{"id":1642,"job_id":3382,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Triage — route 160 (job 3382)\n\nJob: explore/**triage**, general mode, route #160 revision 1 (state `proposed`), origin return #1641.\nAttempt `12b3a1266db3d5c7750825f2d15e6be8`. Decision asked: **is one bounded next experiment\njustified?**\n\n**Decision: `promising`.** One bounded experiment is justified: build the capped Gram pair\n`(M2^cap, M1^cap)` on the T_46 cap and take the top generalised eigenvector as the re-optimised\nwitness. The prior recorded negative certificate is a *calibration* result about one witness, the\ncapped forms are constructible fixed quadratic forms, the direction of the optimum is fixed\na priori, and no inspected source computes a capped witness for this candidate.\n\n## 1. What is being triaged\n\nRoute #160 is a one-question route. The candidate (Althoefer, *H1 <= 216*, eqs. (1)–(4)) restricts\nthe support to T_46 with `A = 2583/10000`, `eps_s = 3/400`, `delta = 3/250`, `B1 = B2 = 3/20`,\n`B_m = 4/25` (`m >= 3`). The source criterion is `46 J(F) > I(F)` on that restricted support, and\nStadlmann Prop. 1 (`arXiv:2608.31126`) fixes its threshold at `1/A = 10000/2583 = 3.8714672861…`\n(#1606). Route #160 asks whether the capped optimum clears it:\n\n> Does some symmetric `F` supported on `T_46` satisfy `M^cap_{46,25/861}(F) > 1/A`?\n\nReturn #1641 answers the preceding question exactly: for the **#1606 uncapped Ritz witness** the\ncapped quadratic form is negative (`-0.10191368629446095`, exact rational, sign Lean-checked). So\nthe uncapped witness is not the right test function for the restricted support. The route is now a\nwitness-optimisation question, and that is exactly what is being triaged.\n\n## 2. Prior-art update (2026-09-25)\n\nFull record in `prior_art.md`. The decisive line: **no inspected source evaluates the capped\nquadratic form for the k = 46 candidate**, and the paper that owns the cap machinery\n(eprint 2026/1893) marks the reproduction of its *own* capped numerator as \"remains pending\"\n(Prop. 4.21). The neighbouring bounds are recorded (Song 236, Stadlmann 240, OpenAI 186, Axiom 212)\nand are a *value* caveat (§5), not coverage of this experiment.\n\n## 3. Weakest assumption\n\nThe one load-bearing borrowed step is that the exact capped form is a **fixed quadratic form** in\n`F`: the corrections are region integrals of `F^2` (numerator `E_C, E_D, E_E`; denominator\n`Delta = int_{s<U, R_r>c_r} F^2`), hence fixed symmetric bilinear forms, so\n`M2^cap = M2 - (region forms)` and `M1^cap = M1 + Delta-form` are well defined and the generalised\neigenproblem is well posed. This holds structurally. The live risk is bookkeeping, not structure:\n#1631 *proved* the source's Appendix-B premise `u - d < ell` fails for this candidate, so the base\ncutoff must be imposed on `C_r, D_r` (#1641's fix). A newly built capped pair must therefore be\nre-validated on the `k = 4` controls (inert caps -> zero correction; a real `k = 4` cap matches\nbrute force to 0.1%; occurrence path == Laplace path exactly; `violation_F2 == I_0 - I_cap`)\nbefore any eigenvector is trusted.\n\n## 4. Mapping the borrowed method onto this problem\n\n* **Borrowed:** 2026/1893's capped-moment identities (Lemma 4.19, Eqs. (76)–(77)), fiber geometry\n  (Lemma 4.20, Eq. (80)), correction regions (Eqs. (98)–(100)), Appendix B; #1631's exact\n  capped-moment instrument; #1641's radial/Laplace transform of `m_nu(X_shifted)` (replacing an\n  occurrence-wise expansion that reaches ~2M terms at `k = 46`) and integer-scaled exact\n  integration.\n* **Assumption that does NOT transfer unmodified:** `u - d < ell` (Appendix B) — proven false here\n  by #1631; the base cutoff is imposed on `C_r, D_r` instead.\n* **Assumption of the candidate:** the `eps = 79/1250` configuration is moot — its uncapped margin\n  is smaller, so it cannot certify once `25/861` fails.\n\n## 5. Recommended experiment (smallest credible check on the uncovered step)\n\n**Cheapest decisive probe first, then the full optimisation.**\n\n1. **Low-dimensional exact probe (cheap).** Evaluate the exact capped form on the span of the\n   uncapped top Ritz vector `c` and the all-small-stratum profile `s` (and, if useful, a\n   two-parameter family `c + a s`), exactly (rational). Because the cap deletes the `r >= 1`\n   strata, the pre-registered prediction is that mass moves toward the all-small stratum. If\n   `c^T Q_cap c` can be pushed above `1/A` inside this tiny family, the full optimisation is\n   vindicated cheaply; if it cannot, that is evidence (not proof) that the route is dead.\n2. **Full capped eigen-optimisation (only if step 1 is positive or inconclusive).** Build\n   `(M2^cap, M1^cap)` by the same machinery (the corrections are additive to the pair), take the\n   top generalised eigenvector, re-evaluate the exact rational form, and state the exact sign.\n\n**Bounded next step (route `next_step`).** question: does some symmetric `F` on `T_46` satisfy\n`M^cap_{46,25/861}(F) > 1/A`? method: build the capped Gram pair (or run a Krylov iteration\nagainst the capped quadratic forms) and take the top generalised eigenvector as the new witness,\nre-evaluating the exact form. success: an exact rational witness with\n`c^T(M2^cap - (1/A)M1^cap)c > 0`. failure: every converged capped witness stays below `1/A`,\nwhich removes the variational route for the H1 <= 216 candidate. budget: **4 h**\n(`ram_gb 16`, `disk_gb 1`; `python3`, `sympy`, `python-flint`).\n\n**Falsifier fixed before any run.** Every eigenvector is judged by the exact rational sign, never\nby a floating Rayleigh quotient; and any newly built capped pair that does not reproduce the\n`k = 4` controls is rejected as an instrument defect (the #1631 class), not reported.\n\n## 6. Scope / not claimed\n\nNo bound is proved or improved here; the twin prime conjecture is open; no asymptotic claim. The\ncapped pair has not been built in this triage. The value `H1 <= 216` is dominated by published\nbounds (Song 2026/1893: 236; OpenAI short_gaps 2026-08-30: 186; the Axiom `bgp212` bundle: 212),\nso a successful certificate calibrates the method rather than advancing the record bound. This is\na value caveat, disclosed, not a refutation of the experiment.\n\nHandle note: 61 of @Benjaminsen's returns wait for a verdict (26 on deepseek-v4-flash).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-25T05:09:49.930Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1606,1610,1631,1641],"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":"promising","route_id":160,"next_step":{"method":"Optimise on the capped support. First a cheap exact probe: evaluate the exact capped quadratic form on the span of the uncapped Ritz vector and the all-small-stratum profile (the cap deletes the r>=1 strata, so the pre-registered prediction is that mass shifts toward the all-small stratum). If positive or inconclusive, build the capped Gram pair (M2^cap, M1^cap) with the #1631/#1641 machinery -- the corrections are region integrals of F^2, hence additive fixed quadratic forms -- or run a Krylov iteration against the capped forms; take the top generalised eigenvector as the new witness and re-evaluate the exact rational form. Re-validate any newly built pair on the k=4 controls before trusting an eigenvector. Compute: ~0.1 CPU-h for the probe, ~4 CPU-h for the full optimisation (16 GB RAM, 1 GB disk); tools python3, sympy, python-flint.","compute":{"ram_gb":16,"disk_gb":1,"cpu_hours":4},"failure":"Every converged capped witness stays below 1/A (the cheap probe cannot push the exact form above 1/A and the full optimisation agrees, no floating sign used), which removes the variational route for the H1<=216 candidate.","success":"An exact rational witness with c^T(M2^cap-(1/A)M1^cap)c > 0, i.e. the capped-support certificate; or, from the cheap probe alone, an exact rational direction in the 2-dimensional family with the same sign, which locates the re-optimisation.","question":"Does some symmetric F supported on T_46 satisfy M^cap_{46,25/861}(F) > 1/A = 10000/2583, i.e. is the capped optimum above the source threshold even though the #1606 witness is not?","budget_hours":3.5,"required_tools":["python3","sympy","python-flint"],"required_sources":[]},"depends_on":[1606,1631,1641],"evidence_md":"# Evidence — triage, route 160 job 3382 (run-2026-09-25-a)\n\n## What the evidence changes\n\nRoute #160 is a one-question route: does some symmetric `F` supported on `T_46` clear\n`M^cap_{46,25/861}(F) > 1/A = 10000/2583 = 3.8714672861…`? The recorded return #1641 answers the\npreceding question exactly: for the #1606 uncapped Ritz witness the capped form is negative,\n`c^T(M2^cap - (1/A)M1^cap)c = -0.10191368629446095` (exact rational; sign Lean-checked in\n`lean/CappedCertificateNegative.lean`). The cap costs ~8.2% of the numerator `J` and ~4.9% of the\ndenominator `I`, so the capped Rayleigh quotient falls to `3.7642958695…`, below `1/A`. Both the\nsource's sufficient route (`J_cap/I_0 > 1/A`) and the exact form fail.\n\nThat is a **calibration** result about one witness, not about the candidate: it removes the\nuncapped Ritz vector as the test function and leaves exactly one uncovered question — whether some\n*other* `F` on the same capped support certifies.\n\n## Why one bounded experiment is justified (specific evidence)\n\n1. **The object is constructible.** Each cap correction is an integral of `F^2` over a region of\n   the support (numerator `E_C, E_D, E_E`; denominator `Delta`), hence a fixed symmetric bilinear\n   form: `M2^cap = M2 - (region forms)`, `M1^cap = M1 + Delta-form`. The capped generalised\n   eigenproblem is therefore well posed. #1631 already built and validated the exact capped-moment\n   instrument: inert caps give zero corrections; a real cap at `k = 4` matches brute-force\n   quadrature to 0.1%; the occurrence and Laplace paths agree exactly on `C, D, E`; and\n   `violation_F2 == I_0 - I_cap` from the independent capped denominator.\n2. **The direction of the optimum is fixed a priori.** The cap deletes the `r >= 1` strata, so the\n   optimum must shift mass toward the all-small stratum. That is a pre-registerable prediction,\n   not a free parameter — and it is exactly what the chosen probe tests.\n3. **The instrument prerequisites are met.** #1641's radial/Laplace transform replaces an\n   occurrence-wise expansion that reaches ~2M terms at `k = 46` and is validated exactly against\n   the occurrence path; integration is integer-scaled. `python3`, `sympy`, `python-flint` are\n   available (used by #1631).\n\n## Weakest assumption (mapped)\n\nThe load-bearing borrowed step is that the exact capped form is a *fixed* quadratic form in `F`.\nThe `E`-regions and `Delta` are region integrals of `F^2`, so this holds structurally; the live\nrisk is bookkeeping, not structure. #1631 proved the source's Appendix-B premise `u - d < ell`\nfails for this candidate, so the base cutoff must be imposed on `C_r, D_r` (#1641's fix). A newly\nbuilt capped pair must be re-validated on the `k = 4` controls before any eigenvector is trusted.\n\n## Domination caveat (scope)\n\nThe value `H1 <= 216` is dominated by published bounds — Song 2026/1893 (`H1 <= 236`), OpenAI\nshort_gaps 2026-08-30 (`H1 <= 186`), the Axiom `bgp212` bundle (212). A successful certificate\nwould calibrate the exact capped-support method, not improve the record bound. Disclosed as a\nvalue caveat, not a refutation.\n\n## Not claimed\n\nNo bound is proved or improved. The capped Gram pair was not built in this triage (0 CPU-h). No\nfloating sign is ever used as the decision. No asymptotic claim; the twin prime conjecture is open.","prior_art_md":"# Prior art — route 160 triage (search 2026-09-25, run-2026-09-25-a)\n\nQuestion searched: has any source evaluated or re-optimised a variational witness on the *capped*\nsupport (the Althoefer `H1 <= 216` support `T_46`) for the `k = 46, eps = 25/861` candidate, and\ndoes any source compute the capped Gram pair `(M2^cap, M1^cap)`?\n\n## Sources inspected\n\n* **eprint 2026/1893**, Z. Song & S. Yue, *Bounded Gaps Between Primes: An Upper Bound of 236*\n  (approved 2026-09-09), <https://eprint.iacr.org/2026/1893> — supplies the exact capped-moment\n  identities (Lemma 4.19; Eqs. (76)–(77)), the fiber geometry (Lemma 4.20, Eq. (80)), the\n  correction regions `C_r, D_r, E_r` (Eqs. (98)–(100)) and Appendix B rules used by #1631/#1641.\n  **Its own Proposition 4.21 states that reproducing its printed capped numerator \"remains\n  pending\" at that revision** — it prints no capped witness.\n* **Althoefer**, *A Checked Candidate Extension of Stadlmann's Method to H1 <= 216*\n  (`outputs/threshold/H1_216_candidate.pdf`) — eqs. (1)–(4) fix `A = 2583/10000`,\n  `eps_s = 3/400`, `delta = 3/250`, `B1 = B2 = 3/20`, `B_m = 4/25` (`m >= 3`).\n* **J. Stadlmann**, arXiv:2608.31126 (`H1 <= 240`) — Prop. 1 fixes the threshold\n  `1/A = 10000/2583` (#1606).\n* **OpenAI**, *Improved short gaps between primes* (2026-08-30), `H1 <= 186`\n  (`cdn.openai.com/pdf/…/short_gaps.pdf`) — abstract/CDF-level only here (PDF not text-extractable\n  in this environment).\n* **Axiom** `bgp212` bundle (212) — as recorded by #1608.\n* Route-record returns: **#1606** (threshold `1/A`; `d = 27` exact certificates), **#1610**\n  (banked restartable Ritz step), **#1631** (exact capped-moment instrument + base-cutoff\n  obstruction), **#1641** (this route's origin: the negative capped certificate), plus #1589/#1599.\n\n## Searches run (2026-09-25)\n\n1. `bounded gaps between primes upper bound 236 capped support certificate variational witness`\n   → the bound papers themselves (2026/1893; OpenAI short_gaps; Stadlmann; Polymath8b; Zhang).\n2. `Ritz eigenvector optimisation capped support certificate Maynard k=46 admissible tuple\n   marginal base cutoff` → numerical-linear-algebra results only (sparse/truncated Rayleigh–Ritz,\n   verified eigen-computation); no number-theory carrier.\n3. `\"capped\" OR \"restricted support\" variational certificate witness re-optimised H1 216\n   Althoefer Stadlmann exact rational` → unrelated results.\n\n## Exact remaining gap\n\nWhether some symmetric `F` supported on `T_46` satisfies `M^cap_{46,25/861}(F) > 1/A`. No inspected\nsource computes it; the paper that owns the cap machinery marks its own capped-numerator\nreproduction as pending; and return #1641 supplies the exact instrument plus the (negative)\ncalibration value for the uncapped witness. The gap is therefore a clean\nwitness-optimisation question on a *constructible* capped pair, not an external-prior-art gap.\n\n## Difference from the closest work\n\n2026/1893 builds capped-moment identities for **its own** (`236`) configuration and leaves its\ncapped numerator uncomputed. This route asks for the capped **optimum** at the Althoefer\n`H1 <= 216` parameters, where #1631 proved the source's Appendix-B premise `u - d < ell` does not\napply and the base cutoff must be imposed on `C_r, D_r`."},"research_route_id":160,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_b81e5ce2fdc81eae0781da8b","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in triage. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/160 and return #1641. Return the ordinary report and transcript plus research: {route_id: 160, 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":"1606","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1631","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1641","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/160","transcript_url":"/projects/twin-primes/return/1642/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}