[{"type":"thinking","thinking":"The user wants me to help write a prompt for a separate, self-contained thread focused on re-proving Lagarias's Theorem 6.1's right part of the horizontal segments — fixing just that specific gap, without redoing the whole Lagarias research.\n\nKey question they're asking: \"Can we do it without doing the Lagarias complete research over? Can we just focus on the issue and re-do that?\"\n\nThe answer is yes — the finding is highly localized. The gap is exactly one piece: the right part [3, 2√n] of the horizontal segments (II)/(IV), handled by the stepping argument in (6.12). Everything else in the proof (left vertical, right vertical, left part of horizontals [-3,3], the contour, the residue identity, the bridge to λ_n(T)) is fine. And the repair T = c√n is verified.\n\nSo the prompt should be for a new thread (a fresh conversation/session) that is self-contained: it needs to tell the new agent/thread enough context to work independently — pointing to the archive files (D-072, E-031, the paper PDF, the letter package), stating the goal precisely, scoping what NOT to do (don't redo the whole paper, don't touch the main ii′-6 line, freeze protocol), and defining deliverables.\n\nLet me think about what the goal of this separate thread is, per the user's words: \"re-proof Lagarias' Theorem 6.1, the right part of the horizontal segments correct as intended.\"\n\nHmm — \"correct as intended\" — so the goal is to produce a correct proof of the horizontal bound. Which version? The user says \"re-proof ... the right part of the horizontal segments correct as intended.\" Given my analysis, there are two levels:\n\n1. The repair (T = c√n): easy, verified. Produce a clean, rigorous, self-contained write-up of the corrected proof of the horizontal bound at T = c√n, c > 1/√(log 2). This makes Theorem 6.1's conclusion correct (with constants depending on c).\n\n2. Possibly also investigate the sharpening question (E-032 quadrature experiment) to know whether the per-side statement at T = √n is false.\n\nThe user asks \"Can we do it without doing the Lagarias complete research over? Can we just focus on the issue and re-do that?\" — Yes. The thread should be scoped narrowly: the only broken piece is the right-tail stepping; the fix touches only that piece plus the final specialization T = √n + ε → T = c√n + ε. A focused re-proof needs:\n\n- The exact statement of what's being proved: a lemma \"H-bound: for T = c√n + ε (zero-avoiding), c > 1/√(log 2), the horizontal integrals satisfy |I_II| + |I_IV| = O_c(√n)\" (with explicit constants if we want — our project needs explicit constants anyway, and the D-071 honest-Bn work already has them: kernel charge e^{4/c²}+1, etc.)\n- The inputs that are NOT in question (state them as black boxes): (6.9)/(6.10) the Dirichlet bound, Lemma 6.1 the L'/L bound O((log T)²) on the zero-avoiding line, the kernel bounds (6.11), the left/right vertical bounds, rectInt'_Jlag residue identity.\n- The proof itself: stepping with ratio e^{1/c²}/2 < 1, geometric series from σ = 3, right tail ∫ bounded by geometric sum → O(1) per unit... produce the O(√n) with explicit constant.\n\nActually — importantly — for OUR project the horizontal piece at ii′-6-c needs explicit constants anyway (B(n) is explicit). So this separate thread could do double duty: produce the corrected lemma in a form that (a) serves as the erratum's repaired proof for the letter/Lagarias, and (b) hands the main line the exact statement + constants to formalize in ii′-6-c. That's efficient: one focused thread, two consumers.\n\nAlso the E-032 experiment (true complex integral vs envelope) — should it be in this thread? The user asked to focus on the issue and re-do that. The E-032 experiment answers whether the ORIGINAL T = √n per-side claim is false. That's part of \"understanding the issue\" and sharpens the letter. I proposed it; the user seems to be following the proposal. I'd include it as Phase 0/1 of the thread since it's cheap and decisive, but clearly mark it as scoping/diagnostic, with the main deliverable being the repaired proof.\n\nWait — let me re-read the user: \"Our main goal is to re-proof largaris' Theorem 6.1, the right part of the horizontal segments correct as intended. Can we do it without doing the Largaris complete reasearch over. Can we just focus on the issue and re-do that?\"\n\nSo the MAIN GOAL of the new thread = re-prove the right part of the horizontal segments correctly. The user is asking whether the prompt can be scoped to just that issue. Answer: yes — and I should write the prompt that way. The prompt should:\n\n1. Set context: project, finding F-D, where things live (archive paths).\n2. State precisely the broken piece and why (so the thread doesn't re-derive the finding from scratch — it reads D-072 + paper pp. 28–29).\n3. Define the deliverable: a rigorous, self-contained repaired proof of the horizontal bound — the right part of (II)/(IV) — at T = c√n, c > 1/√(log 2), with explicit constants, written as a drop-in replacement for the (6.12) stepping passage; plus the verification that the rest of Thm 6.1's proof is untouched (a dependency audit: list every other ingredient and confirm each is unaffected by T = c√n).\n4. Define secondary deliverables: E-032 (true integral vs envelope — decides if per-side claim at T=√n is false); updated consequence audit (Thm 1.1 height factor); recommendation of c for the main line.\n5. Constraints: do NOT redo the whole paper; do NOT touch main-line files (freeze protocol: paperA/paperB/draft/references.bib frozen until ii′-7; existing Lean proofs untouched); work in a new file (e.g., D-073) + new numerics script E-032; honesty tags; end with archive updates per §8.4.\n\nAlso should the thread attempt the T=√n salvage (scope 3)? The user said \"correct as intended\" — hmm, \"as intended\" might mean \"as Lagarias intended\" (i.e., at T = √n)? Let me think. \"re-proof Lagaris' Theorem 6.1, the right part of the horizontal segments correct as intended\" — I read this as: make the right part of the horizontal segments correct, as (the theorem) intended. I.e., restore a correct proof of that piece. Given my previous message explained the T=√n salvage is research-level hard and the repair is the way, and the user is now asking for a focused thread, I think they want the repair done properly — but I should cover both: the thread's primary deliverable is the repaired proof (T = c√n), and the E-032 experiment determines whether anything more (the T=√n question) is even alive. I'll make that explicit in the prompt so the new thread knows the priority order.\n\nShould I ask the user a clarifying question about whether \"as intended\" means T=√n or the repair? Hmm. My previous turn already laid out the three scopes, and the user responded by asking for the prompt focused on re-proving the horizontal right part \"correct as intended\" without redoing everything. The natural reading: they want the focused repair thread. Including E-032 first gives the diagnostic on whether the original intent (T=√n) is salvageable at all. I could add a line in the prompt: \"Primary target: the repaired proof at T = c√n. The T = √n salvage is explicitly OUT of scope unless E-032 shows the true integral is small (hidden cancellation), in which case flag it back to the user before pursuing.\"\n\nThat handles both readings cleanly without another round-trip.\n\nNow — how should the prompt look? It should be copy-pasteable into a new conversation. It should be self-contained: the new agent in this repo will have NGUYEN.md auto-loaded, but the prompt should still point to exact files and state the mission, scope, deliverables, constraints, and end-protocol. Length: substantial but focused. Let me draft it.\n\nStructure of the prompt:\n\n---\n**THREAD MISSION: Repaired proof of the right part of the horizontal segments in Lagarias Thm 6.1 (Finding F-D follow-up)**\n\nContext (read first, in order):\n1. STATE.md — project resume point (session 038 end, Finding F-D).\n2. research/derivations/D-072-... — the finding (broken (6.12) stepping).\n3. research/paper/lagarias-li-coefficients-automorphic-0404394.pdf, pages 27–29 — the actual passage (§6, proof of Thm 6.1).\n4. research/derivations/D-071-... — the honest-Bn design + 8-step protocol (constants machinery this repair feeds).\n5. research/numerics/scripts/E-031 script + data — the envelope evidence.\n\nBackground in one paragraph: ...\n\nMission: Produce a complete, rigorous, self-contained REPAIRED PROOF of exactly one lemma: the bound on the horizontal segments (II) and (IV) in Lagarias's Thm 6.1 proof, including the right part [3, σ₁] where the printed stepping fails. Deliver it as a drop-in replacement passage for the (6.12) paragraph + the final specialization line, at T = c√n + ε, c > 1/√(log 2), with EXPLICIT constants (no un-traced O's), in a form that (a) can be appended to the Lagarias letter as \"the repair,\" and (b) gives the main line (ii′-6-c) the exact statement to formalize.\n\nExplicitly OUT of scope: re-proving any other part of Thm 6.1 (the contour, the residue identity, the verticals, the left part [−3,3], Lemma 6.1, (6.9)/(6.10) are accepted as-is — audit them only for T-independence); the T = √n salvage unless Phase 0 says otherwise; edits to frozen files.\n\nWork plan:\n- Phase 0 (diagnostic, ~cheap): E-032 — compute the TRUE complex horizontal integral I_II at T = √n zero-avoiding for n = 10³, 10⁴ (10⁵ if cheap) by adaptive quadrature (mpmath dps 40; integrand smooth, phase O(1) rad/unit), compare with the E-031 envelope. Deliverable: verdict — Re(I_II) ~ envelope (per-side claim FALSE at T=√n) vs |I_II| ≪ envelope (hidden cancellation). This decides whether anything at T = √n remains to salvage and sharpens the letter.\n- Phase 1 (the repair proof): write the corrected proof. Steps: (i) dependency audit — list every ingredient of the Thm 6.1 horizontal argument and mark T-dependence; (ii) the corrected stepping lemma: for T = c√n, per-step product ratio ≤ e^{1/c²}/2 =: q < 1, hence from σ = 3 the product ≤ (e⁴+1)·|ζ′/ζ(2)|·q^{σ−3}... produce explicit right-tail integral bound ∫₃^{σ₁} ≤ C(c) = (e⁴+1)|L'/L(2)|·q/(1−q)... and the total horizontal bound with explicit constants; (iii) choose recommended c for the main line (2 for margin, 1.5 for range — quantify both, recommend one with reasoning); (iv) consequence audit: exact downstream effect on Thm 6.1's conclusion (T = c√n + ε) and Thm 1.1's height (factor 1/c on the implicit √· — check the actual text); (v) cross-check everything numerically (extend E-031-style validation to the new constants, e.g. E-032b or E-033).\n- Phase 2 (write-up + handoff): file D-073 (the repaired proof, honesty-tagged, with the dependency audit and consequence audit), update the finding-note package if the user wants (do NOT modify the letter without asking — it's user's correspondence), register in research.db + index.md, journal + STATE update per §8.4, git commit.\n\nConstraints:\n- Freeze protocol (user directive 2026-08-21): no edits to research/paper/paperA.tex, paperB.tex, draft.tex, references.bib; no edits to existing Lean files. Work only in new files: research/derivations/D-073-*.md, research/numerics/scripts/E-032*.py (+ data sidecars).\n- Every claim honesty-tagged [PROVED]/[NUMERICAL-VALIDATED]/[HEURISTIC].\n- Numerics: deterministic scripts, CSV+JSON sidecars, cached; mpmath dps ≥ 40 for the quadrature; validate ζ′/ζ against the Dirichlet series where re > 2.\n- The main ii′-6 line continues independently; this thread's D-073 is an INPUT to it, not a branch of it. Don't start Lean formalization — deliver the statement + constants.\n- Session protocol per NGUYEN.md §8: read STATE.md first, end with journal + db + STATE NEXT ACTION + commit.\n\nSuccess criteria:\n1. D-073 contains a complete repaired proof of the horizontal bound with explicit constants and a full dependency audit showing nothing else in Thm 6.1's proof is affected.\n2. E-032 verdict on the T = √n per-side question.\n3. A recommended c with quantified tradeoffs for the main line.\n4. An updated one-paragraph \"repair summary\" suitable for the Lagarias letter (as a suggestion; not applied without user approval).\n---\n\nAlso answer the user's question directly before giving the prompt: \"Can we do it without doing the Lagarias complete research over? Can we just focus on the issue and re-do that?\" — Yes: the gap is fully localized to one paragraph of the proof (the right-tail stepping); the rest of Thm 6.1 is unaffected and the thread only audits rather than re-proves it. The prompt is scoped exactly that way.\n\nOne more consideration: should the new thread be told about the three-scope analysis from my previous message (the circularity argument, why T=√n salvage is hard)? Yes — a compact version, so the thread doesn't burn time re-attempting the salvage naively. Include the \"why hard\" summary in the prompt's background.\n\nAlso the prompt should mention the key structural facts the thread can rely on (so it doesn't re-derive): kernel ratio exact form |1−1/(s+1)²|ⁿ, sharpness of both bounds, phase O(1), the conjugation identity I_II+I_IV = 2i Im(I_II), Re(I_II) unconstrained by the identity. These are the load-bearing insights.\n\nShould I use the AskUserQuestion tool for the c choice? No — the prompt itself defers the recommendation to the thread (Phase 1 (iii)); the user decides at the end. But maybe I should ask the user one thing: do they want E-032 included, or the proof only? They said \"focus on the issue and re-do that\" — E-032 IS part of the issue diagnosis and I proposed it and they didn't object. I'll include it as Phase 0 but note they can trim it. Actually, keep it simple: include it; mention in my reply that Phase 0 is optional if they want proof-only.\n\nAlso: file numbering. Next derivation number: D-073 (D-072 is the finding). Next experiment: E-032. Good.\n\nLength of the prompt: needs to be self-contained but not bloated. Aim ~600-900 words for the prompt block. Use a markdown code block so it's copy-pasteable? A code block is the cleanest for \"copy this prompt.\" I'll put the prompt in a fenced block.\n\nLet me also make sure the prompt includes the exact archive paths (Windows-style? The repo uses forward slashes in tools; relative paths fine).\n\nOne more thing: the user said \"separate, self-contained thread\" — meaning a new conversation. The new conversation will get NGUYEN.md in context automatically (project instructions) and memory. So the prompt doesn't need to re-explain the whole project — just point to files. Good.\n\nDraft the prompt now.\n\nAlso double-check the exact claim about c threshold: kernel ratio ≤ (1+1/T²)ⁿ ≤ e^{n/T²} = e^{1/c²} at T = c√n. Need e^{1/c²} < 2 ⟺ 1/c² < log 2 ⟺ c > 1/√(log 2) ≈ 1.20112. Correct.\n\nExplicit right-tail bound shape: at σ ≥ 3 on the horizontal: |k_n(s)| ≤ |k_n(s)+1| + 1 = |1+1/s|ⁿ + 1. Hmm — careful: the stepping bounds |k_n(s)+1| = |1+1/s|ⁿ. Product with L'/L: |1+1/s|ⁿ·|L'/L(s+1)|. Per-step ratio of |1+1/s|ⁿ is ≤ e^{1/c²}; per-step ratio of |L'/L| is ≤ 1/2 (from (6.10): |L'/L(σ+1)| ≤ 2^{-(σ-2)}|L'/L(2)| — going one unit right halves). So product ratio ≤ e^{1/c²}/2 = q < 1. Initial value at σ = 3: |1+1/s|ⁿ ≤ (1 + (σ+1)/(σ²+T²))ⁿ ≤ e^{n(σ+1)/(σ²+T²)} = e^{4c²/(9c²+1)}·... hmm wait with T² = c²n: n(3+1)/(9+c²n) = 4n/(9+c²n) ≤ 4/c². So kernel at σ=3 ≤ e^{4/c²}. |L'/L(4+iT)| ≤ 2|L'/L(2)| — hmm (6.10) at σ=3: |L'/L(s+1)| ≤ 2^{-(3-2)}|L'/L(2)| = ½·|L'/L(2)|. Whatever — the thread will do the bookkeeping; the prompt just needs to specify \"explicit constants, no untraced O's\" and point to D-071's existing constant machinery (kernel charge e^{4/c²}+1, the (6.10) constant C₀ = |ζ′/ζ(2)|, etc.).\n\nNote: (6.10) as printed bounds |L'/L(s,π)| ≤ 2^{-(σ-2)}|L'/L(2,π)| for σ > 2 — applied with s+1 in the integrand. For ζ: |ζ′/ζ(2)| = Σ Λ(k)/k² ≈ 0.5699... known. Fine.\n\nAlso, one subtlety to flag in the prompt: the printed (6.10) uses the FULL Dirichlet series bound |L'/L|(σ) ≤ 2^{-(σ-2)}|L'/L|(2) — valid since all terms m^{-σ} ≤ 2^{-(σ-2)}m^{-2}·... yes m ≥ 2. Sharp at m=2. Fine.\n\nAnd for the left part [−3, 3]: |k_n| ≤ e^{4/c²}+1? At T² = c²n: (1+4/(16+c²n))ⁿ ≤ e^{4n/(16+c²n)} ≤ e^{4/c²}. With L'/L = O((log T)²) via Lemma 6.1 (zero-avoiding) — the left part stays O((log n)²)·length 6·(e^{4/c²}+1) — that part was never broken. The dependency audit confirms.\n\nThe right vertical: |k_n| ≤ e^{√n/(2c)}+1... at σ₁ = 2√n, T = c√n: |1+1/s|ⁿ ≤ e^{n(σ₁+1)/(σ₁²+T²)} = e^{n(2√n+1)/(4n+c²n)} ≈ e^{2√n/(4+c²)} — times |L'/L| ≤ 2^{-(2√n-2)}|L'/L(2)| — still o(1). Fine, unaffected (actually improved with c>1? kernel exponent slightly smaller — no wait, at c=1 it was e^{2√n/5}; at c=2: e^{2√n/8} = e^{√n/4} — smaller. fine).\n\nOK. Also include in the prompt: \"do not attempt to contact anyone or send anything; the letter is the user's to send.\"\n\nAnd per session protocol: the new thread will read STATE.md first anyway (NGUYEN.md §8.3). The prompt should say: this thread is a side-thread; the main line (ii′-6) is parked at the F-D decision; this thread RESOLVES the decision's technical content.\n\nActually, one important framing: this thread's output D-073 + E-032 + recommended c IS the input the main line needs to execute repair (R) (the 8-step protocol re-run). So the prompt should state that relationship: \"The main line (Phase 5B ii′-6-c) is parked awaiting exactly this thread's deliverables.\"\n\nNow write my reply: short answer to their question (yes, fully localized — explain in 2-3 sentences why: the broken object is one paragraph; everything else audits clean; the repair touches only the final specialization), then the prompt in a code block, then 2-3 usage notes (how to use it, what's optional).\n\nKeep my own text concise; the prompt block can be thorough.","signature":""},{"type":"text","text":"Yes to both questions. The gap is fully localized: it's exactly **one paragraph** of the proof — the right-tail stepping after (6.12) — plus the one-line final specialization \"T = √n + ε\". Everything else in Thm 6.1 (the contour, the residue identity, both verticals, the left part [−3,3], Lemma 6.1, (6.9)/(6.10)) is untouched and only needs a *dependency audit* (confirming each piece is T-generic), not a re-proof. So the thread can be scoped to: diagnose the broken paragraph precisely, re-prove just that paragraph correctly with explicit constants, and audit the rest. Here is the prompt.\n\n```\nTHREAD MISSION — Focused re-proof of the right part of the horizontal\nsegments in Lagarias's Theorem 6.1 (Finding F-D follow-up, side thread)\n\nYou are working in D:\\Python\\Quantum (NGUYEN-RH-1). This is a\nSEPARATE, 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\nREAD FIRST (in order):\n1. STATE.md — resume point (session 038 endpoint, Finding F-D).\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\nBACKGROUND (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\nMISSION DELIVERABLES:\n(D1) E-032 DIAGNOSTIC (cheap, decisive): compute the TRUE complex\n     horizontal integral I_II at T = √n (zero-avoiding) for\n     n = 10³, 10⁴ (10⁵ if cheap) by direct quadrature (mpmath,\n     dps ≥ 40; integrand smooth, phase O(1) rad/unit; validate ζ′/ζ\n     against the Dirichlet series where re > 2). Compare against the\n     E-031 envelope. Verdict: Re(I_II) ~ envelope ⟹ the per-side\n     claim is FALSE at T = √n (only the pair survives); |I_II| ≪\n     envelope ⟹ hidden cancellation exists. Either way, record the\n     numbers; this sharpens the finding note.\n(D2) D-073 — THE REPAIRED PROOF, as a drop-in replacement for the\n     (6.12) paragraph + final specialization line:\n     (i) dependency audit: every ingredient of Thm 6.1's proof listed\n         and marked T-generic / T-dependent (the contour, residue\n         identity, (6.9)/(6.10), Lemma 6.1, left part [−3,3], both\n         verticals) — confirm nothing else is affected;\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\n          series from σ = 3, explicit right-tail bound ∫₃^{σ₁}, total\n          horizontal bound |I_II| + |I_IV| with EXPLICIT constants\n          (no untraced O's — use D-071's machinery: kernel charge\n          e^{4/c²}+1, |ζ′/ζ(2)|, the zero-avoiding (log T)² bound);\n     (iii) recommended c for the main line, quantified: c = 2 (margin\n           comfort, N₁ → (T₀/2)² ≈ 2.25×10²⁴) vs c = 1.5 (range,\n           N₁ ≈ 4×10²⁴); one clear recommendation with reasoning;\n     (iv) consequence audit: the exact downstream effect on Thm 6.1's\n          conclusion (T = c√n + ε) and on Thm 1.1's height (check the\n          paper's actual endgame text for where √-scaling enters);\n     (v) numerical cross-validation of every new constant\n         (E-033-style script + CSV/JSON sidecars).\n(D3) A one-paragraph \"repair summary\" suitable for the Lagarias\n     letter (draft only — do NOT edit the letter or any correspondence\n     file; the user owns those).\n(D4) A one-paragraph verdict statement on the T = √n per-side\n     question, combining E-032 with the circularity analysis, for the\n     finding note (draft only, same rule).\n\nHARD 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 back to the user before going further); Lean work;\n  contacting 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:\n  research/derivations/D-073-*.md, research/numerics/scripts/E-032*.py\n  / E-033*.py + data sidecars.\n- Honesty tag on every claim ([PROVED] / [NUMERICAL-VALIDATED] /\n  [HEURISTIC]); numerics deterministic + cached with CSV/JSON\n  sidecars, registered in research.db via db.py.\n- Session protocol per NGUYEN.md §8: STATE.md read first; journal,\n  index updates, db registration, STATE.md NEXT ACTION, and a git\n  commit at the end of each sub-stage. Sub-stage rule applies (§8.5):\n  land D1 first as its own committed sub-stage.\n\nSUCCESS CRITERIA:\n1. D-073 contains a complete, rigorous repaired proof of the\n   horizontal bound with explicit constants, plus the dependency and\n   consequence audits, all honesty-tagged.\n2. E-032's verdict is recorded with numbers.\n3. The main line can execute repair (R) from D-073 + the recommended\n   c without any further analysis.\n```\n\nUsage notes:\n\n- **Why this is enough**: the thread re-proves one paragraph and audits the rest — it never touches Lagarias's contour setup, residue identity, or vertical bounds, because those were never broken. The dependency audit (D2-i) is what *certifies* that \"just the issue\" suffices.\n- **D1 is optional but recommended** — it's the cheap experiment that tells us whether the printed per-side claim is actually false (not just unproven), which changes the letter's framing. If you want proof-only, delete D1 and D4.\n- **When it's done**: the main line takes D-073 + the recommended c and executes the 8-step protocol re-run (convexity re-proof, base-point re-certification), then resumes ii′-6-c — no further analysis needed.\n- If E-032 surprises us (|I_II| small at T = √n), the prompt tells the thread to stop and come back to you before chasing the hidden-cancellation question.\n\nWant me to adjust the scope (e.g., drop D1, or pin c = 2 now instead of asking the thread to recommend)?"}]