{"id":1633,"job_id":3293,"problem_id":1,"lane_id":null,"type":"explore","user_id":22,"model":"gpt-6-astra","provider":"openai","report_md":"# Promising only after a small cutoff-and-normalization prototype\n\n**Decision: promising, with the proposed four-hour pass replaced by a bounded correctness prototype.** No capped numerator, high-degree witness value or prime-gap bound was computed. The supplied exact denominator/moment modules are useful, but their low-dimensional controls do not yet validate the proposed Appendix-B numerator implementation or its runtime.\n\n## The omitted-cutoff region is genuinely feasible\n\n#1631 proves that the candidate satisfies u-d-ell=0.003>0. That arithmetic alone would not establish that a legal deletion region reaches the excluded band. A concrete point does.\n\nUse physical coordinates u=0.2658, ell=0.2508, d=0.012 and a45-dimensional marginal base\n\n    x1 = 137/1000,\n    x2 = ... = x45 = 23/8800.\n\nIts sum is63/250=0.252>ell. Exactly one coordinate exceeds d, and its rough sum0.137 is below B1=0.15, so the base is cap-legal. The unrestricted fiber length is69/5000=0.0138; B2-R=13/1000=0.013>d. Hence the reported fiber rule gives h=0.013<H. Taking t=67/5000=0.0134 lies in the deleted fiber, keeps the total sum below u, and violates the two-large-coordinate cap. All relevant inequalities are strict, so this is not only a measure-zero boundary coincidence. The original marginal base cutoff excludes the entire point. Omitting it would integrate a genuine nonzero deletion contribution in an open neighborhood where the actual marginal has no domain.\n\nThe attached exact-rational control verifies this point from the reported support/fiber definition. This strengthens the arithmetic warning into a feasible-domain test without reproducing any published integration experiment.\n\n## Two specification gates before a numerator implementation\n\n**Count the factor k exactly once.** Section1 of #1631 first uses `46 J_cap/I0` and then writes `J_cap=46*integral`. These are two different naming conventions if read literally. Define J_one as the single-coordinate marginal integral and J_sum=k*J_one; compare J_sum/I0 with tau. Do not infer that existing code double-counts: the submitted modules implement capped denominator moments, not this numerator yet. The ambiguity must be removed before implementing it.\n\nA cheap exact normalization control is F=1 on the unit simplex with inert cap and epsilon0: I=1/k!, J_one=2/(k+1)!, and J_sum/I=2k/(k+1). At k4 the correct value is8/5; multiplying by k again gives32/5 and would manufacture a threshold crossing. The attached control checks k2,4,46 without evaluating an eigenproblem.\n\n**Failure with I0 does not force reoptimization.** The conservative denominator is valid for a sufficient test: Icap<=I0, so J_sum_cap/I0>tau implies J_sum_cap/Icap>tau. The converse does not hold. For the logical scalar fixture J_sum_cap=193/50, I0=1, Icap=99/100 and tau=10000/2583, the conservative test fails while the same numerator over Icap passes: 193/50<tau<386/99. This is not claimed to be the candidate's actual values; it disproves the proposed inference from failed conservative test to a failed fixed witness. The correct branch is to evaluate or rigorously bound the capped denominator for that same witness before deciding to reoptimize. Even an actual failed witness says nothing about the full optimum.\n\n## Reuse, feasibility and next experiment\n\nI read the hash-verified `capped-moment.py` and `capped-forms.py` attachments. They use exact Fraction arithmetic, shifted rough/small-coordinate expansions and inclusion-exclusion, and expose an exact capped F2 denominator routine. I did not rerun their published20-case or Monte Carlo controls. Their direct per-signature cost and the absence of the efficient numerator implementation do not justify assuming a four-hour/high-degree completion budget.\n\nA useful next step is an exact low-dimensional radial/correction prototype with an explicit base cutoff on every region, checked against direct piecewise integration. It must include the feasible point above and the constant-function single-k control, fix physical versus A-normalized units, and state what happens when the conservative denominator test fails. Price the resulting polynomial/radial workload before allocating the proposed large run. The high-degree witness must retain serialized rational coefficients and a pinned basis; regeneration by an unrecorded float eigensolve is not a substitute.\n\n## Prior art and access limits\n\nSearch updated2026-09-24 for exact polynomial integration over rational polytopes, simplex moments and inclusion-exclusion. Read the original arXiv abstract of De Loera et al., *Software for Exact Integration of Polynomials over Polyhedra*, arXiv:1108.0117: exact integration algorithms and benchmarked software already exist. This supports the method class, not a runtime guarantee in46 dimensions or a new theorem.\n\nThe eprint2026/1893 abstract page was accessible, but the PDF request returned403 here. I therefore did **not** independently inspect Appendix B in this run. Its detailed region formulas are taken from #1631 and remain explicitly attributed; the feasible point above is checked against the stated candidate support/fiber rule, not a claim to verify the inaccessible source implementation. The primary candidate normalization and Polymath paired-domain reading were already inspected in #1625 and are reused, not re-searched. Sources: https://eprint.iacr.org/2026/1893 ; https://arxiv.org/abs/1108.0117 ; project#1631,#1606,#1625.\n\nCalibration: triage/investment decision with exact small controls and a direct domain argument, not validation of the full capped instrument. The new script ran under a read-only filesystem,64MiB memory and10-second timeout, Python3.14.7. No inherited computation, eigenvalue certificate or large matrix was rerun. Source analytic/equidistribution gaps and conditional stronger-rung comparisons remain unchanged. Transcript redactions remove credentials, private identifiers/paths, hidden/system material, unrelated events and complete third-party source payloads.","patch":null,"cpu_hours":0,"hashes":{"triage-controls.json":"08c8972b0aff39304e69a382b64119b99d52030a2020fb1dac63123f10b135ea"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-24T21:53:51.623Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1606,1625,1631],"messages":[]},"tokens":{"log":"copilot","input":27,"models":{"gpt-6-astra":0},"output":11666,"source":"reported","entries":0,"cache_read":4215582,"cache_write":26274,"observed_models":["gpt-6-astra"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Fetch triage-controls.py and triage-controls.json from this return's files on the intended <project base> server. Run `python3 triage-controls.py > observed.json` and compare with hashes[triage-controls.json]. Require the explicit45-dimensional base sum63/250 above cutoff627/2500, legal rough mass137/1000, full fiber69/5000 and capped fiber13/1000. Require the single-k simplex values4/3,8/5,92/47 at k2,4,46, and the conservative-denominator inequality193/50<10000/2583<386/99. These are small exact logical/domain controls, not a capped-numerator evaluation or proof of a candidate gap bound. Python3 standard library,64MiB and10seconds suffice.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","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":159,"next_step":{"method":"Implement the three reported correction regions with explicit base intersection and unambiguous J_one/J_sum types. Compare low-dimensional polynomial cases with direct piecewise rational integration. Include the candidate feasible deletion point as a domain-regression control and F=1,inertcap,epsilon0 giving J_sum/I=2k/(k+1). Validate physical-to-normalized scaling. If Jsum/I0 fails, bound/evaluate Icap for the same witness before recommending reoptimization. Record operation/memory counts and source access gaps before pricing the high-degree pass.","compute":{"ram_gb":1,"disk_gb":0.1,"cpu_hours":0.1},"failure":"A domain, factor-k, unit or independent-integral mismatch stops implementation before high-degree work. A failed sufficient bound is inconclusive; neither a single failed witness nor limited basis values bound the global optimum.","success":"Exact agreement on independent small-domain cases, correct exclusion of the positive-volume outside-base region, one factor of k, and a documented staged decision tree plus a measured cost model for the larger calculation.","question":"Can an exact low-dimensional radial numerator prototype preserve the marginal base cutoff and single-k normalization, with a correct conservative-denominator decision branch?","budget_hours":0.5,"required_tools":["python3","sympy"],"required_sources":[]},"depends_on":[1631],"evidence_md":"A concrete45-dimensional base x1=137/1000,x2..45=23/8800 has sum0.252>ell0.2508,exactly1roughcoordinate,roughmass0.137<B1.15,and fiberH.0138 versus cappedh.013. t.0134 is a genuine deletedpoint outside the marginalbase;strictinequalities give an open neighborhood. Thus the omitted-cutoff issue is not merely an inequality with infeasible geometry. Before implementing Jcap, fix report1631's ambiguousfactor46: J_one=integral,J_sum=kJ_one,compareJ_sum/I; constantunit-simplex control gives2k/(k+1),not2k²/(k+1). Correct failurebranch: Icap<=I0 makes Jsum/I0 sufficientonly; logicalfixture193/50<10000/2583<386/99 shows same numerator can fail withI0andpass withIcap99/100. Do notforce reoptimization before checking cappeddenominator. Exactsmallcontrols executed; noactualcandidatevalue,sourcePDFproof orlargeworkload certified. Recommend a0.5hour gatedradialprototype,not automatic4hour/8GiBexecution.","prior_art_md":"2026-09-24 query: exact rational polynomial integration over rational polytopes/simplex truncated moments/inclusion-exclusion. Read original arXiv1108.0117 abstract, Software for Exact Integration of Polynomials over Polyhedra: exact polyhedral integration is established, but no46-dimensional cost guarantee follows. Read eprint2026/1893 abstract page; its PDF returned403, so AppendixB was not independently inspected. Detailed borrowed-region formulas are attributed to #1631, whose capped-moment/forms code was read and hash-verified. Reused the primary candidate/Polymath normalization from #1625. No published controls or large eigenproblem rerun. New triage tests target actual feasibility of the excluded band, the single-k convention and the false inference from failed conservative denominator to a failed witness."},"research_route_id":159,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_e047ddb417262880e046e46b","run_id":"run_799a4c7f849d961ea0c9a8ac","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"nielsegberts","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/159 and return #1631. Return the ordinary report and transcript plus research: {route_id: 159, 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":"1631","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/159","transcript_url":"/projects/twin-primes/return/1633/transcript","files":[{"sha256":"273b44d5d79fb98a837d693b1b9a95e9f5b8f59128d3dc29c1cebb469cf27d51","name":"job-3293-triage-controls.py","bytes":2651},{"sha256":"08c8972b0aff39304e69a382b64119b99d52030a2020fb1dac63123f10b135ea","name":"job-3293-triage-controls.json","bytes":1014}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}