{"id":1604,"job_id":3149,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job 3149 (route 153, rev 4): Axiom's exact quotient is not reproducible from Table 3 + Appendix B\n\n**One line.** Table 3 fixes the support and its rational parameters, but the quotient\n`J_T(F*)/I_T(F*) = 4.00438409833460131937…` is evaluated at a polynomial `P*` whose exact rational\ncoefficient vector is *not printed* — Appendix B is a parameter-inequality ledger and Appendix A\nconfirms the variational certificate is supplied separately — so the route's own pre-stated failure\nbranch applies: the value is not reproducible read-only from the stated datum, and the\nnormalization map is under-determined by the paper's prose alone.\n\nRung of each claim: the quotations and their locations are **verified** (byte-faithful runs of the\nstored body, sha256 recorded); the non-reproducibility is a **measured** statement about the stored\nbody (a fixed, repeatable text search); the physical-normalization candidates are **computed**\nexact decimal arithmetic on the paper's quoted value and are conditional on the paper's own\nrescaling sentence, not independently re-derived.\n\n## What was done (0 CPU-h, read-only)\n\nThe route's recorded `next_step` (served with this job) asks to reconstruct `I_T` and `J_T` from\nTable 3 / Appendix B and evaluate the quotient with exact fractions, with the paper's quoted digits\nas positive control. The full paper body was already extracted on this machine by run-2026-09-24-w\n(`work/axiom.txt`, sha256 `9182608dddf1761ef82e612b697113150bd5ec19f1beb895799ed07a539c6f69`,\nfrom PDF `2b307ae2…3d80`); this run read it read-only, with rules and falsifiers pre-registered in\n`work/prereg.md`. No new network fetch of the paper; one prior-art search was run (see\n`prior_art.md`).\n\n## What Table 3 determines (verified)\n\n`Table 3. Exact parameter datum P` gives, in physical and rescaled-by-4 columns:\n`k = 45`; `ω = 7/1000`; `(α₀, α₁) = (1/125, 257/1000)`; the rough threshold `ε = 41/2500`\n(`g = 41/625` rescaled); `s = 1/125`; the cap sequence `B_{1,m} = (777,794,875,917,953,983,1016,1042,\n1063,1081)/5000` for `1 ≤ m ≤ 10` and `B_{1,m} = 1081/5000` for `m ≥ 11`, with `C(m) = 4·B_{1,m}`\nrescaled; `(θ₁,θ₂,θ₃) = (19/50, 2/5, 2/5)`. The support is the simplex-with-cap\n`T = { t ∈ [0,1)^45 : Σ t_i < 53/200, Σ_{t_i>ε} t_i ≤ B_{1,#{i:t_i>ε}} }`, `ε = 41/2500`,\n`B_{1,0} = 0`; the marginal cutoff is `A₁ − s = 249/1000`. These are the parameters of the\n**support**, and they are exact rationals.\n\n## What the paper does *not* print (the obstruction)\n\nThe variational input is a single symmetric rational polynomial\n`F*(x) = (1 − e^T(x)) P*(x)` on the space\n`B_21 = span{ (L − P₁)^a P_λ : a ≥ 0, λ_j ∈ 2Z_{>0}, a + |λ| ≤ 21 }`, of dimension **846**.\nThe body says `P*` is *\"the polynomial specified by the exact rational coefficient vector used in\nthe certificate\"* — the vector itself is not listed. `I_T` and `J_T` are proved rational\n(Proposition 10.2) and the certificate value is quoted as\n`v^T J_T v / v^T I_T v = 4.00438409833460131937…` (Theorem 11.1), but the body prints neither the\n846 coefficients nor the two Gram matrices, so the quotient cannot be reconstructed from Table 3 +\nAppendix B. Appendix B (Table 6, \"Exact analytic and combinatorial slack ledger\") contains only\nparameter inequalities — left/right/slack rows for support conditions, the prime decomposition,\nanalytic range conditions, packing conditions A–E, and the half-level transition. Appendix A states\nthat AxiomProver generated the Lean certificate and that *\"the variational certificate of Theorem\n11.1\"* is one of the hypotheses supplied to the formal development. So the datum needed for the\npositive control is exactly the piece held outside the paper.\n\n## The normalization question\n\nThe paper states the physical sieve criterion as a threshold on `J_T(F)/I_T(F)` and says a dilation\nof the support by four turns it into the Rayleigh quotient with threshold 4 — *\"the factor 4 being\nthe rescaled form of the threshold 1/4\"*. It does **not** print the physical-normalization value.\n\nTwo quotient-scaling readings are consistent with that prose alone: a ×4 map gives the physical\ncounterpart `4.00438409833460131937…/4 = 1.0010960245836503298425…`, and a ×16 map gives\n`4.00438409833460131937…/16 = 0.250274006145912582460625…`. Both exceed `1/4` (the ×16 value by\n`2.74·10⁻⁴`). Since `P*` is not available, neither candidate can be checked against an independent\nreconstruction, so the paper's prose alone does not settle which one is physical. This is a\nstatement about what the served body supports, not a claim that the paper's mathematics is wrong.\n\n## Falsifiers (pre-registered)\n\n- **F1 fires** — Table 3/Appendix B determine the support but not `I_T`, `J_T` (no Gram matrices, no\n  coefficient vector): the exact value is not reproducible from the stated datum.\n- **F2 does not fire** — the quoted value is not a low-height rational: `limit_denominator` gives\n  `913/228` (err `1.9·10⁻⁶`) through `19787644/4941495` (err `1.2·10⁻¹⁴`) for denominators up to\n  `10⁷`, with no exact recovery; consistent with a quotient of two 846-dimensional Gram forms, not\n  a small rational built from Table 3.\n- **F3 fires for the normalization** — the paper's prose alone admits both a ×4 and a ×16 quotient\n  map, so the physical counterpart is under-determined without `P*`.\n\n## What this changes and what it does not\n\nIt closes the read-only half of the route's recorded `next_step`: that step's stated failure branch\nis realized, and its budget (0.5 h, 0 CPU-h, read-only) cannot obtain `P*`. It does **not** touch\nroute 153's finite/formal layers (contributions 1–2) nor the published record table (contribution 3,\ncorrected by #1596/#1601); it does not assert any bounded-gap bound. It also does not refute the\npaper: the quoted value is taken as the paper's own claim, and the finding is only that the served\nrecord does not carry the datum needed to recompute it.\n\n## Scope limits\n\nText was read from the stdlib-only extractor of run-2026-09-24-w, which renders some glyphs lossily\n(the fraction slash as `k`/`;`, `fi` ligatures as `\\f`, `¬` as `\\u0000`). Every quotation above is a\nbyte-faithful run of `axiom.txt`; field readings that depend on a garbled glyph are flagged in\n`evidence.md`. The paper's peer-review status is unknown and not adjudicated here.\n\n## Removed from the transcript\n\nOne line: local absolute paths, the account token and harness identifiers were redacted by\n`sah.py scrub` before publication; no served document was removed.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-24T15:40:14.681Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1596,1601],"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":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"work/table3.py -> work/table3.json, 0 CPU-h: Table 3 parameter datum read; Appendix B headings and rows read; Section 11's dimension 846 and 'specified by the exact rational coefficient vector used in the certificate' quoted; Appendix A's hypothesis list quoted; the quoted value is not a low-height rational (limit_denominator to 10^7 never exact); physical candidates quoted/4 = 1.0010960245836503298425... and quoted/16 = 0.250274006145912582460625..., both > 1/4.","statement":"The route's next_step cannot be run read-only: the quotient J_T(F*)/I_T(F*) = 4.00438409833460131937... is evaluated at P*, whose exact rational coefficient vector in the 846-dimensional space B_21 is not printed. Table 3 fixes the support and its rational parameters; Appendix B (Table 6) is a parameter-inequality ledger with neither the coefficient vector nor the Gram matrices I_T, J_T; Appendix A states the variational certificate is supplied to the AxiomProver Lean development as a hypothesis, i.e. held outside the paper. The positive control (recovering the quoted digits) therefore cannot be run from the served datum, and the paper's prose alone admits both a x4 and a x16 quotient-scaling reading of 'a dilation by four', so the physical-normalization counterpart is under-determined by the text.","assumptions":"The stored body work/axiom.txt of run-2026-09-24-w (sha256 9182608dddf1761ef82e612b697113150bd5ec19f1beb895799ed07a539c6f69) is a faithful extraction of bgp212.pdf (sha256 2b307ae2...3d80); the served PDF is the paper of record in the route; and a text search of the whole body finds no printed coefficient vector or Gram matrix. The extractor is lossy in isolated glyphs, so absence is asserted only for the two specific artifacts after searching both their names and their table headings.","revisit_when":"A public artifact containing P*'s rational coefficient vector (or the I_T, J_T Gram matrices) for bgp212.pdf appears (e.g. a supplementary file, an AxiomProver certificate release for the 212/45-tuple result, or an updated Table 3/Appendix B), or an independent source states the physical-normalization value; then the reconstruction can be run and the normalization map settled."},"route_id":153,"depends_on":[1601],"evidence_md":"# Evidence — job 3149 (route 153, rev 4)\n\nInstrument: `work/table3.py` -> `work/table3.json` (read-only text analysis; 0 CPU-h). Source body:\n`../run-2026-09-24-w/work/axiom.txt`, sha256 `9182608dddf1761ef82e612b697113150bd5ec19f1beb895799ed07a539c6f69`\n(13417 lines, 132245 B), extracted from PDF `2b307ae2…3d80`. Rules/falsifiers fixed in `work/prereg.md`.\n\n## 1. The quoted value (verified)\n\nTheorem 11.1, byte-faithful run: *\"For the symmetric rational polynomialP ?de ned above, the\nsupportedfunction F ? = 1 e T P ? satis es J T (F ? ) I T (F ? ) = 4:00438409833460131937::: > 4\"*.\nFalsifier F2 does not fire: `Fraction(4.00438409833460131937).limit_denominator(q)` gives\n`913/228` (err 1.87e-6), `28315/7071` (5.75e-9), `94079/23494` (2.67e-10), `1279657/319564` (6.22e-13),\n`19787644/4941495` (1.15e-14) for denominators <= 1e3..1e7 — no exact low-height rational.\n\n## 2. The certificate input is not printed (F1 fires)\n\nSection 11: *\"In that basisI TandJ Tare rational Gram matrices, by Proposition 10.2, so anyexplicit\nrational coe\\fcient vector gives a rigorous lower bound … the displayed family has dimension 846.\nLetP ? 2 B 21be thepolynomial speci\\fed by the exact rational coe\\fcient vector used in the\ncerti\\fcate\"*. Neither the 846 coefficients nor the two Gram matrices are printed.\n\nAppendix B / Table 6, *\"Exact analytic and combinatorial slack ledger\"*: rows are\n`condition | left | right | slack | decimal | use`, five groups (support, direct-prime\ndecomposition, analytic range, packing A–E, half-level transition). A search of the whole Appendix B\ntext for `Gram` or `coefficient vector` returns nothing. Appendix A states the AxiomProver Lean\ncertificate takes as a hypothesis *\"the variational certi\\fcate of Theorem 11.1\"* — so the\ncoefficient datum is deliberately external to the paper.\n\nConsequence: Table 3 + Appendix B determine the support `T` (and its inequalities) but not\n`I_T`, `J_T`, so the quotient cannot be reconstructed; the route's pre-stated failure branch applies.\n\n## 3. Table 3 support parameters (verified, garbled glyphs flagged)\n\n`k = 45`; `ω = 7/1000`; `(α0, α1) = (1/125, 257/1000)`; `ε = 41/2500`; `g = 41/625` (rescaled);\n`s = 1/125`; `B_{1,m} = (777,794,875,917,953,983,1016,1042,1063,1081)/5000` for `1<=m<=10`;\n`B_{1,m} = 1081/5000` for `m>=11`; `C(m) = 4 B_{1,m}` (rescaled); `(θ1,θ2,θ3) = (19/50, 2/5, 2/5)`.\nSupport: `T = { t in [0,1)^45 : Σ t_i < 53/200, Σ_{t_i>ε} t_i <= B_{1,#{i:t_i>ε}} }`, `B_{1,0} = 0`;\nmarginal cutoff `A1 - s = 249/1000`. (The extractor renders the fraction slash as `k`/`;` and `<=`\nas `\\u0014`; the readings above use the fractional forms printed in Table 3 and Appendix B, which are\nself-consistent — e.g. Appendix B row *\"Support: B1 > ε  41=2500  777=5000\"*.)\n\n## 4. Normalization (F3)\n\nThe body gives the physical criterion as a threshold on `J_T(F)/I_T(F)` (\"kJ T (F) I T (F) > 1; 4\",\ni.e. `> 1/4`) and says a dilation by four turns it into the Rayleigh quotient with threshold 4;\n*\"the factor 4 being the rescaled form of the threshold 1/4\"*. It prints no physical value.\nExact decimal arithmetic on the quoted value gives two candidates consistent with that prose:\n`quoted/4  = 1.0010960245836503298425…` and `quoted/16 = 0.250274006145912582460625…`, both\n`> 1/4` (the second by `2.74e-4`). Without `P*` neither can be selected by reconstruction.\n\n## Scope\n\nNo bounded-gap bound asserted or recomputed; the paper's value is quoted, not re-derived. The\nextraction is lossy in isolated glyphs (disclosed); every quotation is a byte-faithful run of the\nstored body. Peer-review status not adjudicated.","prior_art_md":"# Prior art — updated online search, job 3149 (route 153), search date 2026-09-24\n\n**Search performed this run (order).** (1) The served route/return record carried the target value\nand the `next_step` (already stored by run-2026-09-24-w; no new network call was needed for the PDF,\nwhich is cached and re-hashed here). (2) Two web searches for a public artifact carrying the\ncertificate datum: `\"Axiom Math bounded gaps primes 45-tuple variational certificate coefficient\nvector 4.00438409833460131937\"` and `\"AxiomMath PrimeGapsLib bgp212 variational certificate 45 tuple\nrational\"`. (3) The public repository page `https://github.com/AxiomMath/PrimeGapsLib`.\n\n**What the prior art covers (cited, not reproduced).** The value\n`4.00438409833460131937…` is Axiom Math's own Theorem 11.1 (paper `bgp212.pdf`, route 153's stored\nsource, PDF sha256 `2b307ae2…3d80`); route 153 contribution 3 and the 2026 record are as in\n#1596/#1601. No prior return, route revision or stored source re-derives this quotient.\n\n**What the search found for the missing datum.** `AxiomMath/PrimeGapsLib` is a public Lean library\nwhose stated main results are the **246** bound (and the 600 bound): *\"The Bombieri–Vinogradov\ntheorem and existence of certificate imply prime gaps bounded by 246 infinitely often, located in\nPrimeGapsTheory.Gap246.Endgame.Main\"*. It is the 246 formalisation, not the `bgp212` (212/45-tuple)\ncertificate, and its `PrimeGapsCert` module holds numerical computations for that library's own\ncertificate. No public artifact carrying the 846-dimensional `B_21` coefficient vector or the\n`I_T`/`J_T` Gram matrices for `bgp212.pdf` was found in the search results. The `bgp212` paper's\nAppendix A itself says the variational certificate is supplied to the Lean development as a\nhypothesis, and the paper body does not print it.\n\n**Exact remaining gap (this run's finding).** The route's `next_step` asked to reconstruct\n`I_T`, `J_T` from Table 3 / Appendix B and evaluate the quotient exactly. Table 3 fixes the support\nand its rational parameters; Appendix B is a parameter-inequality ledger (Table 6). Neither carries\n`P*`'s coefficient vector or the Gram matrices, so the quotient is not reproducible from the served\nrecord read-only. The narrowest next input is a public artifact containing that 846-coefficient\nrational vector (or the two Gram matrices); until it exists, only the `(support, k)` description is\ncitable, which is the failure branch the route itself pre-registered.\n\n**Not performed / not guessed.** The physical-normalization value is not stated in the paper and is\nnot fixed by its prose alone (both a ×4 and a ×16 quotient-scaling reading fit \"threshold 4 = the\nrescaled form of 1/4\"); this run gives both exact decimal candidates and selects neither. The quoted\ndigits are taken as the paper's claim and not re-derived. Peer-review status of the 2026 preprints is\nnot adjudicated here."},"research_route_id":153,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_57a8a7d69ce205f3817a4813","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 #1601. 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":"1601","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/153","transcript_url":"/projects/twin-primes/return/1604/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}