagarias\n letter (DRAFT only \u2014 do NOT edit any correspondence file).\n- (D4) A one-paragraph verdict on the T = \u221an per-side question,\n combining E-032 with the circularity analysis (DRAFT only).\n\n**HARD CONSTRAINTS:**\n- OUT OF SCOPE: re-proving any other part of Thm 6.1 (audit only);\n the T = \u221an salvage as a proof goal (D1's verdict settles whether\n anything there remains to pursue \u2014 if |I_II| comes out SMALL, STOP\n and report to the user before going further); Lean work; contacting\n anyone.\n- FREEZE PROTOCOL (user directive 2026-08-21): no edits to\n research/paper/paperA.tex, paperB.tex, draft.tex, references.bib;\n no edits to existing Lean files; no edits to\n research/correspondence/**. New files only: this file,\n research/numerics/scripts/E-032*.py / E-033*.py + data sidecars.\n- Honesty tag on every claim ([PROVED] / [NUMERICAL-VALIDATED] /\n [HEURISTIC]); numerics deterministic + cached, CSV/JSON sidecars,\n registered in research.db via db.py.\n- Session protocol per NGUYEN.md \u00a78; sub-stage rule (\u00a78.5): land D1\n first as its own committed sub-stage.\n\n**SUCCESS CRITERIA:**\n1. This file contains the complete repaired proof of the horizontal\n bound with explicit constants, plus the dependency and consequence\n audits, all honesty-tagged.\n2. E-032's verdict recorded with numbers.\n3. The main line can execute repair (R) from this file + the\n recommended c without any further analysis.\n\n## 1. Progress log (append one entry per sub-stage; this is the in-thread stop-and-go record)\n\n- 2026-08-25: mission agreed (user directive), thread registered in\n NGUYEN.md \u00a75 as ii\u2032-6-R; this file created as the resume anchor.\n NEXT: sub-stage 6-R-0 (E-032 diagnostic).\n\n<!-- Sections to be written by the thread:\n## 2. E-032 diagnostic (D1) \u2014 verdict + numbers\n## 3. Dependency audit (D2-i)\n## 4. The corrected stepping lemma (D2-ii) \u2014 repaired proof, explicit constants\n## 5. The choice of c (D2-iii)\n## 6. Consequence audit (D2-iv)\n## 7. Cross-validation (D2-v, E-033)\n## 8. Drafts (D3 letter paragraph; D4 per-side verdict paragraph)\n-->\n",
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raw content_json (database row)
[{"type":"thinking","thinking":"So D-071 and D-072 are NOT registered in research.db either, and not in index.md. Session 038's end-protocol was partially executed for those (they exist as files, are referenced in STATE.md/NGUYEN.md, but index.md + db registrations were missed — consistent with the session ending at the F-D discovery with the user pausing).\n\nWhile I'm here fixing archival hygiene: I should register D-071 and D-072 in the db + index, AND create the D-073 seed file + index row + db registration. That keeps everything consistent. This is in-scope: the user asked about persistence/tracking, and I'm ensuring the new thread's anchor (plus fixing the two gaps from session 038 that would otherwise confuse the resume).\n\nLet me:\n1. Create research/derivations/D-073-fd-horizontal-repair.md — the mission seed: header (title, date, session, status IN PROGRESS, evidence planned), the mission prompt verbatim (as \"Mission (agreed 2026-08-25)\" section), and a progress log section stub for the thread to append to.\n2. Append rows for D-071, D-072, D-073 to research/derivations/index.md.\n3. Register all three in research.db via db.py add-derivation.\n4. Update STATE.md NEXT ACTION to point at the D-073 mission file (replace \"the thread prompt (agreed 2026-08-25)\" with the concrete path).\n5. Commit.\n\nFor the D-073 file content — the mission prompt verbatim from my earlier message, lightly adapted into file form. Keep it as the exact agreed scope. Also add a \"Progress log\" section at the end (empty, for the thread to fill — that's what makes stop-and-go precise within the thread).\n\ndb.py add-derivation signature: positional id, --title --task --status --file (NO --date flag per handoff notes).\n\nD-071 file: research/derivations/D-071-ii-prime-6-design.md (exists per earlier ls).\nD-072 file: research/derivations/D-072-finding-F-D-right-tail-stepping.md (exists).\nD-073 file: research/derivations/D-073-fd-horizontal-repair.md (to create).\n\nTitles for index/db:\n- D-071: \"ii′-6-0 design: honest-Bn redefinition (4F, (2/3)Z2hat) + 8-step protocol (user-approved); findings F-A/F-B/F-C; perimeter additions ZeroCountGlobalProp + MZeroProp; E-030 ALL CHECKS PASS\" | task 5.1-ii′-6 | complete (design)\n- D-072: \"Finding F-D: the (6.12) stepping in Lagarias Thm 6.1 broken at T = √n (ratio e/2 > 1; right-tail ~e^{0.066√n}; E-031); inheritance into R-004's B(n); what survives; repair (R) T = c√n verified; literature status; open questions\" | task 5.1-ii′-6 | documented\n- D-073: \"F-D repair side thread ii′-6-R: mission + plan — focused re-proof of the (6.12) horizontal right-tail (6-R-0 E-032 diagnostic → 6-R-1 repaired proof with explicit constants → 6-R-2 drafts)\" | task 5.1-ii′-6-R | in-progress\n\nNow write the D-073 file. Content: header + mission (the agreed prompt, verbatim-ish, formatted as the mission) + progress log stub.","signature":""},{"type":"text","text":"Session 038's archive closure missed D-071/D-072 registrations (the session ended at the F-D discovery). I'll fix that while creating the D-073 mission anchor."},{"type":"tool_use","id":"Write_21","name":"Write","input":{"file_path":"D:\\Python\\Quantum\\research\\derivations\\D-073-fd-horizontal-repair.md","content":"# D-073 — F-D repair side thread ii′-6-R: mission, plan, and repaired proof of the Thm 6.1 horizontal right-tail\nDate: 2026-08-25 | Session: 039 (mission agreed; thread START) | Status: IN PROGRESS — sub-stage 6-R-0 (E-032 diagnostic) is next\nEvidence: E-031 (finding), E-032 (planned diagnostic), E-033 (planned cross-validation)\nChecklist anchor: NGUYEN.md §5 Phase 5B, item **ii′-6-R**\n\n## 0. Mission (agreed 2026-08-25, user directive — this section IS the thread prompt, persisted so nothing lives only in conversation)\n\n**MISSION — focused re-proof of the right part of the horizontal\nsegments in Lagarias's Theorem 6.1 (Finding F-D follow-up).** This is\na SEPARATE, SELF-CONTAINED side thread. The main line (Phase 5B, Lean\ntier ii′) is parked at the Finding F-D decision point awaiting exactly\nthis thread's deliverables. Do NOT resume main-line work; do NOT open\nLean formalization.\n\n**READ FIRST (in order):**\n1. STATE.md — resume point.\n2. research/derivations/D-072-finding-F-D-right-tail-stepping.md —\n the finding: mechanism, E-031 evidence, inheritance, what survives.\n3. research/paper/lagarias-li-coefficients-automorphic-0404394.pdf,\n pages 27–29 — the actual Thm 6.1 proof passage ((6.9)–(6.14)).\n4. research/derivations/D-071-ii-prime-6-design.md — the honest-Bn\n constant machinery this repair feeds (kernel charge, bridge, FAlt).\n5. research/numerics/scripts/E-031 script + research/numerics/data\n E-031 sidecars — the envelope evidence.\n\n**BACKGROUND (established; do not re-derive):**\n- The printed stepping (6.12) bounds the kernel ratio by e and (6.10)\n gains a factor 2 per unit step right from σ = 3, so the product\n ratio is ≤ e/2 > 1: the product GROWS to a peak ~e^{0.066√n} at\n σ ≈ 0.36√n (E-031, measured). Both bounds are SHARP (zeta halving =\n Dirichlet m=2 term; kernel ratio |1−1/(s+1)²|ⁿ ≈ e^{n/T²} exact), so\n no side-wise/triangle refinement at T = √n can work.\n- Oscillation cannot rescue T = √n by standard tools: the phase\n Φ(σ) ≈ −n^{3/2}/(σ²+n) has |Φ′| = O(1), so IBP/van der\n Corput/stationary phase gain O(1)–polynomial factors against an\n e^{0.066√n} amplitude range. The only known exact evaluation is the\n residue identity itself, which is circular for this purpose.\n- Conjugation gives I_II + I_IV = 2i·Im(I_II): the closed identity\n constrains ONLY Im(I_II). Re(I_II) is unconstrained by the contour\n argument and may be of order ~envelope — i.e. the printed per-side\n claim |I_II| = O(√n) may be FALSE as a statement at T = √n.\n- The repair is one line inside Lagarias's own framework: he proves\n the horizontal bound for general T with T² ≥ n and only specializes\n T = √n + ε in the final line. Taking T = c√n + ε with\n c > 1/√(log 2) ≈ 1.2012 makes the kernel ratio ≤ e^{1/c²} < 2, the\n stepping works verbatim, and every other ingredient is T-generic\n (verified numerically, E-031 V3, c = 1.5 and 2).\n\n**DELIVERABLES:**\n- (D1) **E-032 DIAGNOSTIC** (cheap, decisive): compute the TRUE\n complex horizontal integral I_II at T = √n (zero-avoiding) for\n n = 10³, 10⁴ (10⁵ if cheap) by direct quadrature (mpmath, dps ≥ 40;\n integrand smooth, phase O(1) rad/unit; validate ζ′/ζ against the\n Dirichlet series where re > 2). Compare against the E-031 envelope.\n Verdict: Re(I_II) ~ envelope ⟹ the per-side claim is FALSE at\n T = √n (only the pair survives); |I_II| ≪ envelope ⟹ hidden\n cancellation exists. Record the numbers either way.\n- (D2) **THE REPAIRED PROOF** (this file, sections to be written):\n (i) dependency audit — every ingredient of Thm 6.1's proof listed\n and marked T-generic / T-dependent;\n (ii) the corrected stepping lemma at T = c√n, c > 1/√(log 2):\n per-step product ratio ≤ q := e^{1/c²}/2 < 1, geometric series from\n σ = 3, explicit right-tail bound ∫₃^{σ₁}, total horizontal bound\n |I_II| + |I_IV| with EXPLICIT constants (no untraced O's — use\n D-071's machinery: kernel charge e^{4/c²}+1, |ζ′/ζ(2)|, the\n zero-avoiding (log T)² bound);\n (iii) recommended c for the main line, quantified (2 margin /\n 1.5 range);\n (iv) consequence audit (Thm 6.1 conclusion + Thm 1.1 height);\n (v) numerical cross-validation of all new constants (E-033).\n- (D3) A one-paragraph \"repair summary\" suitable for the Lagarias\n letter (DRAFT only — do NOT edit any correspondence file).\n- (D4) A one-paragraph verdict on the T = √n per-side question,\n combining E-032 with the circularity analysis (DRAFT only).\n\n**HARD CONSTRAINTS:**\n- OUT OF SCOPE: re-proving any other part of Thm 6.1 (audit only);\n the T = √n salvage as a proof goal (D1's verdict settles whether\n anything there remains to pursue — if |I_II| comes out SMALL, STOP\n and report to the user before going further); Lean work; contacting\n anyone.\n- FREEZE PROTOCOL (user directive 2026-08-21): no edits to\n research/paper/paperA.tex, paperB.tex, draft.tex, references.bib;\n no edits to existing Lean files; no edits to\n research/correspondence/**. New files only: this file,\n research/numerics/scripts/E-032*.py / E-033*.py + data sidecars.\n- Honesty tag on every claim ([PROVED] / [NUMERICAL-VALIDATED] /\n [HEURISTIC]); numerics deterministic + cached, CSV/JSON sidecars,\n registered in research.db via db.py.\n- Session protocol per NGUYEN.md §8; sub-stage rule (§8.5): land D1\n first as its own committed sub-stage.\n\n**SUCCESS CRITERIA:**\n1. This file contains the complete repaired proof of the horizontal\n bound with explicit constants, plus the dependency and consequence\n audits, all honesty-tagged.\n2. E-032's verdict recorded with numbers.\n3. The main line can execute repair (R) from this file + the\n recommended c without any further analysis.\n\n## 1. Progress log (append one entry per sub-stage; this is the in-thread stop-and-go record)\n\n- 2026-08-25: mission agreed (user directive), thread registered in\n NGUYEN.md §5 as ii′-6-R; this file created as the resume anchor.\n NEXT: sub-stage 6-R-0 (E-032 diagnostic).\n\n<!-- Sections to be written by the thread:\n## 2. E-032 diagnostic (D1) — verdict + numbers\n## 3. Dependency audit (D2-i)\n## 4. The corrected stepping lemma (D2-ii) — repaired proof, explicit constants\n## 5. The choice of c (D2-iii)\n## 6. Consequence audit (D2-iv)\n## 7. Cross-validation (D2-v, E-033)\n## 8. Drafts (D3 letter paragraph; D4 per-side verdict paragraph)\n-->\n"},"extra_content":null}]