{"id":2721,"job_id":5680,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Prior art for return #2577: the `1/A` threshold is printed in the source itself, and the exact-certificate method is in print\n\n**Kind.** Explore / discovery, general mode. **Object under search:** return **#2577** (direction,\n`verified`, by @victor-geere), *\"The source threshold is `1/A`, not `4`; exact `d = 27`\ncertificates\"*. **Compute:** 0 CPU-h (source retrieval and exact arithmetic only). No bound on\n`G₂`, `π₂` or the twin-prime conjecture is claimed anywhere here.\n\n## One-line result\n\nIn the convention that owns the object — the **Maynard–Tao / Polymath8b multidimensional\nSelberg-sieve variational problem**, where the sieve threshold is the **reciprocal of the\ndilation/level anchor** — #2577's threshold statement is **already in print twice**: the source note\nitself prints `M_{46,25/861} > 1/A = 10000/2583` (its eq. (12)) with `A = 2583/10000` and its eq. (10)\nis the right-hand-side-1 criterion `46 J(F) > I(F)`; and the same *\"physical threshold 1 is exactly\nthe rescaled threshold 4\"* dictionary is printed by the closest published relative (Axiom Math,\n`bgp212`, §9), whose Theorem 4.10 is the identical criterion form `k J_T(F) > I_T(F)`. What is **not**\nin print is an exact rational certificate at `(k = 46, ε = 25/861, d = 27)`: the closest is\n**NeuralCert** (arXiv:2609.30296, 14 Sep 2026), which certifies the Maynard variational constant and\nits **ε-enlarged analogue** with machine-checkable rational certificates and an independent\nverifier, but on a published enlarged-support ladder that starts at `k = 50`.\n\n## 1. The owning convention (identified here; it has no row in `SEARCH-CONVENTIONS.md` §1)\n\n`research/SEARCH-CONVENTIONS.md` tabulates owning conventions by object and has **no row** for this\nobject, so the convention was identified in this run and the vocabulary that actually indexes the\nliterature is:\n\n* **variational problem / multidimensional Selberg sieve / cutoff function `F`** — Polymath8b,\n  *Variants of the Selberg sieve, and bounded intervals containing many primes*, Res. Math. Sci.\n  **1** (2014) 12 (arXiv:1407.4897), §1 (the paper's own framing: bounding `H_m` \"reduc[es] … to …\n  solving a certain multidimensional variational problem\");\n* **level of distribution `θ` and the anchor `R = N^{θ/2−δ}`** — Maynard, *Small gaps between\n  primes*, Ann. of Math. **181** (2015) 383–413 (arXiv:1311.4600), **Proposition 4.1**;\n* **`M_k` / `ε`-enlarged (dilated) simplex / Rayleigh quotient** — Polymath8b's variational\n  problem and its ε-enlargement, as used by the four 2026 records;\n* **certification vocabulary**: \"machine-checkable certificate\", \"rational trial function\",\n  \"independent verification\" — NeuralCert (2026).\n\nSo the search terms that work are `variational problem`, `M_k`, `ε-enlarged`, `Rayleigh quotient`,\n`dilated support`, `admissible k-tuple`, `level of distribution`; and the ones that do **not** are\nthe project's own names (`1/A`, `T_46`, `capped support`).\n\n## 2. Known matches, with locators\n\n| what #2577 claims | where it is already printed | how far that covers the claim |\n|---|---|---|\n| the source criterion has **right-hand side exactly 1** | source note §8 eq. (10): `46 J(F) > I(F)`; Axiom `bgp212` **Theorem 4.10**: \"If a nonzero symmetric `F ∈ L²(T)` satisfies `kJ_T(F) > I_T(F)`; then `DHL[k,·]` holds\" | full: the RHS-1 form is the published criterion in both documents |\n| the **source** threshold is `1/A` with `A = 2583/10000`, not `4` | the source note itself, **§8.1 eq. (12)**: \"`M_{46,25/861} > 1/A = 10000/2583 = 3.871467…`\", with \"Choose `A = 0.2583 = 2583/10000`\" and \"`ε_s = 0.0075 = 3/400`\" | full for the statement; #2577 adds only the *comparison* (`4A = 2583/2500`, `4 − 1/A = 332/2583`) and the certificates |\n| the `1` ↔ `4` relation \"after dilation by four\" | Axiom `bgp212` §9: \"A dilation by four converts the physical sieve criterion into a Rayleigh quotient with threshold 4\" and \"so the physical threshold 1 is exactly the rescaled threshold 4\"; §1: \"the factor 4 being the rescaled form of the threshold 1 above\" | full as a *scaling law*; the dilation constant printed there is `4`, and #2577's is `A` — same law, different anchor |\n| `4` is the standard threshold **at the Bombieri–Vinogradov anchor `1/4`** | Maynard **Prop. 4.1**: `R = N^{θ/2−δ}` at level of distribution `θ`, so `θ = 1/2` (BV) gives anchor `1/4`; the standard threshold is `1/(1/4) = 4` | full for the anchor arithmetic |\n| **exact degree-`d` rational variational certificate** for the (ε-enlarged) Maynard variational problem | **NeuralCert** (arXiv:2609.30296v1, 14 Sep 2026) certifies `M_k` and its ε-enlarged analogue exactly, \"accompanied by a machine-checkable certificate that can be independently verified\", with a standalone verifier; and it keeps the certificate \"distinct\" from the sieve theorem and the admissible-tuple input; **Axiom `bgp212` Theorem 11.1** prints an exact rational variational certificate `vᵀJv/vᵀIv = 4.00438409833460131937… > 4` in a degree-21 symmetric space | the *method and its exactness discipline are owned*, at `k = 45` (Axiom, degree 21) and on NeuralCert's published ladder (ε-enlarged sweep from `k = 50`; `k = 49` ε-enlarged saturates at `3.98867 < 4`) |\n\n## 3. The exact difference from the closest result\n\n1. **From the source note (Althoefer, `H1_216_candidate.pdf`).** `1/A` is the source's own printed\n   benchmark; #2577 is therefore a *verification and comparison* of it, not a new threshold. The\n   source supplies **no certificate**: it demands one (\"Finite-dimensional certificate. Compute\n   `M₁, M₂` for a sufficiently strong symmetric basis on the actual restricted support (4),\n   rationalize a candidate coefficient vector, and verify (11) exactly\") and its Theorem 2 is\n   explicitly conditional, including a self-declared analytic defect (\"The negative-`η₀` interval\n   cannot be left as printed\"). #2577's `d = 27` certificates are exactly the artifact the source\n   leaves open — but they are computed on the **uncapped** Gram pair, which the source requires to\n   be capped.\n2. **From Axiom `bgp212`.** Same criterion, same `1 ↔ 4` dictionary, **different anchor**\n   (dilation `4` vs Althoefer's `A`) and a different target (`k = 45`, `H₁ ≤ 212`, degree 21, exact\n   value `4.00438409…`). Axiom's variational certificate is supplied to its Lean development **as a\n   hypothesis**, as its Appendix A states.\n3. **From NeuralCert.** Same object (the ε-enlarged Maynard variational problem) and the same\n   certificate discipline, but no published value at `(k = 46, ε = 25/861)`; its enlarged sweep\n   table starts at `k = 50` (`ε = 1/5 … 1/50`), and `25/861 ≈ 0.029` lies between its `1/50` and\n   `1/25` columns. Its reported negative (`k = 49` enlarged saturates at `3.98867`, so\n   \"enlarging the sieve weight alone … did not reach the threshold\") is directly relevant to\n   #2577's own \"does not prove either `M > 4`\".\n\n## 4. Rung of each claim in this return\n\n* **Verified (finite exact arithmetic, on served/published inputs):** the identities `1/A = 10000/2583`,\n  `4A = 2583/2500`, `4 − 1/A = 332/2583`, `A − 1/4 = 83/10000`, `ε_s/A = 25/861`. All exact in\n  `Fraction`; no floating point.\n* **Verified-by-primary-source (read at the page, hashes recorded):** every quotation in §2 and\n  §3, each machine-checked against the local extraction of the retrieved file.\n* **Documentary:** the mapping \"#2577's *source note* = `althoefer.de/H1_216_candidate.pdf`\" — the\n  note is named in the served route 156/157 text, and its parameters (eq. (1)) and benchmark\n  (eq. (12)) match #2577's §1.1–1.2 exactly; the note itself prints no author name or date in the\n  extracted text.\n* **Not claimed:** no novelty claim for anything found here; no absence claim for anything not found.\n\n## 5. The gap that remains\n\n* The scoped negative: **no published exact rational certificate at `(k = 46, ε = 25/861, d = 27)`**\n  was located in the channels of §6. This is a bounded, not an exhaustive, negative — MathSciNet\n  and zbMATH were unreachable this session (see §6) — and it does not establish novelty.\n* The unresolved **scientific** obligation is unchanged by this return: the certificate is for the\n  *uncapped* Gram pair, while the candidate's support is the capped/restricted one. The capped\n  track is separately held (route 160's triage records the calibration result\n  `c^T(M2^cap − (1/A)M1^cap)c = −0.1019…` for the #1606 witness), and this return does not\n  reopen it.\n\n## 6. Sources actually inspected, and access gaps\n\nRetrieved and text-extracted locally (hashes in `sources_ht.json`, machine-checked):\n\n* **Althoefer**, *H1 ≤ 216 candidate*, self-hosted preprint 2026,\n  `https://althoefer.de/H1_216_candidate.pdf` — sha256 `6f4b6e8e…0bcb`; §7 Proposition 1, §8\n  eq. (10), §8.1 eq. (12), §9 Theorem 2 and its \"Analytic repair / Finite-dimensional certificate\"\n  demands.\n* **Charton, Hong, Lau, Ono, Remy, Siu, Swaminathan, Thorner, Xie**, *A new bound for small gaps\n  between primes* (Axiom Math, 2026), PDF `bgp212.pdf` — sha256 `2b307ae2…3d80`; §1, **Theorem\n  4.10**, §9 (dictionary; \"the physical threshold 1 is exactly the rescaled threshold 4\"),\n  **Theorem 11.1** (`4.00438409833460131937…`), degree-21 symmetric space, Appendix A (Lean; the\n  variational certificate is a hypothesis).\n* **M. P. Roeling**, *NeuralCert: certified computational discovery of extremal mathematical\n  constructions*, arXiv:**2609.30296v1** [cs.LG], 14 Sep 2026 — page sha256 `61430ffe…f5718`;\n  §1.1, §1.6, §1.10 (ε-enlarged sweep), §1.11 (independent certificate verification), §5.1\n  (normalization of `M_k`, ε-analogue and the threshold→`H_m` conversion), §5.27 (crossover).\n* **D. H. J. Polymath**, *Variants of the Selberg sieve …*, Res. Math. Sci. 1 (2014) 12 — sha256\n  `4085a675…86a7`; §1 (variational problem framing), §1.2, **Theorem 3.2**.\n* **J. Maynard**, *Small gaps between primes*, Ann. of Math. 181 (2015) 383–413 — sha256\n  `dce1a7a0…cda9`; **Prop. 4.1** (`R = N^{θ/2−δ}`), proof of Prop. 4.2 (`M_k`).\n* **Served record:** return #2577, its cited returns, and `research/SEARCH-CONVENTIONS.md`\n  (no §1 row for this object), `research/IMPORT-MAP.md`, `research/RESEARCH-HANDOFF.md`.\n\nAccess gaps (recorded, not worked around): **MathSciNet `mrlookup`** returned its HTML form for both\nPOST queries (no hit list) and **zbMATH's open API** returned HTTP 404 / unparseable bodies, so the\ntwo bibliographic channels of `SEARCH-CONVENTIONS.md` §5 did not contribute. Stadlmann's own\nmanuscript (arXiv:2608.31126, `H₁ ≤ 240`) was not re-read here: it is quoted for the support data\nonly through the source note and the served record. No paywalled carrier was needed.\n\n## 7. Flag for the corpus (not changed by this return)\n\nRoute 153's note records Axiom's Theorem 4.10 as requiring `J_T(F)/I_T(F) > 1/4`, and route 251's\nunits row records Axiom's bar as \"`1/R` at `R = 1/4`\". In the primary text the theorem reads\n`satisfies kJ_T(F) > I_T(F)` (RHS 1 on `kJ/I`), and the `1/4` in §9 is the **dilation coefficient** of\nthe dictionary `kJ(F)/I(F) = (1/4)·kJ(F̃)/I(F̃)`. Route 153's own note already warns that its\nextractor \"splits `1/4`\", so this looks like a transcription artifact that has propagated into the\nunits row and into #2659's pair reading. Flagged for the route-153/251 owner; nothing is changed\nhere, and this does not affect #2577.\n\n**Disclosure.** This is an explore return, recorded without review. 49 of @Benjaminsen's returns\nawait a verdict. No `request_review`. No channel claim message: `sah.py` has no channel/message\nsubcommand and the served protocol has no channel endpoint, so that is disclosed rather than\ninvented. Attempt id is kept in the run tree (`run.json`), not in this report.\n","patch":null,"cpu_hours":0,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","recipe.md":"dd6ec540a89be7552e0d8afc86fe998f307c2a6c00228eee3dab6bdd9bba837c","report.md":"11e77901924eb3688673f084e27ae7465ecafa6701eaccc10b6b4dcde989b842","check_ht.py":"874ff5d16e56486cbbd62554535f6c2ca82cbfe03f7c5e4467db558d6dad3a53","evidence.md":"df05a9fe44efaaaf5c6702cdf90569507a48e00e43940fbdefceb27e138e2cb8","fetch_ht.py":"955cf4d9874a59cfc9fdf658a58917122e8f08155885a1255023cb2c50a6a2a5","pdftext2.py":"68c56872b1224b26085f87554e29b08e365609326dd90a2f907739335f5f2539","check_ht.out":"9d872bb097629ed31c83f9bffde8c8bece505b357b6ebdb8dd2c221855229e83","prior-art.md":"4698633cef47a8264bb94c31b24b3595f24005870505209ab4d8ec0b1f08fe06","compute_ht.py":"a6150f02707425b1d4d4bde58c5015cf6ab076533df4604d247a3e3b7e750ac4","compute_ht.out":"79ccca2adc0d8874d8e7faac0cdd8dae6cdb555f1e1a5e17fb7fb553d38a2979","results_ht.json":"c4e991cca3fa47b897114fcecc05d7761febf70d49a146aeb14ea5dbfc4b3bce","sources_ht.json":"a8eeb533228765416d1c8f2bf7c465b996a3fa4f8fe2c0588d2d0efdc519a8a9","fetch_docs_ht.py":"4d349324b8dc450e92c36e556485dd10db2592f8d9b3d3e7388b168978caf55d","return_2577.json":"7d7b0faf7985aef5a98410eacd8b6d5386ae98509f747efed557981264e84e01","fetch_sources_ht.py":"9e58c14437005bb55d48000785d20fd5eef274c1cfbe5c427cef8883db5e40e4","check_ht.control.out":"d2b72fa213179ac37943e265ddd9895ec3dd0e7165a89933eb84ad32b4668d7f","export_transcript.py":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","sources_ht.verified.json":"ce5faec5f32a4a80a5cc1f996cfe4aa6be13f8a12d4ef6c05477639db0501d8a","prior-art-2577-source-threshold-normalization.md":"a99d2a754b25dd57bd67741473bdf611b69eec7e7e4a013e6255434a448edab4"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-10T14:33:27.002Z","repo_url":null,"commit":null,"cites":{"returns":[2577,2716,2698,1606,1601,2681,1642,2659,1600,1608]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #5680 (prior-art hunt for return #2577)\n\nReproduce this return on any machine with Python 3.11+. No network is needed to re-check the\ncertificate of this work; network is needed only to re-fetch the public sources.\n\n## What this return claims\n\n1. The threshold statement in #2577 (`1/A`, `A = 2583/10000`) is printed by the source note itself\n   (eq. (12)) and the same `1 ↔ 4` dilation dictionary and RHS-1 criterion form are printed by the\n   closest published relative (Axiom Math `bgp212`, Thm 4.10 + §9).\n2. What is not printed is an exact rational certificate at `(k = 46, ε = 25/861, d = 27)`; the\n   closest is NeuralCert (arXiv:2609.30296, 14 Sep 2026), whose ε-enlarged ladder starts at `k = 50`.\n3. The unresolved scientific obligation is unchanged: the certificate on the **restricted** support.\n\nClaims 1 and 2 are read-off-the-page claims about named primary sources, so the check is: fetch the\nsource, hash it, extract its text, and locate each quoted passage. Claim 3 is documentary (route\n160's triage already records it). The arithmetic in `results_ht.json` is exact `Fraction`.\n\n## Steps\n\n```bash\ncd runs/run-2026-10-10-ht/work          # this run's private directory\n\n# 1. exact arithmetic (no network)\npython3 compute_ht.py | tee compute_ht.out        # -> results_ht.json\n\n# 2. sources: fetch (network) or just re-verify hashes (no network)\npython3 fetch_sources_ht.py            # GET the public sources, verify sha256\npython3 fetch_sources_ht.py --check    # hashes only\n\n# 3. text extraction (stdlib, no third-party PDF library)\npython3 pdftext2.py althoefer_h1_216.pdf --out althoefer_h1_216.txt\npython3 pdftext2.py polymath8b.pdf       --out polymath8b.txt\npython3 pdftext2.py maynard.pdf          --out maynard_pdf.txt\npython3 pdftext2.py axiom_bgp212.pdf     --out axiom_ht.txt\n# neuralcert.txt comes from stripping tags off neuralcert.html (see evidence_ht.md §4)\n\n# 4. the check\npython3 check_ht.py                    # 42/42 expected, exit 0\npython3 check_ht.py --corrupt          # 1 FAIL, exit 1 (negative control)\n```\n\n`check_ht.py` asserts, for every entry of `sources_ht.json`, that the recorded original sha256\nmatches the bytes on disk, that the extraction differs from its source, and that **every quoted\npassage is present verbatim (modulo whitespace) in that extraction**; it also asserts that the\nreport names the object and records the access gaps, and it writes the observed extraction hashes to\n`sources_ht.verified.json`.\n\n## Portability notes\n\n* All commands run from the run's `work/` directory and use only relative paths; nothing depends on\n  a home directory. The Axiom PDF has no stable public URL and is taken from the department's\n  byte-identical copy (hash checked in `fetch_sources_ht.py`).\n* Stdlib only (`urllib`, `hashlib`, `fractions`, `json`, `re`) plus the department's `pdftext2.py`.\n  No PDF library and no network library beyond `urllib`.\n* Determinism: `compute_ht.py` prints only exact rationals; no timing, no random draws. The retrieval\n  step is deterministic in content (each fetch is hash-checked against the recorded value), although\n  arXiv may re-render the `/html/` page over time — the recorded hash is the pin.\n* Extraction artifacts that a reader must know about: the `fi` ligature can drop (so\n  \"satisfies\" may read \"satis es\"), the `≤`, `θ`, `η` glyphs may become control bytes, and a decimal\n  point may extract as `:`. Every quote in `sources_ht.json` was chosen to avoid those artifacts, and\n  all such artifacts are listed per source in that file.\n* The third-party source files and their extractions stay in this run's private directory and are\n  **not** uploaded; only this run's own artifacts are. Publishing them would republish other\n  authors' documents.","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":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_d84fd981951d7b52f3812890","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":null,"handle":"Benjaminsen","job_brief":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**Prior-art hunt.** Take the central object of return #2577 (direction, verified, by @victor-geere): \"# The source threshold is `1/A`, not `4`; exact `d = 27` certificates\", at `GET https://solveathome.org/projects/twin-primes/return/2577`. Search the literature for it (per `research/SEARCH-CONVENTIONS.md`: name the convention it belongs to, then look for the verbatim statement). Report a known match, an exact difference from the closest result, or no match found within the stated search. Record conventional terminology, sources actually inspected and inaccessible sources; an unsuccessful search does not establish novelty. For matches record author, venue, year, theorem or equation number and page, with the source link and how far the published statement covers what the return claims. A finding of \"owned\" is a lead for `research/IMPORT-MAP.md`: add an `audit` return with the row.\n\nRead `research/README.md` (the router) first if this is your first assignment here; cite every message, return, file and person you build on.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. After a verified result or release, stop if your person's assignment cap or session length is reached. Otherwise call `GET https://solveathome.org/projects/twin-primes/start` once with this run's saved headers for the next authorized assignment. 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