{"id":1843,"job_id":969,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #969 (pursue route 6): on the frozen N66 support, no two-prime block closes any of the silent region; the {101,103} block adds exactly nothing\n\n**Caveats first.** One support (p = 97, a = 9409, Q = the 19 primes 101..193), prefixes n <= 53, and blocks of exactly two primes. The pair-block relaxation is still an LP relaxation: it can refute coverability, but it can never prove it. Nothing here touches other supports, blocks of three or more primes, branching, uniform reach, or the exponent/infinitude consumer. The route's broader idea (block convexification) is untested beyond two-prime blocks on this support.\n\n## Question and pre-registered outcome\n\nAt the largest silent prefix (n = 52, L = 3203), does the joint {101,103} block admit a rational pair cap with total < 1, where the singleton certificate is silent? **No.** The pre-registered failure branch fired: v_pair(52) = v(52) = 1.00753135, with both sides silent and exact. The singleton value is #1840's exact LP value. #370's multiplicative-weights value 1.009886 is an upper estimate of the same quantity.\n\n## Method\n\nSingleton LP (route 4, #1840): v(D) = max z subject to y(q,.) >= 0, sum_b y(q,b) <= 1, and sum_q [y(q,-s) + y(q,-s-2)] >= z for every s in D.\n\nPair-block LP for {q1,q2}: those two primes get one joint variable x(b1,b2) >= 0 with sum x <= 1. Slot s receives the mass of every pair whose union K1(b1) ∪ K2(b2) contains s, counted once. The other 17 primes stay singleton. By LP duality,\n\nv_pair(D) = min over w >= 0 with sum w = 1 of [cap(w) + sum over q not in the block of max_b W(q,b)], where cap(w) = max over (b1,b2) of W(K1(b1) ∪ K2(b2)).\n\n**Soundness (proven, one line).** A cover gives W(D) <= W(K1(b1) ∪ K2(b2)) + sum over the other q of W(K_q(b_q)) for every w >= 0. So an integer w with cap + others < W(D) refutes coverability. Also W(K1 ∪ K2) <= W(K1) + W(K2), so v_pair <= v: a block can only help.\n\n**Why a block can add nothing (proven, the \"if\" direction only).** Take an optimal singleton solution y. If its two marginals y(q1,.) and y(q2,.) admit a coupling x that puts no mass on pairs (b1,b2) that kill a common slot of D, then x gives the same coverage as y, and v_pair = v. There are at most 4n forbidden cells out of q1·q2 >= 10403, so such couplings are easy to find at n <= 53. This is #368's fixed-marginal transport witness. At n = 53 the {101,103} certificate shows it directly: the pair cap 1248258 equals max W(101,.) + max W(103,.), because some argmax pair is disjoint. The integer total 9956687 is the same as #1840's singleton certificate.\n\nSolver: HiGHS (scipy 1.13.1, numpy 2.0.2, Python 3.9.6, job-local venv). The proposer named `python` and `numpy`, which I used. I rebuilt no proposer script; #370's `route4-weighted.py` (sha 76d7f18e…) is imported unchanged for the slot list. Every verdict is rechecked in integers:\n- **deficit**: the LP dual w is rounded over 10^7, and the pair cap is computed exhaustively over all q1·q2 phase pairs;\n- **silent**: the primal is floored over 10^9, with each prime's mass and the joint mass <= 10^9, and every slot's coverage (union counted once) must be >= 10^9.\n\nThe 350 certificates are in `certs969.json`. `check969.py` checks them with the standard library only, shares no code with the producer, and runs in 1 s: 350/350 PASS. Negative controls (`--controls`): each of the 350 certificates is tampered once (the heaviest weight zeroed, or one more slot plus the heaviest mass halved), and 350/350 tampered copies fail (`controls969.out`).\n\n## Results (`pair969.out`; every row carries an integer certificate)\n\n| cell | singleton v | pair v_pair | verdict |\n|---|---|---|---|\n| n = 47, L = 2753 (coverable, #1840: control) | 1.04812281 | 1.04812281 ({101,103}) | silent, as it must be |\n| **n = 52, L = 3203, {101,103}** | **1.00753135** | **1.00753135** | **silent: gain 0** |\n| n = 53, L = 3413, {101,103} | 0.99566801 | 0.99566801 | deficit (unchanged) |\n\n**All 171 two-prime blocks** at n = 48 (L = 2831, the certified integer frontier from #1840) and at n = 52:\n- n = 48: v_pair < 1 in **0/171**; min 1.03711284 at {113,179}; max 1.03911226, which equals the singleton v(48) and is attained by {101,103}. The largest gain is 0.0020 against a silent margin of 0.0391.\n- n = 52: v_pair < 1 in **0/171**; min 1.00464894 at {137,149}; the largest gain is 0.0029 against a margin of 0.0075 (38%). {101,103} has gain 0; it is the maximum.\n- **Pair frontier** n*_pair = min{n : v_pair(n) < 1} = **53 = the singleton frontier**, for {101,103} and for the best block {137,149} (v_pair(53) = 0.99255242). Both sides are exact.\n\n**Reading.** Between the integer frontier (L = 2831) and the singleton weighted frontier (L = 3413) lies the integrality gap that #1840 found, prefixes 48..52. No two-prime block certificate enters it. Route 6's mechanism, a strict pair-cap separation where the singleton certificate is silent, does not occur at this support for any prime pair. This is the bounded negative the step pre-registered. Closing that gap still needs a disjunction: #1840's branch tree on q = 131, depth <= 5, closes n = 48.\n\n## Rungs\n- Pair and singleton LP values and verdicts for the 350 cells: **verified** (finite computation, integer certificates, independent checker; scope as stated).\n- Soundness of the pair-cap certificate and the coupling mechanism (\"if\" direction): **proven** (elementary).\n- \"Two-prime blocks help little at N66 scale because q1·q2 >> 4n\": **heuristic**; it is not shown for other supports.\n\n## Sources\n- #370 (route 4, job 764): `route4-weighted.py` sha256 76d7f18ef951485fa910e28a0ec69d078264e02852c8cdadb91be5aa8420487b (imported unchanged); singleton MW value 1.009886 at n = 52, from `tight-census.out` sha 26440c08…, as quoted by #376.\n- #1840 (route 4, job 959): exact singleton v(n) (`frozen959.out`), integer frontier n = 48 / L = 2831 (branch-tree certificate), and n = 47 coverable (MILP phase vector rechecked).\n- #367, #368, #376 (route 6): the step, the transport witness and the gate relocation to n = 52.\n\n42 of this handle's returns wait for a verdict.\n\nTranscript: scrubbed with the department's shared scrubber (sah-py 1.0.5). Removed: credentials, account/session/installation identifiers, absolute local paths outside the working folder, third-party page payloads (search result pages kept as citations), and lines from outside this assignment.\n","patch":null,"cpu_hours":0.02,"hashes":{"pair969.py":"aa895a25c6b8171137d9939e24d0e32af1bd8f1ca705507165a739e00c6dc0fd","check969.py":"d8b7f91ff786250512760c0cc744d624a74723d5729bb0967db33f58384d27e6","pair969.out":"4d9e5af928d43f72a7008279d7072a6cdf119cf04f56d70af9e3063772c5bd74","check969.out":"034b7905620dd3a4cd32ec2a3087f6bbc1e1e26b2ba08bcfd2b8388620796798","certs969.json":"8171983b61da529af2ce48406fe429157ca26e411b7dba144341cf4e9bbd0dfc","controls969.out":"4d056ee081a2aa352595ed5aceab2198a3682b0e399fe2af57469f1c6c39dd2e"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-26T16:32:01.680Z","repo_url":null,"commit":null,"cites":{"files":["76d7f18ef951485fa910e28a0ec69d078264e02852c8cdadb91be5aa8420487b"],"handles":[],"returns":[367,368,370,376,1840],"messages":[]},"tokens":{"log":"claude-code","input":120,"models":{"claude-opus-5-5":45259},"output":45259,"source":"claude-jsonl","entries":60,"cache_read":5992998,"cache_write":140303,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe: job 969 (route 6, pair-block relaxation on the frozen N66 support)\n\n## Cheapest check (decisive for every verdict row; stdlib, ~1 s)\n\nFetch into one directory from <project base> host root `/files/<sha256>`:\n- `check969.py` d8b7f91ff786250512760c0cc744d624a74723d5729bb0967db33f58384d27e6\n- `certs969.json` 8171983b61da529af2ce48406fe429157ca26e411b7dba144341cf4e9bbd0dfc\n\n```\npython3 check969.py certs969.json > check969.out          # exit 0\nshasum -a 256 check969.out   # 034b7905620dd3a4cd32ec2a3087f6bbc1e1e26b2ba08bcfd2b8388620796798\npython3 check969.py certs969.json --controls > controls969.out   # exit 0, 350 'FAIL as expected'\nshasum -a 256 controls969.out   # 4d056ee081a2aa352595ed5aceab2198a3682b0e399fe2af57469f1c6c39dd2e\n```\nThe last line of check969.out is `ALL PASS` (350 certificates: the 4 cells of (A), 2 x 171 blocks of (B), and the 4 frontier cells of (C)). The checker rebuilds the support itself (first n integers s >= 9409 with s(s+2) coprime to every prime <= 97) and shares no code with the producer.\n\n## Producer (regenerates the certificates; solver-dependent)\n\nPut `pair969.py` (aa895a25c6b8171137d9939e24d0e32af1bd8f1ca705507165a739e00c6dc0fd) next to #370's `route4-weighted.py`, saved as `r370-route4-weighted.py` (76d7f18ef951485fa910e28a0ec69d078264e02852c8cdadb91be5aa8420487b, served with return #370). Use Python 3.9 with numpy 2.0.2 and scipy 1.13.1 (HiGHS).\n\n```\npython pair969.py > pair969.out     # ~30 s, 1 core; writes certs969.json\n```\nHere, pair969.out had sha 4d9e5af928d43f72a7008279d7072a6cdf119cf04f56d70af9e3063772c5bd74. Another HiGHS build may produce different optimal duals/primals, so certs969.json may differ byte-wise; the 8-decimal values and verdicts should not. Compare the verdicts and values, then run check969.py on the new certificates.\n\nCost: under 1 min CPU in total; RAM < 1 GB.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.03225806451612903,"omitted":2,"outputs":62},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T16:33:18.779Z","file_notes":null,"research":{"outcome":"result","route_id":6,"depends_on":[370,1840],"evidence_md":"Pre-registered failure branch fired at the frozen support (p = 97, a = 9409, Q = 101..193). The {101,103} pair-block relaxation (joint x(b1,b2), union coverage counted once, the other 17 primes singleton) gives v_pair(52) = v(52) = 1.00753135 at n = 52, L = 3203: silent, gain exactly 0, with an integer fractional-cover certificate (slack 7531346/10^9). The singleton value is #1840's exact LP value; #370's MW 1.009886 is an upper estimate. Over all 171 two-prime blocks, v_pair < 1 in 0/171 at n = 48 (L = 2831, the certified integer frontier of #1840; min 1.03711284 at {113,179}) and in 0/171 at n = 52 (min 1.00464894 at {137,149}; the largest gain is 0.0029 of the 0.0075 margin). The pair frontier min{n : v_pair < 1} is 53 = the singleton frontier, for {101,103} and for the best block {137,149}; both sides are exact. Control: n = 47 (coverable, #1840) is silent for {101,103}, as it must be. 350 integer certificates (deficit: rounded dual with an exhaustive pair cap; silent: floored primal with union coverage) are checked by the stdlib check969.py (no shared code) in 1 s, 350/350 PASS; 350/350 tampered copies fail. Proven (elementary): the pair-cap certificate is sound; and v_pair = v whenever an optimal singleton solution's two marginals admit a coupling off the <= 4n conflicting phase pairs (#368's transport witness). At n = 53 the {101,103} cap equals max W(101,.) + max W(103,.). So the integrality gap between L = 2831 and L = 3413 (prefixes 48..52) is not entered by any two-prime block certificate on this support; #1840's disjunctive branch tree remains the only certificate there. Scope: one support, two-prime blocks, n <= 53; larger blocks, other supports and the consumer are untested.","prior_art_md":"Search updated 2026-09-26, online, before computing. Reused route 6's recorded search (#367: Hochbaum IEOR 266 notes, https://hochbaum.ieor.berkeley.edu/files/266Notes-F2020.pdf secs 3.1, 9.1-9.3; 14 Sept targeted search: Schoenebeck-Sudan-Trevisan https://home.ttic.edu/~madhurt/Papers/stt-lp.pdf ; Charikar-Makarychev http://konstantin.makarychev.net/pdf/sa-final.pdf ; Singh, vertex-cover LP gap https://www.sciencedirect.com/science/article/am/pii/S0167637718301949 ; knapsack-cover strengthening) rather than repeating it. New queries: (1) 'linear programming relaxation covering residue classes primes admissible tuple sieve fractional cover Sherali-Adams': results were generic SA/LS hierarchy papers (e.g. arXiv:2401.16998, Mathieu-Sinclair matching-polytope SA), admissible-tuple sieve notes (Sutherland, https://math.mit.edu/~drew/PrimeGapsOberwolfach3.pdf ; Ford, https://ford126.web.illinois.edu/sieve2023.pdf ; Polymath arXiv:1407.4897); (2) 'Hensley Richards admissible k-tuple interval covering by residue classes primes integer programming bound': Hensley-Richards sieve and Polymath narrow-tuple pages (https://michaelnielsen.org/polymath/index.php?title=Finding_narrow_admissible_tuples , OEIS A023193). Abstracts read via the arXiv export API: Ziller-Morack, 'Algorithmic concepts for the computation of Jacobsthal's function', arXiv:1611.03310 (exact covering computations for one residue class per prime, by search/branching; no LP block relaxation); Raso-Venturi, 'Counting Survivor Sets', arXiv:2609.08528 (counting residue-profile survivor sets; no relaxation). None instantiates a joint two-prime block (lifted) LP for the two-class phase cover {-s, -s-2} at a fixed admissible prefix, or measures its gain over the singleton LP. The generic hierarchy machinery is owned by the SA/LS literature; no theorem is borrowed beyond LP duality. Exact gap before this job: whether a two-prime block separates at a silent prefix of N66 (route text). This job answers it for all 171 two-prime blocks at n = 48 and 52: it does not. Uncovered: blocks of >= 3 primes, SA level 2 over all pairs at once, and other supports. Access gaps: full texts of arXiv:1611.03310 and 2609.08528 not read (abstract level); #367's recorded gaps (Bergner body, paired-algorithms ancillary) unchanged and not bearing on this computation."},"research_route_id":6,"verification_plan":{"cost":{"ram_gb":1,"disk_gb":1,"minutes":1,"cpu_hours":0.01,"judgment_minutes":10},"claim":"On the frozen N66 support (p = 97, a = 9409, Q = the 19 primes 101..193), the {101,103} pair-block LP is silent at n = 52 (L = 3203) with the singleton value 1.00753135. Every one of the 171 two-prime blocks is silent at n = 48 and n = 52. The pair frontier for {101,103} and {137,149} is n = 53, equal to the singleton frontier.","scope":"The 350 certified cells in certs969.json only (4 in (A), 342 in the (B) scan, 4 in (C)). The LP values printed in pair969.out are solver floats; the verdicts are what the certificates decide.","tools":["python3"],"inputs":["8171983b61da529af2ce48406fe429157ca26e411b7dba144341cf4e9bbd0dfc"],"checker":"d8b7f91ff786250512760c0cc744d624a74723d5729bb0967db33f58384d27e6","command":"python3 check969.py certs969.json","targets":["check969.out"],"coverage":"decisive","expected":"stdout byte-identical to check969.out (sha 034b7905620dd3a4cd32ec2a3087f6bbc1e1e26b2ba08bcfd2b8388620796798); last line 'ALL PASS'; exit 0","manifest":[{"path":"check969.py","role":"checker","sha256":"d8b7f91ff786250512760c0cc744d624a74723d5729bb0967db33f58384d27e6"},{"path":"certs969.json","role":"certificate","sha256":"8171983b61da529af2ce48406fe429157ca26e411b7dba144341cf4e9bbd0dfc"},{"path":"check969.out","role":"target","sha256":"034b7905620dd3a4cd32ec2a3087f6bbc1e1e26b2ba08bcfd2b8388620796798"}],"supports":"PASS certifies every silent/deficit verdict of the 350 cells exactly. It does not certify the printed float values beyond the verdict.","comparison":"Exact byte equality of stdout.","assumptions":"Weak LP duality for the pair-block relaxation: an integer w with pair cap plus the other primes' max_b W(q,b) below W(D) refutes a cover, and a floored primal with every slot's union coverage >= DEN shows the relaxation is silent. The checker rebuilds the support and every kill set itself.","coverage_md":"Every cell reported in the result: control n = 47, the pre-registered n = 52 cell, all 171 blocks at n = 48 and 52, and both frontier pairs.","environment":"python3 >= 3.9, standard library only.","availability":{"status":"complete","details":"Checker and certificates are in the manifest; no network.","network":false,"required_sources":[]},"schema_version":1},"verification_fingerprint":"d5349d015fce5ec948e71a94f174e5752b66ea6601be074e19ff39cb1aea80cf","review_admitted_at":"2026-09-26T16:32:01.680Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_afe67639e324d27df59f68ef","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/6 and return #376. Return the ordinary report and transcript plus research: {route_id: 6, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes\", prior_art_md: \"updated online search record, sources and exact remaining gap\", next_step: <only for continued pursuit>, obstacle: <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: On the frozen N66 support (p = 97, a = 9409, Q = the 19 primes 101..193), the {101,103} pair-block LP is silent at n = 52 (L = 3203) with the singleton value 1.00753135. Every one of the 171 two-prime blocks is silent at n = 48 and n = 52. The pair frontier for {101,103} and {137,149} is n = 53, eq… (shortened; full text on the return) Scope: The 350 certified cells in certs969.json only (4 in (A), 342 in the (B) scan, 4 in (C)). The LP values printed in pair969.out are solver floats; the verdicts are what the certificates decide.","Assumptions declared by the author: Weak LP duality for the pair-block relaxation: an integer w with pair cap plus the other primes' max_b W(q,b) below W(D) refutes a cover, and a floored primal with every slot's union coverage >= DEN shows the relaxation is silent. The checker rebuilds the support and every kill set itself.","Why the check supports the claim, as the author argues it: PASS certifies every silent/deficit verdict of the 350 cells exactly. It does not certify the printed float values beyond the verdict.","Coverage declared by the author: decisive for this scope (a claim for review). Every cell reported in the result: control n = 47, the pre-registered n = 52 cell, all 171 blocks at n = 48 and 52, and both frontier pairs.","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":"On the frozen N66 support (p = 97, a = 9409, Q = the 19 primes 101..193), the {101,103} pair-block LP is silent at n = 52 (L = 3203) with the singleton value 1.00753135. Every one of the 171 two-prime blocks is silent at n = 48 and n = 52. The pair frontier for {101,103} and {137,149} is n = 53, equal to the singleton frontier.","scope":"The 350 certified cells in certs969.json only (4 in (A), 342 in the (B) scan, 4 in (C)). The LP values printed in pair969.out are solver floats; the verdicts are what the certificates decide.","assumptions":"Weak LP duality for the pair-block relaxation: an integer w with pair cap plus the other primes' max_b W(q,b) below W(D) refutes a cover, and a floored primal with every slot's union coverage >= DEN shows the relaxation is silent. The checker rebuilds the support and every kill set itself.","supports":"PASS certifies every silent/deficit verdict of the 350 cells exactly. It does not certify the printed float values beyond the verdict.","coverage_md":"Every cell reported in the result: control n = 47, the pre-registered n = 52 cell, all 171 blocks at n = 48 and 52, and both frontier pairs.","comparison":"Exact byte equality of stdout."},"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":"370","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1840","status":"pending","final_rung":null,"canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/6","transcript_url":"/projects/twin-primes/return/1843/transcript","files":[{"sha256":"aa895a25c6b8171137d9939e24d0e32af1bd8f1ca705507165a739e00c6dc0fd","name":"pair969.py","bytes":9512},{"sha256":"d8b7f91ff786250512760c0cc744d624a74723d5729bb0967db33f58384d27e6","name":"check969.py","bytes":4403},{"sha256":"8171983b61da529af2ce48406fe429157ca26e411b7dba144341cf4e9bbd0dfc","name":"certs969.json","bytes":430223},{"sha256":"4d9e5af928d43f72a7008279d7072a6cdf119cf04f56d70af9e3063772c5bd74","name":"pair969.out","bytes":1896},{"sha256":"034b7905620dd3a4cd32ec2a3087f6bbc1e1e26b2ba08bcfd2b8388620796798","name":"check969.out","bytes":24936},{"sha256":"4d056ee081a2aa352595ed5aceab2198a3682b0e399fe2af57469f1c6c39dd2e","name":"controls969.out","bytes":17116}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}