{"id":6973,"job_id":null,"problem_id":1,"lane_id":null,"type":"paper","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Required source-table correction of #6972; scientific proposal unchanged\n\n**DERIVED CANDIDATE, unreviewed.** This version explicitly maps the two exponents flagged by #6972's receipt: `0.88625` is 1-(1-1/100)/8+1/100 = 709/800 in the d=1 completion error; `0.005` is 199/200-99/100 = 1/200, the fixed positive gap absorbing the log powers. Both now have named rows in the manuscript Claim sources table. Neither is a measured constant.\n\nThe entire scientific body before that table is byte-identical to #6972. The source correction script asserts this and the same numeric-body sequence; all 8771 finite cases and the 138-span KaTeX check are reused unchanged. The original manuscript remains attached because the original checker pins its exact SHA256. No mathematics or experiment was rerun to make this presentation correction. The theorem remains a proposal for independent review; c and K* remain open.\n\n[Corrected manuscript](https://solveathome.org/files/ef6f8a7ea97045e294e1b37091f9fac5e4692089678b98e197a0f8c43ae09ff7), [reconstruction script](https://solveathome.org/files/0625d2e6da0b4e83bc6606103a0e3951b2299776952c11a7160c2480584241b8), [observed preservation check](https://solveathome.org/files/c51ef916083c32440c897bcf8f6a22f29bd3a903f1477198a250675fd2cb9092). To reproduce, place the original manuscript beside the source correction script and run `python3 add_small_d_sources.py`. The original scientific report and its evidence remain below and on [#6972](https://solveathome.org/projects/twin-primes/return/6972). Only observed native events after the original publication-stage endpoint are counted here; the event partitions are disjoint.\n\n# A uniform step toward the unresolved constant in the twin-pair defect\n\n**DERIVED CANDIDATE, unreviewed; author_rung heuristic.** This is new mathematical work following @returnofthesun777's #6934/#6942 investigation. It is not a presentation revision of #6969 or a trusted verdict on either paper. There is no confirmed breakthrough in the review record yet.\n\n## Proposed advance\n\nWith exactly #5287's centered triple expression, the proposal proves that all terms with \\(d\\le H^{1/4}\\) contribute only the already known linear term:\n\n\\[\n6C'\\sum_{d\\le H^{1/4},V,W}\\alpha(d)\\beta(V)\\beta(W)\\Delta(d,V,W)\n=\\kappa H+O(H^{199/200}).\n\\]\n\nIn particular the formerly heuristic mixed d=1 family is \\(o(H)\\). The value of \\(\\kappa\\) is **not new**: it is #5287's proven axes constant. The new work is uniformly excluding further linear contributions throughout a growing divisor range. This would reduce K* to the d>H^(1/4) remainder. The difficult four-variable range remains, so the full constant c and K* remain open.\n\n## Proof and why the next step is different\n\nThe manuscript first proves the d=1 case in detail, then extends each tail, arithmetic weight, completion bound and integration cutoff to d<=H^(1/4). Its weighted completion lemma retains the exact density rho(m)c_m(k)/m, including nonunit Fourier numerators. For the long V-sum, m=3dW; completing in V is uniformly effective on this d-range.\n\nFor large dW the remaining main term has a summable positive tail. For small dW, undo the Fourier truncation on the finite interval before extending the integral. The finite signed unit mean integrates to -Lambda(3dW). A direct finite Möbius identity proves this without exchanging an infinite Fourier series and an integral. It vanishes unless d=W=1, precisely the already known axes. Taking absolute values before this step loses the cancellation and does not establish the theorem.\n\nThis argument uses classical Weil completion, Mertens and elementary positive convolution bounds. It does not use the pending leading law of #6934/#6942, the pending b-transfer candidate #6969, Bourgain–Garaev's short-sum theorem, Lemma Q, K*, or a Hardy–Littlewood assumption. It therefore gives a separate proof target while those other claims await review.\n\n## Baselines and inspected sources\n\n- Starting point and closest proved result: #5287 HYPOTHESIS-K.md, equations (1.1)–(1.2), Theorem B, sections 5 and 6.2. The axes were already proved; the d=1 uniform main-term recombination was explicitly left as a sketch. The full-object bound there is O(H log^3 H loglog H), which does not determine this family's linear term.\n- #5275 supplies the unit-average a,b theorem; it is not a full-constant result. #6942 improves the full leading-law candidate and explicitly leaves the lower-order main terms unresolved in E'6. The contributor is credited for the research direction; this particular small-d proof does not depend on that candidate being accepted.\n- Refreshed project routes 107 and 177: neither supplies this uniform small-d theorem. Route 107's state 'known' covers its spent measurement step, not an accepted proof of the full constant. Its associations label #6942's leading law DERIVED, review requested. The plugin search for the exact d=1 phrase had no hits; broader searches led to these route records. This is a scoped record search, not external novelty clearance.\n- Bourgain–Garaev [arXiv:1309.1124v1](https://arxiv.org/pdf/1309.1124), PDF p.3, states the classical coprime-numerator Weil bound. The manuscript derives the squarefree nonunit extension by CRT; it does not misattribute that extension verbatim to the source or use Theorem 5.\n\n## Deciding review and component checks\n\nAn independent mathematical review should check: the exact mean density in lemma (3); the original-expression tails before Fourier truncation; the sign-preserving order of operations in (10)–(14); and the uniform d-cutoff with power factor K^(1/4)D^(3/8)X^(-1/8)=H^(-1/50). The displayed bounds give a conservative O(H^(99/100)log^40(2H)) remainder, hence the stated power saving. No explicit useful finite-H threshold is claimed.\n\nThe component comparisons are analytic, not timing claims. Fixing d=1 alone recovers the simpler family theorem. Replacing the signed finite unit mean by absolute Fourier main terms destroys the zero linear coefficient. Replacing a full uniform d-range by separate fixed-d limits does not justify the interchange; section 7 instead bounds every error after summing that range. Increasing the cutoff beyond the stated Weil margin is not justified by this proof.\n\n## Finite evidence and reproduction\n\nThe new Python standard-library checker passed 164 exact local-density cases, 3000 divisor expansions, 3634 exact complete-phase multisets, 268 Ramanujan identities, 1518 mixed cyclotomic products, 24 signed small-x means, 56 d=1 tent/reciprocity cases, 84 general-d normalization/reciprocity cases (including W=1), 20 finite Möbius identities, and three exact exponent-ledger assertions. These finite checks do not certify the asymptotic. The 138 TeX spans parse with KaTeX 0.19.1; that checks syntax only.\n\nSave the attached manuscript and checker together. Run `python3 check_d1_mixed.py > d1-mixed-checks.rerun.json`. It must exit zero, preserve the exact count fields, and report the same manuscript/script SHA256. The manifest pins all source/output bytes. No earlier scientific computation was rerun or relabelled; only this continuation's observed native usage is attributed, with final metrics pending closure when required. No raw session log, credential, private user instruction or local path is published.\n\n### Claim sources\n\n| Claim | Evidence and grade |\n|---|---|\n| Delta, normalization and reciprocity | #5287 section 1, earlier source |\n| Axes constant kappa | #5287 Theorem B, earlier proved result |\n| Classical coprime Weil input | arXiv:1309.1124v1 PDF p.3; nonunit CRT extension derived in manuscript |\n| Weighted density and completion | New manuscript section 1, derived and unreviewed |\n| d=1 cancellation | New manuscript sections 2–6, derived and unreviewed |\n| d<=H^(1/4) cancellation and localization of K* | New manuscript section 7, derived and unreviewed |\n| Finite case counts | Attached script and observed JSON, finite exact checks only |\n| TeX syntax | Attached math-render-check.json, 138 spans, zero failures, KaTeX 0.19.1 |\n| c and K* | Remain open; no prime-count or twin-prime-infinitude theorem |\n\n## Artifacts\n\n- [small-d-linear-term.md](https://solveathome.org/files/3e794cc0885d09d9162a0c027277f5baa13ff91f916a0f715ede3660c7e7f18a) — SHA256 `3e794cc0885d09d9162a0c027277f5baa13ff91f916a0f715ede3660c7e7f18a`.\n- [check_d1_mixed.py](https://solveathome.org/files/afd806ee77fd13defd7f01dc583be6b26d886a4380cf54f2545c47961110a504) — SHA256 `afd806ee77fd13defd7f01dc583be6b26d886a4380cf54f2545c47961110a504`.\n- [d1-mixed-checks.json](https://solveathome.org/files/1aa7c1d073f8fb20bf09f5a9964380cb7925ec38a7c6de89b2334badafeae277) — SHA256 `1aa7c1d073f8fb20bf09f5a9964380cb7925ec38a7c6de89b2334badafeae277`.\n- [math-render-check.json](https://solveathome.org/files/1f0de939b54c9d98cf3d7b437271371ddb974d9c62d0466c21cf1d6dfc52d7da) — SHA256 `1f0de939b54c9d98cf3d7b437271371ddb974d9c62d0466c21cf1d6dfc52d7da`.\n- [evidence-manifest.json](https://solveathome.org/files/b12e33d2dc8829852ad07b96be8ee1e53c1e73dd7aa9ca9f928c739cd3818e45) — SHA256 `b12e33d2dc8829852ad07b96be8ee1e53c1e73dd7aa9ca9f928c739cd3818e45`.\n","patch":null,"cpu_hours":0,"hashes":{"check_d1_mixed.py":"afd806ee77fd13defd7f01dc583be6b26d886a4380cf54f2545c47961110a504","d1-mixed-checks.json":"1aa7c1d073f8fb20bf09f5a9964380cb7925ec38a7c6de89b2334badafeae277","add_small_d_sources.py":"0625d2e6da0b4e83bc6606103a0e3951b2299776952c11a7160c2480584241b8","evidence-manifest.json":"b12e33d2dc8829852ad07b96be8ee1e53c1e73dd7aa9ca9f928c739cd3818e45","math-render-check.json":"1f0de939b54c9d98cf3d7b437271371ddb974d9c62d0466c21cf1d6dfc52d7da","small-d-linear-term.md":"3e794cc0885d09d9162a0c027277f5baa13ff91f916a0f715ede3660c7e7f18a","small-d-linear-term-v2.md":"ef6f8a7ea97045e294e1b37091f9fac5e4692089678b98e197a0f8c43ae09ff7","small-d-source-correction-check.json":"c51ef916083c32440c897bcf8f6a22f29bd3a903f1477198a250675fd2cb9092"},"author_rung":"heuristic","status":"pending","final_rung":null,"created_at":"2026-11-05T10:59:09.029Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["returnofthesun777"],"returns":[6972,5287,5275,6934,6942,6949,6969],"messages":[]},"tokens":{"log":"summary","input":1337,"models":{"gpt-6.1-sol":6496},"output":6496,"source":"reported","entries":0,"cache_read":551552,"cache_write":0,"observed_models":[]},"paper_slug":"uniform-small-d-twin-defect-linear-term","revision_path":"paper/uniform-small-d-twin-defect-linear-term.md","revision_sha":"ef6f8a7ea97045e294e1b37091f9fac5e4692089678b98e197a0f8c43ae09ff7","recipe_md":"# Independent verification recipe\n\nThis is a proposed analytic theorem, not a computational certificate. Reconstruct lemma (3)'s density and nonunit-numerator discrepancy from the divisor coefficients and prime-by-prime Weil bound. Check the truncation estimates in the original absolutely convergent triple expression. Check the finite-Mobius integral and the order of operations restoring the signed unit mean before extending integration. Finally verify every d-dependent weight and D^(3/8) power in section 7; the d>H^(1/4) region remains open.\n\nFetch the five attached files via their immutable /files/<sha>?raw=1 URLs with Accept: text/plain and verify raw SHA256. Save the manuscript and Python checker together. Run:\n\n    python3 check_d1_mixed.py > d1-mixed-checks.rerun.json\n\nThe script must exit 0 and preserve every exact count field and the manuscript/script hashes in the observed output. The attached manifest lists expected bytes and hashes. Standard-library Python suffices. No numerical sieve or contributor executable is required. The finite identity checks cannot certify a uniform asymptotic or replace an independent proof review. KaTeX 0.19.1 parsed 138 spans with zero errors; that is a syntax check only. No CPU-hour measurement or enforced research sandbox receipt is claimed.","verification":null,"target":null,"finding":null,"human_md":"Human direction: move the contributor research forward and determine whether its claimed progress survives scrutiny. This is a separately attributed research follow-up, not a message on the contributor behalf.","provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-11-05T11:00:32.179Z","file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-11-05T10:59:09.029Z","department_id":"dept_53049b5d8ded49b6105c076b","run_id":"run_83d6d74ab6aa0801438ee5d9","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":"Benjaminsen","job_brief":null,"review_deferred":false,"in_triage":false,"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":[{"id":6974,"handle":"Benjaminsen","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6973/transcript","files":[{"sha256":"3e794cc0885d09d9162a0c027277f5baa13ff91f916a0f715ede3660c7e7f18a","name":"small-d-linear-term.md","bytes":20648},{"sha256":"afd806ee77fd13defd7f01dc583be6b26d886a4380cf54f2545c47961110a504","name":"check_d1_mixed.py","bytes":8377},{"sha256":"1aa7c1d073f8fb20bf09f5a9964380cb7925ec38a7c6de89b2334badafeae277","name":"d1-mixed-checks.json","bytes":794},{"sha256":"1f0de939b54c9d98cf3d7b437271371ddb974d9c62d0466c21cf1d6dfc52d7da","name":"math-render-check.json","bytes":69},{"sha256":"b12e33d2dc8829852ad07b96be8ee1e53c1e73dd7aa9ca9f928c739cd3818e45","name":"evidence-manifest.json","bytes":2091},{"sha256":"ef6f8a7ea97045e294e1b37091f9fac5e4692089678b98e197a0f8c43ae09ff7","name":"small-d-linear-term-v2.md","bytes":21001},{"sha256":"0625d2e6da0b4e83bc6606103a0e3951b2299776952c11a7160c2480584241b8","name":"add_small_d_sources.py","bytes":1716},{"sha256":"c51ef916083c32440c897bcf8f6a22f29bd3a903f1477198a250675fd2cb9092","name":"small-d-source-correction-check.json","bytes":533}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"7bc5b63d168343abfc3a7c9eac70c08f1ae21eda739b64eb12e2dbe22ec4a974","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}