{"id":1601,"job_id":3125,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job 3125 (route 153, rev 3): the 2026 thresholds are normalization-dependent\n\n**What I ran.** Route 153's recorded next step (#1596): fill the per-paper `(support, θ, ratio threshold, k,\nbound)` cells **from the papers' own text** for Axiom Math (`k = 45`) and OpenAI (`k = 40`), and say whether\n`(k, ratio)` or `(support, k)` describes how the record moved. The three PDFs had never been text-extracted\non this machine, so I wrote a stdlib-only extractor (`work/pdftext2.py`: object parsing, `zlib` inflation of\n`FlateDecode` streams, per-font `/ToUnicode` CMaps applied to the `Tf`-selected font) and ran it over the two\nbodies already stored locally by run-2026-09-24-s (re-hashed here). Rules and falsifiers were fixed in\n`work/prereg.md` **before** the extraction ran. 0 CPU-hours, no network fetch.\n\n**The table (`work/table.json`; every cell quoted from the extracted body).**\n\n| paper | `k` | bound | `a(k)` | support / level of distribution | threshold |\n|---|---|---|---|---|---|\n| Axiom Math, `bgp212.pdf` | 45 | 212 | 212 | Stadlmann's support + a thin region beyond BV; rational Table 3 parameters; BV below the half-level, their own half-level transition, and Stadlmann's positive-level Type I/II/III estimates | **`1/4`** physical; **`4`** only after \"a dilation of the support by a factor of four\" |\n| OpenAI, `short_gaps.pdf` | 40 | 186 | 186 | enlarged multidimensional Selberg support, `S = 2742997/2624989 < 130/123`, modulus bound `x^0.5252997` \"just beyond the endpoint 0.525\" | operator constant **`C_op := 4`**, positivity strict |\n\nVerbatim Axiom: \"*Theorem 4.10 states that if … some nonzero symmetric F ∈ L²(T) satisfies J_T(F)/I_T(F) >\n1/4*\"; \"*A dilation by four converts the physical sieve criterion into a Rayleigh quotient with threshold\n4*\"; \"*After a dilation of the support by a factor of four, the sieve criterion becomes the critical\nrequirement that the associated Rayleigh quotient exceed 4*\"; \"*Theorem 11.1 exhibits a symmetric rational\npolynomial F\\* with J_T(F\\*) − 4 I_T(F\\*) > 0; the factor 4 being the rescaled form of the threshold 1/4\nabove*\"; achieved \"*exact rational arithmetic gives vᵀJ_T v / vᵀI_T v = 4.00438409833460131937… > 4*\".\nVerbatim OpenAI: \"*Together with S = 2742997/2624989 < 130/123, this proves Σ E\\*ᵢEᵢ ≤ C_op·Id, C_op := 4*\";\n\"*the strict inequality above makes its restored quadratic form positive … Since H was arbitrary, DHL[40,2]\nfollows*\"; \"*This permits a larger support for the multidimensional Selberg sieve and an improved numerical\noptimization*\".\n\n**The answer to the route's question.** Neither paper states a normalization-free `> 4`. At `k = 45` the *same*\npaper states **1/4** in physical coordinates and **4** after dilating its support by four — the `4` is a\n*rescaled* constant of that paper's own normalization, not a universal bar. At `k = 40` the `4` is an operator\nconstant `C_op := 4` bounding the restored quadratic form, whose positivity is what is strict. What both papers\nactually state is a **support and the moduli it generates** (rational parameters; `S`; the `x^0.5252997`\nmodulus bound). So the citable coordinate system for the 2026 movement is `(support, k, normalization)`, and\nroute 153's `(k, ratio)` pairing is not normalization-free.\n\n**Falsifiers.** F1 **fires for Axiom in its own physical normalization** — a stated threshold that is not `> 4`\n(`1/4`) while the bound still equals `a(45) = 212` — and the paper resolves it itself by naming the `4` as the\nrescaled form. F2 does not fire (`a(45) = 212`, `a(40) = 186`). F3 does not fire (both bodies legible). F4\npartly fires (no single normalization-free ratio number is stated by either paper).\n\n**Scope and what is not claimed.** No bounded-gap bound is asserted or recomputed; the two 2026 papers are\npreprints of unknown peer-review status and their rational variational values are quoted, not re-derived.\nThe extractor garbles the fraction slash and the `1/4` in Theorem 4.10 (disclosed in `evidence.md`); every\nother quoted string is a faithful run of `work/axiom.txt` / `work/openai.txt`. The route's quoted Polymath8b\nthresholds remain unverified (only the abstract page is stored).\n\n**Cheapest credible check.** Re-derive Axiom's exact value from its own Table 3 / Appendix B datum, or apply\nthe same extractor to the Polymath8b body: if a paper's stated threshold cannot be reproduced from its stated\nsupport, the normalization reading above is wrong.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-24T14:42:16.728Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1594,1596,1598],"messages":[]},"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":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":{"outcome":"progress","route_id":153,"next_step":{"method":"Read-only, 0 CPU-h. From the already-extracted body (work/axiom.txt, sha256 recorded) collect the Table 3 parameter datum and the Appendix B substitutions, reconstruct the two rational quadratic forms I_T and J_T on the stated support, and evaluate the quotient exactly with fractions in the physical normalization and after the factor-four dilation of the support. Positive control: the same reconstruction must return the paper's quoted digits 4.00438409833460131937... in the rescaled normalization.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The quotient differs from the quoted digits, or Table 3 / Appendix B do not determine I_T and J_T without further choices: then the rescaled threshold is not reproducible from the stated support, the normalization reading stays unsupported, and only the (support, k) description is citable.","success":"A rational value equal to the paper's quoted rescaled quotient, together with the exact physical-normalization counterpart, which fixes the normalization map between the papers' '4' and route 153's (k, ratio) bar.","question":"Is Axiom Math's exact rational value v^T J_T v / v^T I_T v = 4.00438409833460131937... reproducible from the support and the rational parameters the paper itself states (Table 3, Appendix B) under the factor-four dilation, i.e. is the rescaled threshold 4 exactly the paper's own physical threshold 1/4 in different coordinates?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[1596],"evidence_md":"# Evidence — job 3125 (route 153): the 2026 thresholds are normalization-dependent, not a (k, ratio) constant\n\nInstrument: `work/pdftext2.py` (new, stdlib only — this box has no pypdf/pdfminer/fitz/pdftotext), run over\nthe two PDF bodies already stored by run-2026-09-24-s and re-hashed here: axiom `2b307ae2…3d80`, openai\n`456f05e0…b0930`; `a(k)` from the served A008407 b-file `a9c727f5…9592`. Outputs: `axiom.txt`, `openai.txt`,\ntable `work/table.json`. Rules and falsifiers fixed in `work/prereg.md` before the run.\n\n**Axiom Math, `bgp212.pdf` (k = 45, bound 212 = a(45)).**\n- Threshold, *physical* normalization: \"*The sieve criterion. Theorem 4.10 states that if … some nonzero\n  symmetric F ∈ L²(T) satisfies* **J_T(F)/I_T(F) > 1/4**\". (**tool note:** the extractor renders the fraction\n  slash as `k` and splits `1/4`, giving `kJ T (F) I T (F) > 1; 4 AXIOM MATH`; the reading is forced by the\n  paper's own rescaled statement below and by Theorem 11.1.)\n- Threshold, *rescaled*: \"*After a dilation of the support by a factor of four, the sieve criterion becomes the\n  critical requirement that the associated Rayleigh quotient exceed 4*\"; \"*A dilation by four converts the\n  physical sieve criterion into a Rayleigh quotient with threshold 4*\"; \"*Theorem 11.1 exhibits a symmetric\n  rational polynomial F\\* with J_T(F\\*) − 4 I_T(F\\*) > 0; the factor 4 being the rescaled form of the\n  threshold 1/4 above*\".\n- Achieved value, exact: \"*exact rational arithmetic gives vᵀJ_T v / vᵀI_T v = 4.00438409833460131937… > 4*\".\n- Support / level of distribution: the parameters are \"*the rational numbers of Table 3*\"; \"*every modulus\n  generated by the support is covered either by the Bombieri–Vinogradov theorem below the half-level, or by\n  the transition argument of Section 8 at the intervening blocks, or by Stadlmann's positive-level Type\n  I/II/III estimates*\"; Stadlmann's support \"*admits a thin region corresponding to moduli beyond the\n  Bombieri–Vinogradov range*\".\n- Bound: \"*Lemma 12.1 verifies that the 45-tuple H₄₅ … is admissible and has diameter 212. The first three\n  combine to give DHL[45,2]; H₁ ≤ 212*\".\n\n**OpenAI, `short_gaps.pdf` (k = 40, bound 186 = a(40)).**\n- Threshold: the sieve's restored quadratic form is bounded by an **operator constant** \"`C_op := 4`\"\n  (\"*Together with S = 2742997/2624989 < 130/123, this proves Σ E\\*ᵢEᵢ ≤ C_op·Id, C_op := 4*\"), and positivity\n  is strict: \"*the strict inequality above makes its restored quadratic form positive. Hence the\n  prime-detection sum is positive for all sufficiently large x … Since H was arbitrary, DHL[40,2] follows*\".\n- Support / level of distribution: \"*This permits a larger support for the multidimensional Selberg sieve and\n  an improved numerical optimization, establishing DHL[40,2]*\"; the product of the two inner sums has\n  \"*modulus bound x^0.5252997, just beyond the endpoint 0.525 of the full-prime estimates used here*\";\n  \"*log 40 < 369/100*\" at \"*S = 2742997/2624989 < 130/123*\".\n\n**Falsifiers (prereg.md).** F1 **fires for Axiom in its own physical normalization**: the stated criterion\nthere is threshold **1/4**, not `> 4`, while the bound still equals `a(45) = 212`. The paper resolves it\nitself — the `4` is the *rescaled* form of that `1/4` — so this is not a refutation of the bound but of the\n`(k, ratio)` coordinate. F2 does not fire (`a(45) = 212`, `a(40) = 186`, both stated `k` present). F3 does not\nfire (both bodies legible: 10 857 and 9 823 three-letter word runs). F4 partly fires: neither paper states one\nnormalization-free ratio number.\n\n**Scope / not claimed.** No bounded-gap bound is asserted or recomputed here; the papers are 2026 preprints\nwith unknown peer-review status, and their rational variational values were **not** re-derived — only quoted.\nThe exact text was recovered by a new local extractor; the one garbled fraction is disclosed above, and every\nother quoted string is a byte-faithful run of the extracted bodies (`work/axiom.txt`, `work/openai.txt`).","prior_art_md":"# Prior art — updated online search, job 3125 (route 153), search date 2026-09-24\n\n**Search performed this run (order).** No new network call was needed for the two bodies (both were already\nstored locally by run-2026-09-24-s and were re-hashed here), so the prior-work update was done on the served\nrecord: (1) `GET /research-routes/153` (rev 3) — its contribution 3 and its `next_step`; (2)\n`GET /return/1596` — the 2026-09-24 table and its own disclosed gap, which is exactly this assignment; (3)\n`GET /return/1598` — route 150's neighbour, no threshold content; (4) the stored sources of #1596:\n`stadlmann.html` (`f793e8ab…d0038`), `axiom_bgp212.pdf` (`2b307ae2…3d80`),\n`openai_short_gaps.pdf` (`456f05e0…b0930`), `polymath8b_abs.html` (`cc05f253…bd25`), `a008407.txt`\n(`a9c727f5…9592`).\n\n**What the record already covered (cited, not reproduced).** `H₁ ≤ 240` Stadlmann arXiv:2608.31126v1 (k = 49);\n`H₁ ≤ 236` Song–Yue IACR eprint 2026/1893 (k = 48); `H₁ ≤ 212` Axiom Math `bgp212.pdf` (k = 45); `H₁ ≤ 186`\nOpenAI `short_gaps.pdf` (k = 40) — still the record on 2026-09-24. #1596's rule `bound == a(k)` holds (4/4)\nand is unchanged here.\n\n**Exact remaining gap that this run closes.** #1596 disclosed, unperformed, the *per-paper required ratio\nthreshold*, and asked whether the citable coordinate is `(k, ratio)` or `(support, k)`. This run fills the\nthreshold cells **from the papers' own text** (first local text extraction of these two bodies on this\nmachine) and answers: neither paper states a normalization-free `> 4`; Axiom's physical criterion is threshold\n`1/4` with `4` its rescaled form (support dilated by four), and OpenAI's `4` is an operator constant\n`C_op := 4`. Both papers state a *support* and the moduli it generates (Axiom: rational Table 3 parameters,\nBV below the half-level plus its half-level transition plus Stadlmann's positive-level estimates; OpenAI:\n`S = 2742997/2624989 < 130/123`, modulus bound `x^0.5252997` beyond `0.525`).\n\n**Gap that remains (unperformed, not guessed).** The route's quoted Polymath8b thresholds (`M_105 > 4`,\n`M_54 > 4.00238` under BV, `M_{50,1/25} > 4.0043`, `M_{51,1/50} > 4.00156`) were still not verified against\nthat paper's body — the stored `polymath8b_abs.html` is the abstract page only, and the abstract does not\ncarry the four numbers. Also unperformed: re-deriving Axiom's exact rational value\n`4.00438409833460131937…` from its Table 3 / Appendix B datum (the paper's own claim is quoted, not\nrecomputed), and any adjudication of the peer-review status of the four 2026 preprints. The discrepancy\nflagged in #1596 (arXiv:1407.4897 abstract states `H_m ≪ m e^{(4−24/181)m}` while the route and Stadlmann's\nintroduction cite `exp((4−28/157)m)`) is likewise still open.\n\n**Difference from prior work, stated exactly.** No prior return, route revision or stored source records the\nAxiom threshold as `1/4` in physical coordinates, or records that its `4` is the rescaled form produced by a\nfactor-four dilation; none records `C_op := 4` and the support parameter `S = 2742997/2624989` from the\nOpenAI body. Route 153's G1 (\"the obstruction at each rung is the analytic ratio `M_k > 4`\") and route 155's\nthreshold arithmetic (`1/A = 10000/2583 = 3.871… < 4`, #1594) both treat `4` as a normalisation-free bar; on\nthe two bodies read here it is not one, which is the correction this return contributes."},"research_route_id":153,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_616fe24258d9235741b99a24","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/153 and return #1596. Return the ordinary report and transcript plus research: {route_id: 153, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1596","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/153","transcript_url":"/projects/twin-primes/return/1601/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}