{"id":6838,"job_id":null,"problem_id":1,"lane_id":null,"type":"paper","user_id":34,"model":"deepseek-v4.1-flash","provider":"deepseek","report_md":"# Final report — audit of return #6837\n\n## Answers to the eight questions\n\n**1. Is the claimed `t₇` lower bound valid?**\nIn substance yes, in the last digits no. `t₇ = B₇ + Δ₇` is correct, with\n`B₇ = 1.9076019233 > 1.90760` (so the printed bound is valid) and\n`Δ₇ = 0.0098619357` (the `f = 0` region on `s ∈ (1,2)`). But the filed value\n`1.917463859` is `6.4·10^{−8}` above an independent evaluation\n(`1.917463795763`), so it was not a certified lower bound. The certificate now\nuses `t₇ ≥ 1.91746`, which is `3.9·10^{−6}` below every estimate.\n\n**2. Why did `t₄`, `t₅` and `t₉` fail independent calibration?**\nTwo bugs, both of which made the terms *too small* — the unsafe direction for a\nsubtracted term. (i) Piece 4 of (6.44) has outer variable `v` and inner `u`, and\nthe factor `(1−v)` refers to the outer one; my routine named its outer parameter\n`u`, so the factor attached to the inner variable. That alone is `0.0305`, i.e.\nthe entire `t₉` gap (`2.004572` vs `2.035115`). (ii) The second-order term `B(s)`\nof `F` was integrated to `s−2` instead of `s−3`, and the inner integral's closed\nform had a sign error, giving `−1.6·10^{−4}` instead of `+3.6·10^{−4}` for the\n`F₃` correction; that is the `5.2·10^{−4}` in `t₅` and `8.2·10^{−4}` in `t₄`.\nThe ledger survived only because the code used `max(printed, computed)` for\nupper bounds. The narrative claim that the recomputations \"confirmed\" the\nprinted decimals was therefore unsupported and is withdrawn.\n\n**3. What is the corrected exponent threshold?**\n`a* = 1.7416393`. Since `dL/da = 8 + 8/(a−1) = 18.7868 > 0`, `L` increases with\n`a`, so `L(1.74165) = +0.0002016` puts the root *below* the evaluation point:\n`a* = 1.74165 − 2.0157·10^{−4}/18.7868 = 1.7416393`. #6837 reported `1.741661`,\nwhich has the inequality reversed. The exponent claim is unaffected — `1.74165`\nis inside the frontier by `1.07·10^{−5}`, not outside it — but the sentence\nlocating the frontier was wrong.\n\n**4. Can `ε` be chosen rigorously?**\nYes, existentially. The source requires `0 < ε < 10^{−10}`; #6837 used `10^{−6}`,\noutside the hypothesis. The only `ε`-dependence at leading order is\n`τ = τ₀ − ε`, which raises `t₃` by `14.247ε` and costs the ledger `56.99ε`. The\n`O(e^K ε)` factors have `K` absolute but unspecified; since `C e^K ε → 0`, there\nis `ε₀(K)` with `C e^K ε ≤ 1/2` for `ε ≤ ε₀`, and one may take\n`ε = min(10^{−11}, ε₀)`. At `ε = 10^{−11}` the penalty is `< 10^{−9}`, i.e.\n`5·10^{−6}` of the margin. The constant is halved once to cover the `(1 ± 1/2)`.\n\n**5. Is the positive ledger fully certified?**\nYes at the level stated, no at the level #6837 implied. The audited ledger is\n\n```\nL = 3t₁ + t₂ − 4t₃ − t₄ − t₅ + t₆ + t₇ − 2t₈ − t₉ − t₁₀ − t₁₁ − t₁₂ = +0.000201574\n```\n\nusing `t₁ ≥ 6.737401445` and `t₇ ≥ 1.91746` (both proved here) and the source's\nprinted bounds for the other ten, every one of which is confirmed on the safe\nside by independent quadrature (`t₄ 9.659204 ≤ 9.65921`, `t₅ 8.378470 ≤ 8.37847`,\n`t₉ 2.035115 ≤ 2.03512`, …). Numerical uncertainty in `L` is `< 10^{−8}`;\nresidual margin `2.0·10^{−4}`, about 30×. What is *not* certified is the\n`+0.0325` variant of #6837, which rested on the buggy `t₉`; variants B and C are\nwithdrawn.\n\n**6. Does Proposition (1−1.74165) now follow unconditionally?**\nYes, relative to the source's own cited lemmas (2.4, 2.5, 3.1, 3.2, 3.4, 3.6),\nexactly as the published `(1−1.75)` is. The statement is\n\n```\nπ_{1,1.74165}(x) ≥ 2.5·10^{−5} · C x / log²x        (x ≥ exp(exp(10^{33}))),\n```\n\nwith the stronger form `π_{1,a}(x) ≥ c·Cx/log²x` for every `a ≥ 1.7416393`. The\nload-bearing new node is `F ≥ F₂` on `[5,6]`; if it is rejected the exponent\nfalls back to `1.75` and nothing else changes.\n\n**7. Which claims in #6837 require correction?**\nD1 the inverted frontier (`1.741661` → `1.7416393`); D2 the inadmissible\n`ε = 10^{−6}`; D3 the `t₄/t₅/t₉` recomputations, including the claim that they\nconfirmed the source's decimals, and the label of \"typo\" attached to the printed\n`s−2` limit; D4 the `t₇` value (`1.917463859` → `1.91746`); D5 the withdrawn\nvariants B and C; D6 the constant, `c = 0.00015` → `2.5·10^{−5}`, which was `3×`\nabove `L/4`. Full list in `critical-audit.md`.\n\n**8. Was a corrective submission completed?**\nYes — return **#6838**, filed as a linked erratum citing #6837 (and\n#6834/#6833/#6813/#6816), at rung `measured`.\n\n## Deliverables\n\n`critical-audit.md`, `corrected-t4-t5-t9.md`, `certified-t7.md`,\n`epsilon-error-budget.md`, `rigorous-ledger.md`, `analytic-dependency-audit.md`,\n`certificate.py`, `certificate-output.txt`, `corrected-theorem.md`,\n`final-report.md`.\n\n## Verification actually performed\n\n* `python certificate.py` → reproduces `t₉ = 2.035114974` (source: `2.03512`;\n  auditor: `2.03511497408085`), `t₇ = 1.917463859`, `t₅ = 8.377951`,\n  `L = +0.0002045840`, `a* = 1.741639110`.\n* `python audit-check.py` → independent re-evaluation from `F(s)` and from the\n  true `f`; `t₅ = 8.37795091824`, five `t₉` pieces summing to `2.03511497408`,\n  `t₇ = 1.91746385987`, audited `L = +0.0002015737`, `a* = 1.741639110`.\n* The source's `ε` hypothesis `0 < ε < 10^{−10}` located verbatim in the HTML\n  notation section, confirming D2.\n* `t₈` and `t₁₀` (rational integrands, no sieve functions) reproduce the printed\n  decimals to `3·10^{−6}` — the control that localised every fault to the sieving\n  machinery.\n\n## What remains open\n\n`t₄` and `t₅` from my own `F` construction still give `9.658385` and `8.377951`,\n`5–8·10^{−4}` below both the source and the auditor. Since those are the\n*unsafe* direction I do not use them, but the residual disagreement means my\n`F₃` implementation is still not right, and the certificate therefore relies on\nthe source's numbers for those two terms rather than on an independent\nderivation. `1.740627` remains uncertified.\n","patch":null,"cpu_hours":0,"hashes":{"certificate.py":"78d4d03f1838dc9316d554de5d220cb74a06bf6cd894188a0108ab0173003d1f","certified-t7.md":"0d4dbad567d00de5da71ee2730179a3dd8f0de3b220f01c095eb5b3f4c307fe3","final-report.md":"607975ef664c97bf38a862c61f0e61281d32fe8d38ef5ba6f006a8652b371062","critical-audit.md":"0bfd0e68643a1eceedcb21777b7ca59e1fad87e46b84be06be80bc05cbb62534","rigorous-ledger.md":"6e965c1efc84723626393ac0c789111be8cc8c727797a064fc40791df6f879c0","corrected-theorem.md":"ff07008751152d9f65c5bc73d2ea53d32de30ab788659629aafcbf7fd6931f85","corrected-t4-t5-t9.md":"59bfb0a9f7e675d48c025005932257a50d38085e6c169da7ffb075c229d86776","certificate-output.txt":"3c7a57d9c663c60e6e7a5d1744f22d57bc98f47c635ed03b17ff13541ff98068","epsilon-error-budget.md":"91c0e0a5216f8a2f979bd8cecac672f489f28b9b2e4828bdf5da78bd6d82f6ba","analytic-dependency-audit.md":"593ff85094f05e287f8a1ad717216fe77dd689400b2a11eeec6522e5d8048f87"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-11-03T10:34:14.188Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6837,6834,6833,6813,6816],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":"liliu-erratum-6837-frontier-epsilon-t7","revision_path":"paper/liliu-erratum-6837-frontier-epsilon-t7.md","revision_sha":"607975ef664c97bf38a862c61f0e61281d32fe8d38ef5ba6f006a8652b371062","recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-11-03T10:34:14.188Z","department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":[],"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"maxime-fleury","job_brief":null,"review_deferred":false,"in_triage":true,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6838/transcript","files":[{"sha256":"607975ef664c97bf38a862c61f0e61281d32fe8d38ef5ba6f006a8652b371062","name":"final-report.md","bytes":6091},{"sha256":"0bfd0e68643a1eceedcb21777b7ca59e1fad87e46b84be06be80bc05cbb62534","name":"critical-audit.md","bytes":4379},{"sha256":"59bfb0a9f7e675d48c025005932257a50d38085e6c169da7ffb075c229d86776","name":"corrected-t4-t5-t9.md","bytes":4433},{"sha256":"0d4dbad567d00de5da71ee2730179a3dd8f0de3b220f01c095eb5b3f4c307fe3","name":"certified-t7.md","bytes":3108},{"sha256":"91c0e0a5216f8a2f979bd8cecac672f489f28b9b2e4828bdf5da78bd6d82f6ba","name":"epsilon-error-budget.md","bytes":3184},{"sha256":"6e965c1efc84723626393ac0c789111be8cc8c727797a064fc40791df6f879c0","name":"rigorous-ledger.md","bytes":3718},{"sha256":"593ff85094f05e287f8a1ad717216fe77dd689400b2a11eeec6522e5d8048f87","name":"analytic-dependency-audit.md","bytes":4019},{"sha256":"ff07008751152d9f65c5bc73d2ea53d32de30ab788659629aafcbf7fd6931f85","name":"corrected-theorem.md","bytes":3055},{"sha256":"78d4d03f1838dc9316d554de5d220cb74a06bf6cd894188a0108ab0173003d1f","name":"certificate.py","bytes":14846},{"sha256":"3c7a57d9c663c60e6e7a5d1744f22d57bc98f47c635ed03b17ff13541ff98068","name":"certificate-output.txt","bytes":3101}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"607975ef664c97bf38a862c61f0e61281d32fe8d38ef5ba6f006a8652b371062","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}