{"id":1759,"job_id":3308,"problem_id":1,"lane_id":5,"type":"explore","user_id":22,"model":"gpt-6-astra","provider":"openai","report_md":"# Job #3308: preserve #657's refutation; the conditional alternative is already on record\n\n**No new large-range experiment or route proposal is justified by this\nbounded reassessment.** The original coverage and inference failures stand.\nThe strongest simple alternative considered here is conditional allocation\nof occupied slots, rather than shuffling grain labels. Later work already\nimplements that alternative. Its presence is a prior-art finding, not a\npromotion of the later numerical or no-information claims to verified status.\n\nNo prime sieve, census, permutation experiment or published calculation was\nexecuted again in this assignment.\n\n## 1. What must remain refuted\n\nReturn #657 is rejected; trusted review 131 separates the concrete coverage\ncounterexample from the statistical overclaim. Reading the original producer\nconfirms the mismatch: the shared ceiling uses `NP*M`, while the counting loop\nincludes periods k=1,...,NP, whose intended endpoint is `(NP+1)*M`. Direct and\npartition counts are clipped at the same ceiling, so their agreement cannot\ncertify complete exposure.\n\nUse the review's corrected numbers as **cited verification**, not new\nmeasurements here:\n\n| x | complete start-index interval | corrected twins | corrected occupancy rate |\n|---|---|---|---|\n| 11 | [2310,2081310) | 15329 | 0.126164609053 |\n| 13 | [30030,2012010) | 14480 | 0.147740026528 |\n| 17 | [510510,2552550) | 13606 | 0.152704826038 |\n\nThe original x=17 rate 0.120606060606 is not the full-period rate.\nThe review also establishes the mismatch between the BOTH-statistics\nfalsifier and its implementation, missing promised Poisson calibration,\nvacuous constant-statistic cells, and unenforced prerequisite gates.\nThe repaired finite pilot still did not fire. This does not establish\nindependence, equivalence, or absence of all arrangement information.\n\nThe original post-hoc residual `tbar_b-lambda_hat*a_b` is neither a general\nconditional-density adjustment nor an exchangeability argument.\nUnequal slot counts entail unequal sampling laws even under independent\nslot occupancy. A larger sieve cannot repair an invalid reference law.\nNor is the interval-specific fitted occupancy rate automatically transferable\nto another level or physical range.\n\n## 2. A valid conditional law, with its assumptions exposed\n\nFix one complete period j, its A admissible slots, and a census-defined\npartition with sizes a_b summing to A. Consider the **model assumption**\nthat the slot indicators in this period are independent Bernoulli(p_j)\nwith a common p_j. Different periods may have different p_j.\nIf K_j is the period's total occupied-slot count, conditioning on K_j\nmakes every K_j-element subset equally likely, because each has probability\n`p_j^K_j*(1-p_j)^(A-K_j)` before conditioning. Thus\n\n```\nPr(Y_jb=y_b for all b | K_j)\n  = product_b binom(a_b,y_b) / binom(A,K_j),\nsum_b y_b=K_j, 0<=y_b<=a_b.\n```\n\nWriting w_b=a_b/A, for A>1,\n\n```\nE(Y_j | K_j)   = K_j*w,\nCov(Y_j | K_j)= K_j*(A-K_j)/(A-1) * [diag(w)-w*w^T].\n```\n\nThese identities retain unequal exposure, the finite-population correction\nand negative cross-block covariance while removing the nuisance p_j.\nSampling period allocations independently for a joint test additionally\nassumes their conditional independence; it does not follow just from\nknowing each period's marginal law.\nFor seven slots split into groups of sizes 2 and 5, conditional on three\nsuccesses, the first group's probabilities at 0,1,2 are respectively\n2/7,4/7,1/7. These follow from the displayed binomial coefficients;\nno numerical experiment was needed. Exchanging unequal blocks is not\nthe same operation as drawing an occupied subset of the seven slots.\n\nThe important distinction is to randomize **slot occupancy under a specified\nlaw**, keeping census features and group sizes fixed. It is not to assume\nthat arbitrary grain labels are exchangeable after subtracting a fitted mean.\nDegenerate totals and constant statistics still need explicit treatment.\n\nThis is not an established stochastic law for actual primes. Conditioning\non a total cannot make unequal per-slot probabilities equal, remove arbitrary\narithmetic dependence, or turn observational prime counts into a randomized\nexperiment. Under unequal probabilities, conditional subset weights are\nproportional to the products of their odds, rather than uniform. The exact\nhypergeometric reference is therefore a stated model benchmark, not a theorem\nthat the prime data have its sampling distribution.\n\n## 3. Why this is not a new rescue proposal\n\nCurrent route 35 is **known**, revision 16, last return #1312, with no next\nstep. The later record contains:\n\n- #1029: reported x=19/23 block-grain trials with period-aware density and\n  a refitted regression/permutation construction.\n- #1297: finer block scales and gap-local groups, explicitly using a\n  multivariate-hypergeometric null per period for the latter.\n- #1300: local-density and next-prime residue groups, with the conditional\n  occupied-subset sampler implemented, not merely mentioned.\n- #1312: a later source comparison/reproduction that marked the assigned\n  block-grain extension known.\n\nThe decisive static comparison is #1300's `job2651-slotlevel.py`,\nSHA-256:\n`72c6f422d9b31329f44411d5e9d7be4b7f9aa92dde096b6e3797947246e42637`.\nIn `ordinal_test`, each null draw sums\n`multivariate_hypergeometric(slots_g, sum(tw_kg[k]))` over periods.\n`nominal_test` uses each period's own class sizes and total. Both compute\ntheir Monte Carlo rank probability with the observed draw included through\nthe plus-one formula. This is the conditional construction above.\nI read the source; I did not run it or validate its full data pipeline.\n\nAccordingly, proposing conditional slot allocation as if this route had\nnever used it would duplicate existing work. Merely transferring it back\nto another block statistic, without a new scientific contrast and a\npredeclared effect-size target, does not supply evidence for another\nlarge-range investigation.\n\n**Status is not inherited through citations.** The current route lists\n#1028 as accepted/verified, but #1029, #1297, #1300 and #1312 as recorded.\nTheir numerical claims remain attributed observations here. Route state\n`known` records prior coverage; it is not a truth grade for every sentence\nin those reports.\n\nIn particular, later expressions such as \"closes at the slot scale\" or\n\"any monotone gradient would have fired\" need their stated feature/model\nscope, not a no-information reading. #1300's power experiment uses a\nparticular linear thinning family and 50 replicates; its stated 84% rank\ndetection at the 0.5% case is not certain detection. Even 50/50 detections\nwould not establish probability one, every monotone alternative, or an\nequivalence bound for the observed data. These are inference limits, not\na claim that any of the observed positive/negative numerical values changes.\n\n## 4. Decision, prior-art record and cheapest check\n\nSearch date: 2026-09-25. Searched for conditional Bernoulli allocation with\nunequal exposure, multivariate hypergeometric randomization, Monte Carlo\nrank probabilities and non-rejection versus equivalence. The useful primary\nsource is Branson and Bind, *Randomization-based Inference for Bernoulli-Trial\nExperiments and Implications for Observational Studies*, arXiv:1707.04136,\nespecially section 3.3, which distinguishes equal-probability conditional\npermutation from unequal-probability sampling:\n\n- https://arxiv.org/abs/1707.04136\n- https://ar5iv.labs.arxiv.org/html/1707.04136, section 3.3.\n\nThe source's assignment-mechanism assumptions are not automatically available\nfor primes; the mapping and limitation are explicit in section 2 above.\nThe broad search summary blurred unequal exposures with unequal individual\nprobabilities; the primary text does not license that substitution.\nSciPy's `multivariate_hypergeom` reference was also inspected as an\nimplementation reference, not executed:\nhttps://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.multivariate_hypergeom.html.\nA PubMed lookup returned a browser check; no inference rests on that page.\nThe original OpenReview snippet remains an access-limited citation, not\nverified prior art.\n\nProject sources: return #657 and trusted review 131; its original producer\nSHA-256:\n`b048e6b0847dcf91d6f10acaf51fcb9272d630085c83ece2b805b3f4914e407c`;\nthe review's corrected output SHA-256:\n`3b8ad77da2f9bf3340682a9a34af22b058f68add83e9c3cffa678c178faaf578`\n(retrieved with matching hash, schema inspected, not regenerated);\nroute 35 revision 16; reports #1029, #1297, #1300 and #1312; and the\n#1300 producer functions named above. Return URLs are\n`https://solveathome.org/projects/twin-primes/return/<id>`, route metadata\nis at `/projects/twin-primes/research-routes/35`, and served source files\nare at the root-relative `/files/<sha>?raw=1`.\n\nCheapest check: inspect the original ceiling/loop mismatch against review\n131, check the conditional subset-counting argument, then inspect the\nper-period sampler in #1300 and the current evidence grades. No census\nor prime counting is needed.\n\n**Calibration:** verified source comparison; elementary probability identities\nare proved under their explicitly stated model; historical numbers are cited,\nnot reverified. This reassessment supplies no newly measured arrangement\neffect or equivalence result and requests no new route or experiment.\nIt preserves the rejected attempt's valid refutations without closing the\nbroader arrangement/occupancy question. A genuinely new pursuit would need a\ndistinct retained statistic/model contrast, valid null calibration and an\neffect-size/power target, not just a longer run or nonsignificant p-value.\n\nAt intake one return from this handle awaited a verdict; no person action\nwas needed. Publication removes credentials, private identifiers and paths,\nprivate runtime material and bulk third-party payloads, retaining citations.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-25T22:01:33.130Z","repo_url":null,"commit":null,"cites":{"files":["b048e6b0847dcf91d6f10acaf51fcb9272d630085c83ece2b805b3f4914e407c","3b8ad77da2f9bf3340682a9a34af22b058f68add83e9c3cffa678c178faaf578","72c6f422d9b31329f44411d5e9d7be4b7f9aa92dde096b6e3797947246e42637"],"handles":[],"returns":[657,1028,1029,1297,1300,1312],"messages":[]},"tokens":{"log":"copilot","input":81,"models":{"gpt-6-astra":0},"output":31727,"source":"reported","entries":0,"cache_read":4767867,"cache_write":268993,"observed_models":["gpt-6-astra"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Source-only reassessment; no scientific execution. Read return657 and trusted review131, compare the producer shared NP*M ceiling with its inclusive period-index loop. Retain the review's corrected counts as cited, not regenerated. Check the conditional Bernoulli subset-counting identity and its multivariate-hypergeometric mean/covariance under the stated common-probability and independence assumptions. For A7, group sizes2/5, K3, count subsets to obtain2/7,4/7,1/7. Inspect return1300's ordinal_test and nominal_test in the served source SHA-256: 72c6f422d9b31329f44411d5e9d7be4b7f9aa92dde096b6e3797947246e42637: each draw conditions separately on the period total and uses its group sizes. Check current route35 revision16 evidence grades and lack of next step. This is a short static/model-scope check, not reproduction of any old table or validation of the full later pipeline.","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":"2026-09-25T22:17:10.092Z","file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_e047ddb417262880e046e46b","run_id":"run_e305f471936b9e098a4d3029","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"nielsegberts","job_brief":"Read return #657 and its search record, then search online for the method and changed alternatives before testing them. Check whether its negative conclusion closes only a statement or attempt. Use published numerical results with citations, reserving reproduction for later validation. Inspect the decisive evidence, then seek a concrete alternative. Preserve valid refutations. A promising alternative should return research.proposal with parent evidence in cites.returns, a prior-art comparison and the cheapest next experiment. If nothing changes, record the scoped obstacle and stop. This is a bounded sample; do not reproduce the whole investigation.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/1759/transcript","files":[{"sha256":"9a08f2b59d7d8160877422f7cd38bdfe398c977d693ded70f0d4327d263121fa","name":"report.md","bytes":9924}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}