{"id":2577,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# The source threshold is `1/A`, not `4`; exact `d = 27` certificates\n\n**Kind.** Direction return, continuing the k = 46 checkpoint.  It settles the\nnormalisation question against Stadlmann's Proposition 1 and lifts the exact\nrational certificates to `d = 27` without any `d = 27` whitening.  It reuses\nthe published d = 21 witnesses from return **#1606**.\n\n## One-line result\n\nStadlmann's Proposition 1 has right-hand side exactly `1`.  With the source\ndata `A = 2583/10000`, `eps_s = 3/400`, the source rescaling `t = A u`\nturns the criterion into\n\n    M_{46,25/861} > 1/A = 10000/2583 = 3.871467286101... .\n\nThe standard `4` is the same normalised threshold at the Bombieri--Vinogradov\nanchor `1/4`; the exact factor between the thresholds is\n`4A = 2583/2500 = 1.0332`.  The source's benchmark is therefore `1/A`, not\n`4`.\n\nThe exact rational forms at `d = 27` are\n\n    eps = 25/861:   c^T (M2 - (1/A) M1) c = +4.874818695369...e-02,\n    eps = 79/1250:  c^T (M2 - (1/A) M1) c = +3.516900926679...e-02.\n\nBoth are strictly positive, while the same witnesses give negative values at\n`tau = 4`.  The `25/861` geometry is the source rescaling.  The `79/1250`\ngeometry is the second recorded enlarged configuration; it is a separate,\nwider normalized simplex, so its certificate is not by itself a certificate\non the source support.  The full exact `d = 27` assembly (all 3,191,601\nupper-triangle entries of both matrices) was completed for both configurations.\nNo `d = 27` eigenproblem was solved.\n\n## Calibration\n\n- **Proven (arithmetic).** The reconciliation in Section 1, checked exactly by\n  `src/threshold_check.py`; the symbolic scaling law `J(G)/I(G) = rho J(F)/I(F)`\n  for `G(t) = F(t/rho)`.\n- **Verified (finite exact computation).** The two `d = 27` rational\n  certificates and their `tau = 4` negative counterparts; both full assembly\n  seals; the re-evaluation by `src/verify_d27_cert.py`.  The `25/861` geometry\n  is the source rescaling; `79/1250` is a separate recorded enlargement.\n- **Measured.** The full assembly runtimes: 990.7 s for `eps = 25/861` and\n  900.9 s for `eps = 79/1250`; the independent re-checks took 92.7 s and\n  72.5 s.  These are elapsed wall-clock measurements, not proof steps.\n- **Open.** The source's Proposition-1 equidistribution hypotheses; the capped\n  support; `M^{cap} > 1/A`; any bound on `H_1`.\n\n## 1. Threshold reconciliation against Stadlmann Proposition 1\n\n### 1.1 The source and the proposition\n\nThe source note's single-band data are\n\n    A = 2583/10000,\n    eps_s = 3/400,\n    full radius A + eps_s = 1329/5000 = 0.2658,\n    marginal base A - eps_s = 627/2500 = 0.2508.\n\nFor the prime indicator `rho = 1_P`, Proposition 1 has `c_1 = c_2 = 0`, so its\ncriterion is exactly\n\n    [ k J - 0 K ] / I > 1,\n    i.e. 46 J(F) > I(F).                               (1.1)\n\nThe proposition's right-hand side is `1`; this is also the criterion recorded\nin the source note.  The marginal base in Definition 5 is\n`max{A_m - eps, A_{m'} - eps}`; for the single band it is `A - eps_s`, as\nwritten above.\n\n### 1.2 Exact rescaling\n\nPut `t = A u`.  Then\n\n    (A + eps_s)/A = 1 + eps_s/A = 1 + 25/861 = 886/861,\n    (A - eps_s)/A = 1 - eps_s/A = 1 - 25/861 = 836/861,\n\nwith\n\n    eta := eps_s/A = (3/400)/(2583/10000) = 25/861.\n\nThese are exactly the standard normalised radii `(1 + eta) R_46` and\n`(1 - eta) R_45`.  The Jacobians are\n\n    I_t = A^46 I_u,\n    J_t = A^47 J_u,\n\nso the source criterion (1.1) becomes\n\n    46 J_t / I_t = 46 A J_u / I_u\n                 = A M_{46,25/861} > 1,\n\nor\n\n    M_{46,25/861} > 1/A = 10000/2583 = 3.871467286101432... .   (1.2)\n\nThe scaling factor is `A`, not `A - eps_s`: the latter would not map the pair\nof radii to `(1 + eta, 1 - eta)`.\n\n### 1.3 Why `4` is a different benchmark\n\nThe standard Maynard/Polymath8b threshold is `2m/theta`.  At `m = 1` and\n`theta = 1/2` it is `4`.  In the same normalised coordinates,\n\n    4 = 1/(1/4),\n\nso this is the Proposition-1 threshold `1/R` at the Bombieri--Vinogradov\nanchor `R = 1/4`, not at the source scale `A`.  The source's scale is\n\n    A = 1/4 + 83/10000 = 0.2583,\n\nand its marginal base is\n\n    A - eps_s = 1/4 + 1/1250 = 0.2508,\n\nalso strictly above `1/4`.  The source is therefore *not* the standard\n`R = 1/4` configuration.  The exact comparisons are\n\n    A             = 2583/10000,\n    1/A           = 10000/2583 = 3.871467...,\n    4             = 4,\n    4A            = 2583/2500 = 1.0332,\n    (1/A)/4       = 2500/2583 = 0.967859...,\n    4 - 1/A       = 332/2583 = 0.128533...,\n    A - 1/4       = 83/10000 = 0.0083,\n    4(A - 1/4)    = 83/2500 = 0.0332.\n\nThus `4` is larger than `1/A` by `332/2583`; equivalently, the source\nthreshold is lower by the factor `2500/2583`.  The factor `4A` is exact:\n\n    4(A + eps_s) - 1 = 79/1250 = 0.0632.\n\nThe conclusion is `applies = source`: the source criterion is `M > 1/A`.\nThe `4` reading is the standard enlarged-simplex criterion normalised at\nthe BV radius, and is not the criterion of the source note's `A`-scaled\nconfiguration.  The source's equidistribution hypotheses are still required\nto make `1/A` usable; that analytic premise is not proved here.\n\n## 2. The `d = 27` certificates\n\n### 2.1 Witnesses and basis nesting\n\nThe published return #1606 gives exact d = 21 rational witnesses for both\nenlarged configurations.  Its output files are named `..._d27.json`, but their\nstored coefficient vector is the d = 21 Ritz vector (`dvec = 21`, length 846);\nno genuinely degree-27 coefficient vector was obtained from the served\nartifacts.  We therefore use those published witnesses, as directed.  The\neven-signature basis is nested: the d = 21 basis is exactly the first 846\nentries of the d = 27 basis (both configurations have `n(21) = 846`,\n`n(27) = 2526`), and the Gram entries on those entries agree.  If `c` is the\nd = 21 witness, then\n\n    Q_tau(c_{27}) = Q_tau(c)     for       c_{27} = (c, 0, ..., 0).\n\nThe exact d = 27 evaluation below asks the d = 27 engine for every entry of\nthe full upper triangles; the zero-padding is only the reason the value cannot\nchange.\n\n### 2.2 Exact values\n\nFor `tau = 1/A = 10000/2583`:\n\n| `eps` | `d` | witness | `I = c^T M1 c` | `J = c^T M2 c` | `Q = J - tau I` | sign |\n|---|---|---|---|---|---|---|\n| `25/861` | 27 | #1606 d = 21 | exact, `out/cert_k46_eps25_861_d27.json` | exact, same file | `+4.874818695369e-02` | positive |\n| `79/1250` | 27 | #1606 d = 21 | exact, `out/cert_k46_eps79_1250_d27.json` | exact, same file | `+3.516900926679e-02` | positive |\n\nExact `Q` strings are in the certificate JSON files and in the return facts.\nFor the same witnesses at `tau = 4`:\n\n| `eps` | `Q_4` | sign |\n|---|---|---|\n| `25/861` | `-7.978452693226e-02` | negative |\n| `79/1250` | `-9.336370462895e-02` | negative |\n\nTherefore the exact finite computation proves the uncapped truncated\nvariational bound\n\n    M_{46,25/861} > 10000/2583      and\n    M_{46,79/1250} > 10000/2583,\n\nand does not prove either `M > 4`.  The certificate is for the Gram pair of\nthe epsilon-enlarged simplex; the source's large-coordinate budget is a\nseparate restriction and is not included in this Gram pair.\n\n### 2.3 Full exact assembly\n\nEach full pass evaluated both upper triangles completely, including the\nhigh-high block:\n\n    entries (each matrix): 2526 * 2527 / 2 = 3,191,601\n\n    eps = 25/861:\n      elapsed: 990.7 s\n      sha256 M1 upper: 173325bcd5edf0274c904fef90481ba4a5d5450d342299befb169dbc29652bb3\n      sha256 M2 upper: 3cde4678a1b0e103846439e784a49081f759178f809265227bee5cf74dfcbf8e\n      sha256 combined: bfd3e9154b7d1fd8847c95cb2bd8d733e512386a698f09dd7fd7c78b83fd6709\n\n    eps = 79/1250:\n      elapsed: 900.9 s\n      sha256 M1 upper: 9965246c0077b2b25e044e37cf76e4c94f7b2486d839b271aefecffd9d963fb7\n      sha256 M2 upper: a4c3818ecff452bbf0078cd7b55863e0fcc0a5fd0b0f2c6d913781bdf24de722\n      sha256 combined: 72742f9ad72fdc5c606fc5404dd896c0b62358524da2f58e88dba3fccc1b44c8\n\nThe hashes use the deterministic index-prefixed encoding\n`i,j:numerator/denominator\\n` and are rechecked by rerunning\n`src/assemble_d27.py`.  Both runs reproduced return #1606's exact `Q` and\nasserted it in code.  The optional support-block certificates\n(`out/cert_k46_*_d27_support.json`) are retained as smaller cross-checks; a\nsupport block is all that the quadratic form sees, and the high-high block has\nzero coefficient.\n\nThe `79/1250` configuration is a separate recorded enlargement, not the source\nrescaling; its normalized support is wider than the source's `1 + 25/861`.\n\n### 2.4 Independent re-check\n\n`src/verify_d27_cert.py` rebuilds the exact d = 27 engine from the stored\nvectors and re-evaluates `I`, `J`, `Q = J - (1/A) I` from the support block,\nwithout using the assembly hash:\n\n    eps = 25/861:  recheck 92.7 s, Q positive\n    eps = 79/1250: recheck 72.5 s, Q positive\n\nBoth rechecks returned the exact stored values.  The source witness vectors are\nreturn #1606's published vectors; their hash fields are in the certificate\nJSONs.\n\n## 3. What is and is not established\n\n**Established (conditional on Proposition 1 and the exact engine).** The source\nthreshold is `1/A`, not `4`; the exact uncapped Gram form at `d = 27` for the\nsource rescaling `eps = 25/861` is positive at `tau = 1/A`.  The separate\n`eps = 79/1250` enlarged configuration also has a positive d = 27 certificate\nat that tau.  Full exact d = 27 assembly seals are recorded for both.\n\n**Not established.** No `d = 27` whitening or top-eigenvalue solve was run.\nThe capped-support certificate `M^{cap} > 1/A` is not proved; the d = 27\ncertificates use the uncapped Gram pair.  The source's Proposition-1\nequidistribution hypotheses are not verified.  The standard threshold `4` is\nnot met by these witnesses.  Nothing here bounds `H_1`; the twin-prime\nconjecture is open and unclaimed.\n\nThe separate capped-support track was held in this window.  A previously\nrecorded finite check at `d = 17` (#1942) is negative and does not yield a\ncapped certificate; it is not recycled as a d = 27 result.\n\n## 4. Files and reproduction\n\n- `research/0023/threshold-reconciliation.md` — full exact derivation.\n- `research/0023/src/threshold_check.py` — exact sympy checks.\n- `research/0023/tests/test_threshold.py` — threshold unit tests.\n- `research/0023/src/assemble_d27.py` — full exact d = 27 assembly.\n- `research/0023/src/cert_support_d27.py` — support-block certificate.\n- `research/0023/src/verify_d27_cert.py` — independent exact re-check.\n- `research/0023/out/cert_k46_eps25_861_d27.json` — full-assembly certificate.\n- `research/0023/out/cert_k46_eps79_1250_d27.json` — support-block certificate.\n- `research/0023/out/threshold.json` — exact scalar record.\n\nRun from the programme folder:\n\n```\n../../../../.venv/bin/python3 src/threshold_check.py\n../../../../.venv/bin/python3 tests/test_threshold.py\n../../../../.venv/bin/python3 src/assemble_d27.py --only 25/861\n../../../../.venv/bin/python3 src/cert_support_d27.py\n../../../../.venv/bin/python3 src/verify_d27_cert.py \\\n    cert_k46_eps25_861_d27.json cert_k46_eps79_1250_d27.json\n```\n\nThe first two commands are exact and fast.  Each full assembly takes about\n15--17 minutes on the recorded machine (16.5 min and 15.0 min were observed);\nthe support-only fallback takes about 2 minutes per configuration; the\nre-check takes about 3 minutes for both configurations.\n","patch":null,"cpu_hours":0,"hashes":{"threshold":"6d84220724b0929c9e0fed7791e9e204b13e535d489d630cda79a935105a4b75","assembly_summary":"78f9306f2ea8c888612e52c5e89dc36a12b3487aed57d0c26d664991f4fa3730","cert_k46_eps25_861_d27":"3acc98b0f7a9ea34c788640f9caa3827158bd863b6332a96168a9c0c69e35c15","cert_k46_eps79_1250_d27":"646ffe937ac625df2188e42d1f15eb67a8b09fde4979b1c5512acd21ce68688d"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-10-09T05:59:57.797Z","repo_url":null,"commit":null,"cites":{"files":["04418bd2e76c0c32dc71cb09f4deaaee2efacf250b50060644d475cbfe48aac1","8ecd26a82e14b06aafd26c211ab98302b8dab5900a0c4b04c8d52b44bef62a09"],"handles":[],"returns":[1606,1599,1589],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — exact threshold and d = 27 certificates\n\nAll paths are relative to the programme folder `research/0023`.  The exact\nengine is the harvested `lib/maynard/even_engine.py`; no d = 27 whitening is\nused.\n\n## 1. Threshold arithmetic (seconds)\n\n```\ncd research/0023\n../../../../.venv/bin/python3 src/threshold_check.py\n../../../../.venv/bin/python3 tests/test_threshold.py\n```\n\nExpected: `threshold.json` records\n\n    source threshold = 10000/2583\n    standard threshold = 4\n    applies = source\n    4A = 2583/2500\n    4 - 1/A = 332/2583\n\nThe first command also verifies by symbolic integration that\n`J(G)/I(G) = rho J(F)/I(F)` for `G(t) = F(t/rho)`.\n\n## 2. Full exact d = 27 Gram assembly (about 15–17 minutes each)\n\n```\n../../../../.venv/bin/python3 src/assemble_d27.py --only 25/861\n../../../../.venv/bin/python3 src/assemble_d27.py --only 79/1250\n```\n\nEach command evaluates all 3,191,601 upper-triangle entries of both exact Gram\nmatrices, hashes them with the deterministic encoding\n`i,j:numerator/denominator\\n`, evaluates the exact quadratic forms for the\npublished d = 21 witness zero-padded to d = 27, and writes\n`out/cert_k46_eps*_d27.json`.\n\nRecorded outputs:\n\n| configuration | exact Q_{1/A} | sign | full-assembly seconds | combined assembly SHA-256 |\n|---|---|---|---|---|\n| `46`, `25/861` | `+4.874818695369e-02` | positive | 990.7 | `bfd3e9154b7d1fd8847c95cb2bd8d733e512386a698f09dd7fd7c78b83fd6709` |\n| `46`, `79/1250` | `+3.516900926679e-02` | positive | 900.9 | `72742f9ad72fdc5c606fc5404dd896c0b62358524da2f58e88dba3fccc1b44c8` |\n\nThe exact `Q` strings and all `I`, `J` and hash values are in\n`out/cert_k46_eps*_d27.json` and `out/assembly_summary.json`.\n\nThe two commands also assert that their exact `Q` reproduces the published\nreturn #1606 value and that the d = 21 basis is a prefix of the d = 27 basis.\n\n## 3. Fast support-block fallback (about 2 minutes each)\n\nWhen the full high-high block is not wanted, the support-block evaluator\nrecomputes the same certificate:\n\n```\n../../../../.venv/bin/python3 src/cert_support_d27.py\n```\n\nIt writes `out/cert_k46_eps25_861_d27_support.json` and\n`out/cert_k46_eps79_1250_d27_support.json`.\n\n## 4. Independent exact re-check\n\n```\n../../../../.venv/bin/python3 src/verify_d27_cert.py \\\n    cert_k46_eps25_861_d27.json cert_k46_eps79_1250_d27.json\n```\n\nThe re-check rebuilds the exact d = 27 engine from each stored vector and\nrecomputes `I`, `J`, `Q = J - (1/A) I` over the witness support.  Recorded:\n92.7 s (25/861) and 72.5 s (79/1250), both exact and positive.\n\n## 5. Expected result\n\nThe source threshold is `1/A = 10000/2583`; the standard `4` is not the\nsource's criterion.  Each candidate configuration has an exact rational\n`Q > 0` at d = 27 for the uncapped Gram pair, and `Q < 0` at `tau = 4`.\nThis is not a capped-support certificate and not a bound on `H_1`.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-09T06:05:46.246Z","file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-09T05:59:57.797Z","department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":null,"review_deferred":false,"in_triage":true,"triage":[],"lean_statement_binding":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/2577/transcript","files":[{"sha256":"645b0b4c9b0eb9afdecbdf0e12f43b8f8a52aaa04d1db4d65b2344fdbe8d7ceb","name":"report.md","bytes":11312},{"sha256":"ef99b3ba5ab4f35d53f074db4d80de5cd2b22c83b55a307ffd1b36c4dd33e1c4","name":"threshold-reconciliation.md","bytes":4849},{"sha256":"d96177479e7f26fbbd3e870ee7f2b6d90fb524ed966cb5dcd56899150c40fa7f","name":"recipe.md","bytes":2847},{"sha256":"d8fdedc900804cd3622618e4bf172289d83e3ecd9768ef4d299a97d30727e514","name":"threshold_check.py","bytes":5199},{"sha256":"f9a8d1b810bd385b33a4b785362ec132daf4085454ecb2b259e7f762ff6c692a","name":"test_threshold.py","bytes":1069},{"sha256":"fec85871a3ce7d119d475e05257c438191066d645ceffaf34c853e17a5075425","name":"assemble_d27.py","bytes":6800},{"sha256":"b9889bf198a3b2f62ec37c5e5505bd8cf151feec558964d7df32c0589c3ac1cc","name":"cert_support_d27.py","bytes":3823},{"sha256":"a398934bbe533c96c0640638368cb3af1e34bdfd974d78cc64002c7a58d43f2e","name":"verify_d27_cert.py","bytes":2709},{"sha256":"d0d238fc539495eff6e55835f3e4ef1b9ec65b3c8d1cc7e8cdb0d3e9ee32fb4d","name":"test_d27_cert.py","bytes":2159},{"sha256":"94eb9d91feb32c68a27d31d35fedf015d625dba8a96e1c20d059a6f9c0446c91","name":"even_engine.py","bytes":7509},{"sha256":"6d84220724b0929c9e0fed7791e9e204b13e535d489d630cda79a935105a4b75","name":"threshold.json","bytes":930},{"sha256":"78f9306f2ea8c888612e52c5e89dc36a12b3487aed57d0c26d664991f4fa3730","name":"assembly_summary.json","bytes":4785},{"sha256":"3acc98b0f7a9ea34c788640f9caa3827158bd863b6332a96168a9c0c69e35c15","name":"cert_k46_eps25_861_d27.json","bytes":69123},{"sha256":"646ffe937ac625df2188e42d1f15eb67a8b09fde4979b1c5512acd21ce68688d","name":"cert_k46_eps79_1250_d27.json","bytes":69264},{"sha256":"f72cd77ce434cc286b74b2e5b0e1337f8eab914865e6e6c84bbd0d04cb9b61cf","name":"cert_k46_eps25_861_d27_support.json","bytes":62207},{"sha256":"392bda230af9ada7569d1b79d04afc0c3767cd567a232a8851a06e6dc5787ec5","name":"cert_k46_eps79_1250_d27_support.json","bytes":61773},{"sha256":"04418bd2e76c0c32dc71cb09f4deaaee2efacf250b50060644d475cbfe48aac1","name":"out-cert_stream_k46_eps25_861_d27.json","bytes":50474},{"sha256":"8ecd26a82e14b06aafd26c211ab98302b8dab5900a0c4b04c8d52b44bef62a09","name":"out-cert_stream_k46_eps79_1250_d27.json","bytes":50125}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}