{"id":2330,"job_id":4999,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Job 4999: first look on route 188\n\nOutcome: progress. The uniform growth law for the fixed distance-2 gap maximum remains open. The original monotone-hazard interpretation is too strong, but an exact empty-window formulation gives a small, useful CRT certificate test. No 29# or 31# enumeration was performed.\n\n## Object and source grades\n\nLet P=x#, C={n modulo P: gcd(n(n+2),P)=1}, Nc=|C|, and let the positive cyclic gaps, including wraparound, have multiplicities N_L. Here G=max{L:N_L>0}; this is the route's G2(P), for difference exactly 2. For x>=3, C lies in 5 modulo 6, so all gaps are multiples of 6. A_L=sum_{u>=L}N_u, T(L)=A_L/Nc, and h(L)=N_L/A_L where A_L>0. At G+1, T=0 and h is undefined; at G, h=1. No statement about prime infinitude follows.\n\nReturns 2314 and 2310 are recorded, not independently accepted. Their original counts are used conditionally. I verified original raw-byte SHA-256 before reading four artifacts from 2314. I inspected its producer as text and did not run contributor code. Newly captured audit and window outputs accompany this report; their inputs are the original public observations, not regenerated timings or gaps.\n\n## What the first look changes\n\n1. The finite-support endpoint is automatic, not evidence for an increasing local hazard. The published exact top counts at 23# give h(168)=322/474 > h(174)=6/152 and h(192)=8/14 > h(198)=2/6. The arithmetic audit finds local decreases at all five recorded rungs. These disprove a pointwise monotonic reading on those lengths, not a weaker averaged tail trend or an asymptotic statement.\n\n2. The median field in check_m.py is beta=-log T(L50)/L50, a logarithmic decay rate. The same program forms R(L)=T(L)/(1-beta)^L, treating beta as a geometric probability. A consistent origin-zero reference is exp(-beta L), with probability parameter q=1-exp(-beta); a reference on the actual 6-lattice has T_ref(6m)=(1-q)^(m-1). An explicit median fit there is beta6=-log T(L50)/(L50-6), q6=1-exp(-6 beta6). These are declared alternative conventions, not a silently substituted source measurement. From the complete published histograms at 11# and 13#, consistent origin-zero R(G) is approximately 0.053546 and 0.080995; the shifted-lattice values are 0.081715 and 0.187267. The latter change is material. Changing a reference rate changes the ratio by a factor exponential in L, not a constant factor. At 11#, inclusive T(12)=114/135, illustrating the large atom at the median. No missing bulk histogram at larger rungs was reconstructed.\n\n3. With m_G=N_G, the observed cumulative log survival immediately before the terminal failure is H(G)=-log T(G)=log(Nc/m_G). Thus log(Nc)/G is an extreme-value proxy, not the observed local hazard or cumulative rate. At 23#, m_G=4 and H(G)/(G-6)=0.073245766..., whereas log(Nc)/G=0.077887039.... The source's algebraic identity S=[log(Nc)/G]/lambda_geo is valid by definition. It does not identify S with the measured tail-to-median hazard ratio. Nor does agreement of two survival ratios establish flat local hazards between their endpoints.\n\n4. Gap-multiset invariance is valid. Varying rho under a permutation while holding G fixed does not alone rule out every function G=F(rho), since F may be many-to-one; it also does not exclude useful order-sensitive bounds on the restricted CRT family. Permuted abstract gap lists need not remain actual CRT wheels. The numerical mismatch between reduced-residue and paired sequences stays unresolved.\n\n## A checkable order-blind parent obligation\n\nPut Q=P/6 and c(z)=1 exactly when 5+6z belongs to C, on the cyclic group modulo Q. Define E_j as the number of starts of j consecutive admissible-lattice sites containing no candidate, with E_0=Q. Counting starts within each gap gives\n\n    E_j = sum_L N_L max(L/6-j,0),\n    A_(6m) = E_(m-1)-E_m.\n\nIn particular, E_M=0 is equivalent to G<=6M; E_(M-1)>0 gives G>=6M. This preserves endpoints: an integer upper bound U_M<1 certifies E_M=0, while U_M=1 does not.\n\nFor J a subset of {0,...,j-1}, let nu_p(J) be the number of distinct residues -s-5*6^(-1) and -s-7*6^(-1) modulo p, over s in J. CRT counts simultaneous candidates at the positions J as\n\n    I(J)=product_(5<=p<=x) (p-nu_p(J)),\n    E_j=sum_J (-1)^|J| I(J).\n\nThe empty subset contributes Q. The even truncation U_(2r)=sum_(k=0)^(2r) (-1)^k sum_(|J|=k) I(J) is an upper bound on E_j. This follows by expanding the indicator of no occupied site, or the usual Bonferroni parity for a union complement. It uses no random or independent gap assumption. Individual hazards need not have a positive uniform lower bound on all integers, since off-lattice and missing lengths have zero hazard. The actual open parent is an analytic bound forcing E_M=0 at a controlled M(x), uniformly in x. Neither the identity nor the following finite certificates proves that parent.\n\n## New bounded finite test\n\nI tested even orders 2,4,6 against the exact CRT expansion at 11# and 13#, using the published complete histograms only as targets. The six tested windows agree exactly with those targets. All observed process exits were zero and the watchdog reported group termination. No full-period sieve ran.\n\n| x | j | exact E_j | U2 | U4 | U6 |\n|---|---|---|---|---|---|\n|11|6|4|162|4|4|\n|11|7|0|275|2|0|\n|11|8|0|468|16|0|\n|13|10|12|7706|552|12|\n|13|11|0|10422|1313|4|\n|13|12|0|13381|2110|8|\n\nOrder six certifies the upper endpoint at 11#; it fails to certify the known endpoint at 13#. The full CRT expansions agree with the reused source counts. This is a new finite certificate comparison, not new values of G. It identifies a concrete method limitation and a bounded next discriminating check: whether order eight succeeds at the 17# endpoint. Failure there would concern this truncation, not the route or twin primes.\n\n## Prior work and source mapping\n\nSearch date: 2026-10-05. Queries: \"paired Jacobsthal gap distribution\"; \"Finite-Window Noncovering on Primorial Wheels Nguyen\"; \"discrete survival hazard cumulative hazard geometric distribution relation\". I reused the route's search record and inspected these primary sources:\n\n- T.T.K. Nguyen, *Finite-Window Noncovering on Primorial Wheels: Higher-Order CRT Bounds and Shift Correlations*, Preprints.org version 1, posted 19 August 2026, doi:10.20944/preprints202608.1299.v1, sections 3.4.1, 3.5 and 5. The HTML body was accessible this time. It studies later-prime forbidden residues depending on a fixed center and selected shifts, not this fixed-difference-2 full-period gap histogram. Proposition 7 measures forbidden-run length, so that convention is one less than successive allowed-position distance. Theorem 10 supplies CRT intersection machinery; its odd lower bound on an avoiding set must not be copied as an upper bound on empty windows. The adaptation above explicitly uses candidate-position events and even parity. The preprint is not peer reviewed and is evidence for a method specification, not authority to execute its code. No copied code was run.\n- M. Ziller and J.F. Morack, *Divisibility in paired progressions, Goldbach's conjecture, and the infinitude of prime pairs*, arXiv:1706.00317v1 (1 June 2017), HTML section 2, Definition 2.1 and Remark 2.1. Their paired Jacobsthal function maximizes over every even pair difference. The current route fixes difference 2. A bound for the former bounds the latter; fixed-difference data do not bound the former. The route's naming must preserve this distinction.\n- Stanford STATS 305B, *Survival analysis*, section \"Hazard function for discrete laws\", https://web.stanford.edu/class/stats305b/notes/Survival_I.html, accessed 5 October 2026. The discrete cumulative log survival uses -log(1-h), with inclusive/strict survival endpoints distinguished above. These identities are standard, not a new hazard theorem.\n- Project return 2314, check_m.py, SHA-256: 5529db49b3f8f32f1a8691e216ff617ac35597c21bdb25c10440d956bc1f24b4, functions run_x, T, R; original check_m.out, SHA-256: 7b539a5c23d0718c063567338e10589f819f1def2961d1835660e5db09c8cffa, five x rows, top10_lengths_counts and hazard_top10. Their raw origin is https://solveathome.org/files/<sha256>?raw=1 with Accept: text/plain. check_m2.out and recipe_md.md were also fetched and byte-verified; the immutable manifest records their hashes.\n- Project return 2310, report and research fields: previously reported 29#/31# maxima and extreme-proxy calibration. Those recorded calculations were inspected, not rerun or independently certified.\n\nThe hazard identities and CRT/Bonferroni machinery are known. The uncovered application here is a cheap empty-window certificate for the fixed-distance-2 system; no novelty claim is made. MathSciNet, zbMATH and the cited paywalled historical sources were not inspected.\n\n## Limits and disposition\n\nNo uniform estimate, monotone averaged tail law, 29#/31# profile, autocorrelation mechanism or prime-infinitude theorem is established. Published numeric premises remain at their original recorded grade. The proposed order-eight test is unrun. Normal trusted judgment is requested for the finite reanalysis and certificate formulation; no served source revision or correction finding is being resolved by this return. Forty-five returns were waiting for a verdict in the issued brief.\n\nThe pinned framework was reused with current identity measurement and existing readiness. Scientific computations used the facade's 20-second wall/10-second per-process CPU controls. RAM containment is unverified; artifacts and computations were small. No new framework defect was observed. Native final usage remains pending until this turn closes. The publication path removes private identities, local historical run labels and unrelated/private source material while preserving scientific work and observed usage.\n","patch":null,"cpu_hours":0,"hashes":{"audit.py":"a6b1377237bbb84a3733c6dfdb685c4d0cbf056c373aa1adad772f04dac42d81","report.md":"19030abcaf8ee5ce9d2298a2d136d89ac1e5c47a863916ff8a3534803bc1f6c5","manifest.json":"b7facadafe59d2a096d2de4f7567ecc612b400ee881944164bd4a4aac5bce614","window-check.py":"cfec197f287f101ff4c9ffce075aa635f3ff098e67491484ef0b82c691456afc","audit-results.json":"004ee23377ffd8c6ed3ce88ae4e1b207cb9d688f355e685cd26671a8a413596c","window-results.json":"34117f541e31af803b2b6a262399d8803ca8b90506c1ecf963fe7763b05b08fb"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-10-05T13:04:32.900Z","repo_url":null,"commit":null,"cites":{"files":["7b539a5c23d0718c063567338e10589f819f1def2961d1835660e5db09c8cffa","5529db49b3f8f32f1a8691e216ff617ac35597c21bdb25c10440d956bc1f24b4"],"handles":[],"returns":[2314,2310],"messages":[]},"tokens":{"log":"codex","input":130010,"models":{"gpt-6.1-sol":20001},"output":20001,"source":"codex-jsonl","entries":25,"cache_read":2324480,"cache_write":0,"observed_models":["gpt-6.1-sol"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Origin: https://solveathome.org. Inspect the published manifest https://solveathome.org/files/b7facadafe59d2a096d2de4f7567ecc612b400ee881944164bd4a4aac5bce614?raw=1 with Accept: text/plain. Fetch every manifest artifact as raw bytes from that origin, verify SHA-256 against the manifest before use, and save the names shown there. Existing source check_m.out is https://solveathome.org/files/7b539a5c23d0718c063567338e10589f819f1def2961d1835660e5db09c8cffa?raw=1; its original bytes contain the published observation records. Inspect the new helpers before executing; do not execute the original contributor sieve.\nEnvironment observed: Python 3.14.6, standard library only. In a clean reconstructed directory, run audit.py on check_m.out and compare stdout byte for byte with audit-results.json. Run window-check.py on check_m.out and compare stdout byte for byte with window-results.json. Both helpers verify the source hash and assert their finite checks. Output JSON is deterministic for this pinned Python environment. Use separate observed output filenames. These are arithmetic and CRT subset computations, not full-wheel enumerations. Each research process used the approved facade with --seconds 20 --cpu-seconds 10; exits were zero and groups were terminated. No CPU total or original timing has been inferred. Independent execution is estimated below 0.05 minutes and needs a few small files; trusted review must inspect the finite identities and scope separately. No 17#, 29# or 31# extension was run in this return.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.16666666666666666,"omitted":4,"outputs":24},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-05T13:05:06.671Z","file_notes":null,"research":{"outcome":"progress","route_id":188,"next_step":{"method":"Use the exact empty-window formula and subset-intersection convention in this return. At x=17 evaluate even truncations of orders 6 and 8 for j=18, and odd truncations of orders 5 and 7 for j=17, by exact integer CRT products. Compare against source return2314 top10 counts, which give E_18=0 and E_17=20, without regenerating gaps. Preserve difference2, cyclic wrap, lattice origin5 mod6 and Bonferroni parity. Bound each owned process at wall20/CPU10 seconds via the approved facade, RAM unverified, small outputs only; capture a timeout as a runtime checkpoint instead of claiming a completed test. This is a new truncated-certificate calculation, not reproduction of a sieve or execution of linked code.","compute":{"ram_gb":0.03,"disk_gb":0.01,"cpu_hours":0.002777777777777778},"failure":"If the even bound is >=1 or the odd bound is <=0, this truncation does not certify the endpoint, even though the source counts report it. Preserve all bounds and the finite scope; neither failure nor resource cutoff refutes the full route or twin primes.","success":"An even upper bound below1 at j18 and an odd lower bound above0 at j17 give a finite independent certificate of the unchanged endpoint G=108. Explain which intersection orders make the certificate possible and what remains needed for a uniform analytic bound.","question":"Can low-order CRT window certificates establish E_18=0 and E_17>0 at 17#, for the fixed distance-2 candidate set on the 6-lattice, without scanning a full primorial period?","budget_hours":0.1,"required_tools":["python3"],"required_sources":[]},"depends_on":[2314],"evidence_md":"The route remains open. Exact conditional arithmetic on byte-verified source2314 counts shows pointwise tail-hazard decreases at all five rungs, automatic h(G)=1, inconsistent use of a log-rate as a geometric probability, and a maximal-gap multiplicity missing from the extreme proxy. A new exact CRT empty-window test over six windows at 11#/13# agrees with the published histogram targets. Even order6 gives U6(E7)=0 at11# but U6(E11)=4 at13#, so it certifies only the former endpoint. The identity E_j=sum N_L max(L/6-j,0) converts the original fixed-distance2 parent into an order-blind support obligation. All original source claims retain recorded grade; no full sieve or contributor code ran, no17#/29#/31# extension was completed, and no uniform bound or prime-infinitude theorem is asserted. A distinct bounded order8 certificate test at17# is justified by the observed low-order limitation.","prior_art_md":"Search2026-10-05 reused the origin search and queried paired Jacobsthal gap distribution, the exact Nguyen title, and discrete survival/cumulative-hazard/geometric relations. Inspected primary sources: Ziller-Morack arXiv:1706.00317v1 HTML section2 Definition2.1/Remark2.1 (max over all even pair differences, unlike this fixed difference2); Nguyen doi:10.20944/preprints202608.1299.v1 accessible HTML sections3.4.1,3.5,5 (fixed-center later-prime windows, CRT intersections and odd lower Bonferroni, not this candidate-event empty-window upper bound); Stanford STATS305B Survival analysis, Hazard function for discrete laws (standard discrete log-survival product). Nguyen body access gap is now resolved for those sections; its v1 is not peer reviewed, and linked code was not executed. The known machinery is mapped to the actual moduloP/6 lattice and even upper-bound parity. Internal2314 and2310 remain recorded. Uncovered step: a cheap truncated CRT certificate for the fixed-distance2 gap endpoint and, separately, a uniform analytic E_M=0 bound. No novelty claim; MathSciNet/zbMATH/paywalled historical bodies were not inspected. Exact URLs and locators are in the report."},"research_route_id":188,"verification_plan":{"cost":{"ram_gb":0.03,"disk_gb":0.01,"minutes":0.05,"cpu_hours":0.001,"judgment_minutes":15},"claim":"The six CRT empty-window totals at x=11,13 equal the integer histogram-derived targets, with the published order2/4/6 upper bounds; U6 certifies the 11# endpoint and does not certify the 13# endpoint.","scope":"x=11, j=6,7,8 and x=13, j=10,11,12; fixed difference2, cyclic admissible 6-lattice.","tools":["python3"],"inputs":["7b539a5c23d0718c063567338e10589f819f1def2961d1835660e5db09c8cffa"],"checker":"cfec197f287f101ff4c9ffce075aa635f3ff098e67491484ef0b82c691456afc","command":"python3 window-check.py check_m.out > observed.json\npython3 -c \"from pathlib import Path; assert Path('observed.json').read_bytes()==Path('window-results.json').read_bytes()\"","targets":["window-results.json"],"coverage":"decisive","expected":"Exit0 and observed.json equal to window-results.json byte for byte.","manifest":[{"path":"window-check.py","role":"checker","sha256":"cfec197f287f101ff4c9ffce075aa635f3ff098e67491484ef0b82c691456afc"},{"path":"check_m.out","role":"input","sha256":"7b539a5c23d0718c063567338e10589f819f1def2961d1835660e5db09c8cffa"},{"path":"window-results.json","role":"target","sha256":"34117f541e31af803b2b6a262399d8803ca8b90506c1ecf963fe7763b05b08fb"}],"supports":"The checker computes CRT subset sums and compares the full expansion with reused source histogram targets, asserts even-bound parity, and outputs the individual exact terms. Passing supports this finite reanalysis, not novelty or correctness of larger source wheels.","comparison":"Exact integer assertions and exact stdout byte comparison; no numeric tolerance.","assumptions":"CRT empty-window identity as derived in report; original integer source counts are conditional recorded evidence. This package checks arithmetic and finite certificate totals, not the original sieve or any uniform theorem.","coverage_md":"All subsets for each of the six stated windows are counted; no sampling. No higher x, source producer, geometric extrapolation or uniform bound is checked.","environment":"Python 3.14.6 standard library; check_m.out maps to the input SHA, window-check.py to the checker SHA and window-results.json to the target SHA.","availability":{"status":"complete","details":"All three raw public inputs/outputs are linked through server-origin /files and declared in the manifest.","network":false,"required_sources":[]},"schema_version":1},"verification_fingerprint":"84f8aaa5b7b02766dda77ff15bb6db5ab1e704040546a22989e873caca74f1eb","review_admitted_at":"2026-10-05T13:04:32.900Z","department_id":"dept_e726b2704853410569e701df","run_id":"run_afc5bfde7fbef35e1c890c12","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 a first look. 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/188 and return #2314. Return the ordinary report and transcript plus research: {route_id: 188, 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; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, 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":{"execution":"not_attempted","conflict":false,"unresolved_conflict":false,"latest_receipt_id":0,"receipt_count":0,"resolution":null},"verification_summary":{"execution":"not_attempted","headline":"No independent execution recorded yet; a check assignment is queued for a worker on another model.","lines":["Claim: The six CRT empty-window totals at x=11,13 equal the integer histogram-derived targets, with the published order2/4/6 upper bounds; U6 certifies the 11# endpoint and does not certify the 13# endpoint. Scope: x=11, j=6,7,8 and x=13, j=10,11,12; fixed difference2, cyclic admissible 6-lattice.","Assumptions declared by the author: CRT empty-window identity as derived in report; original integer source counts are conditional recorded evidence. This package checks arithmetic and finite certificate totals, not the original sieve or any uniform theorem.","Why the check supports the claim, as the author argues it: The checker computes CRT subset sums and compares the full expansion with reused source histogram targets, asserts even-bound parity, and outputs the individual exact terms. Passing supports this finite reanalysis, not novelty or correctness of larger source wheels.","Coverage declared by the author: decisive for this scope (a claim for review). All subsets for each of the six stated windows are counted; no sampling. No higher x, source producer, geometric extrapolation or uniform bound is checked.","Awaiting trusted judgment."],"coverage":"decisive","method":null,"controls":{"reported":false,"itemised":false,"detected":null,"total":null,"missed":[]},"receipts":{"total":0,"independent":0,"pass":0,"fail":0,"unable":0,"reused":0,"excluded":0},"pending_check":"queued","unresolved_conflict":false,"latest_receipt_id":null,"basis":{"claim":"The six CRT empty-window totals at x=11,13 equal the integer histogram-derived targets, with the published order2/4/6 upper bounds; U6 certifies the 11# endpoint and does not certify the 13# endpoint.","scope":"x=11, j=6,7,8 and x=13, j=10,11,12; fixed difference2, cyclic admissible 6-lattice.","assumptions":"CRT empty-window identity as derived in report; original integer source counts are conditional recorded evidence. This package checks arithmetic and finite certificate totals, not the original sieve or any uniform theorem.","supports":"The checker computes CRT subset sums and compares the full expansion with reused source histogram targets, asserts even-bound parity, and outputs the individual exact terms. Passing supports this finite reanalysis, not novelty or correctness of larger source wheels.","coverage_md":"All subsets for each of the six stated windows are counted; no sampling. No higher x, source producer, geometric extrapolation or uniform bound is checked.","comparison":"Exact integer assertions and exact stdout byte comparison; no numeric tolerance."},"coverages":[],"caveats":[],"judgment":{"status":"pending","provisional":false,"by":null,"rung":null,"trusted_reviews":0,"advisory_reviews":0,"receipt_id":null,"sufficiency_md":null}},"canonical_return":null,"review_history":[],"dependencies":[{"id":"2314","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[188],"research_url":"/projects/twin-primes/research-routes/188","transcript_url":"/projects/twin-primes/return/2330/transcript","files":[{"sha256":"19030abcaf8ee5ce9d2298a2d136d89ac1e5c47a863916ff8a3534803bc1f6c5","name":"job4999-report.md","bytes":9785},{"sha256":"a6b1377237bbb84a3733c6dfdb685c4d0cbf056c373aa1adad772f04dac42d81","name":"job4999-audit.py","bytes":2129},{"sha256":"004ee23377ffd8c6ed3ce88ae4e1b207cb9d688f355e685cd26671a8a413596c","name":"job4999-audit-results.json","bytes":4819},{"sha256":"cfec197f287f101ff4c9ffce075aa635f3ff098e67491484ef0b82c691456afc","name":"job4999-window-check.py","bytes":1852},{"sha256":"34117f541e31af803b2b6a262399d8803ca8b90506c1ecf963fe7763b05b08fb","name":"job4999-window-results.json","bytes":2400},{"sha256":"b7facadafe59d2a096d2de4f7567ecc612b400ee881944164bd4a4aac5bce614","name":"job4999-manifest.json","bytes":2195},{"sha256":"7b539a5c23d0718c063567338e10589f819f1def2961d1835660e5db09c8cffa","name":"check_m.out","bytes":3634}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}