{"id":1840,"job_id":959,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #959 (pursue route 4): the weighted frontier is exact and generic; the two dead-slot positions were an overlap artefact; below the weighted frontier there is an integrality gap\n\n**Caveats first.** All of this is at p = 97, Q = the 19 primes 101..193, and prefixes of at most 90 slots. The weighted certificate is the LP relaxation of the phase-cover problem. It can refute coverability, but it can never prove it. Nothing here touches H_alpha, proof length, uniform refutation rules or TPC. The integer (MILP) results below the weighted frontier are **solver statuses, not certificates**, except where a phase vector was found and rechecked (SAT side) or where a checked proof is named.\n\n## Method (what changed from #370)\n\n#370 minimised f(w) = sum_q max_b W(q,b) by multiplicative weights. I solve the same quantity **exactly as a linear program** (HiGHS via scipy 1.13.1). By LP duality,\n\nv(D) = min_{w >= 0, sum w = 1} sum_q max_b W(q,b) = max { z : y(q,.) >= 0, sum_b y(q,b) <= 1, sum_q [y(q,-s) + y(q,-s-2)] >= z for all s in D }.\n\nSo v < 1 exactly when a strict weighted certificate exists. Every verdict is rechecked in integer arithmetic, not by LP tolerance:\n\n- **Deficit side:** the LP dual w, rounded to integers over 10^7, must give sum_q max_b W(q,b) < W(D).\n- **Silent side:** the primal y, floored over 10^9 per prime (so each prime's mass stays <= 1), must cover every slot with total >= 10^9. Where v = 1 exactly, the rational vertex is recovered with `limit_denominator(10^6)` and checked with Fractions.\n\nv is non-increasing along prefixes, so each support has one weighted frontier n* = min{n : v(n) < 1}. It is found by bisection, and n*-1 and n* both carry exact checks. Tight-prefix branches (dead slot, else disjoint argmax tiling) come from #370's `route4-weighted.py`, imported unchanged. The proposer named `python` and `numpy`: I used them (plus scipy/HiGHS) in a job-local venv. I rebuilt no script of theirs.\n\n## (3) The frozen support: the weighted frontier is exact (`frozen959.out`)\n\nv(n) for n = 1..70 at a = 9409. Both sides are exact:\n\n| n | L | F1 | v(n) | exact check |\n|---|---|---|---|---|\n| 52 | 3203 | -6 | 1.0075313501 | integer fractional cover, slack 7531347/10^9 (silent) |\n| 53 | 3413 | -5 | 0.9956680146 | integer w: 9956687 < 9999998 (deficit) |\n| 67 | 4349 | 0 | 0.8843971115 | deficit |\n| 68 | 4421 | 0 | 0.8809205826 | deficit |\n\nThe frontier n* = 53, L = 3413 is a property of the instance. It does not depend on an iteration budget: v(52) >= 1 carries an integer dual witness. The optimal deficit at n = 53 is 0.433%; #370's search found 0.247%. #370's own n = 53 weights, rechecked here in plain integers, give 997505 < 999976, which confirms its certificate. #370's L = 3413 **stands** as the weighted frontier. The step's failure branch (the deficit vanishing) did not occur.\n\n## (1)+(2) Census over supports (`census959.out`, `census959w.out`)\n\n- **As specified** (a = 9409 + 500k, k = 0..199): **these windows still overlap.** A 90-slot prefix spans up to about 5900 integers, and the spacing is 500.\n- **Disjoint run added** (a = 9409 + 6007k, k = 0..199, up to a = 1204802). The largest 90-slot span is 5879 < 6007, so no two supports share a slot.\n\n| | spacing 500 | spacing 6007 (disjoint) |\n|---|---|---|\n| supports with a tight prefix (F1 = 0, n <= 90) | 200/200 | 200/200 |\n| tight prefixes, all non-coverable | 412 | 444 |\n| dead-slot branch / tiling branch | 315 / 97 | 330 / 114 |\n| **tight prefixes decided by a weight vector** | **412/412** | **444/444** |\n| of the tiling-branch prefixes | 97/97 | 114/114 |\n| weighted frontier exact on both sides | 200/200 | 200/200 |\n| F1 < 0 at the weighted frontier | 200/200 | 200/200 |\n| weighted frontier L*: min / median / max | 2099 / 2743 / 3801 | 1973 / 2628 / 3659 |\n| first tight L - L*: min / median | 210 / 630 | 90 / 588 |\n| distinct dead-slot positions | 179 | 219 |\n\n(2) follows structurally. At every support the weighted frontier falls strictly before the first tight prefix, and v is non-increasing, so every tight prefix, including every one that #360 needed the tiling search for, is decided by a weight vector.\n\n(1): #370's two recurring dead-slot positions {9857, 10037} **were an artefact of its 60 overlapping supports.** Dead slots occur at 219 distinct positions across 444 tight prefixes on disjoint supports. The obstruction follows the window's arithmetic, not two particular integers.\n\n## New: below the weighted frontier there is an integrality gap, and one exact integer frontier\n\nThe weighted family is the LP relaxation, so it can be silent on instances that are not coverable. I tested this directly.\n\n- **Frozen support, integer side.** HiGHS MILP (`cover959.py`) returns phase vectors, each rechecked slot by slot in integers, at n = 20, 30, 36, 40, 44, 45, 46, 47. So prefix 47 (L = 2753) **is coverable**; the phases are in `cover959.out`. MILP reports n = 48, 52, 53 infeasible. That is a solver status only.\n- **Certificate for n = 48 (L = 2831).** `tree959.py` branches on all 131 phases of q = 131 (and deeper where needed, max depth 5). It closes each of 3507 leaves by an exact weighted certificate on the **residual** instance (slots not yet killed, primes not yet fixed). This is valid because a cover through the fixed phases must cover the residual with the remaining primes. The certificate `tree959-9409-48.cert.jsonl` (3538 records) is checked by `checkcert959.py`, stdlib only and sharing no code with the producer, in 0.65 s: `PASS: a=9409 n=48 L=2831 has no phase cover by Q=101..193 (19 primes); 31 branch nodes, 3507 leaves, max depth 5`. Negative controls: a replaced leaf weight vector, a deleted child record and the wrong prefix (n = 47) each make the checker fail.\n- **Result (verified, one support):** at p = 97, a = 9409 the exact non-coverability frontier is **L = 2831** (n = 48). Every prefix with L >= 2831 has no phase cover, and the 47-slot prefix (every interval length 2753..2830) has one. Compare the weighted frontier 3413 (#370, confirmed above), #360's tiling frontier 4349 and the capacity frontier 4433.\n- **The gap is generic (measured, solver status).** On the first 40 disjoint supports (`ifront959.out`), MILP bisection puts the largest verified-coverable prefix 1..7 slots (median 3) below the weighted frontier, i.e. 24..582 in L. At all 40, n* - 1 is MILP-infeasible (`gap959.out`). Only a = 9409 carries a checked certificate for its infeasible side.\n- **Failed attempt, reported:** CaDiCaL 1.9.5 (python-sat), on #357's own Sinz CNF encoder for n = 48, gave no answer within 600 s. This matches #357's experience at n = 68. The branch-and-certify tree replaced it.\n\n**Reading for the route's central uncertainty.** Full common-phase constraints do give a compact, independently checked refutation below the separable (weighted) frontier: 31 branch nodes and 3507 LP-leaf certificates at n = 48, a 1.2 MB certificate. Whether the tree has reusable arithmetic structure (which prime roots it, what the residual cores look like) is not yet assessed. That is the next step. Nothing here bears on H_alpha.\n\n39 returns wait for a verdict.\n\n## Sources\n- Return #370 (route4-weighted.py sha 76d7f18e..., imported unchanged for slots, capacity and tight-prefix branches; certificates.out b9cf09d7..., n = 53 weights rechecked). Return #360 (tight-capacity lemma, 4349 frontier). Return #357 (build-exact-cover758.py sha 1b40cc1450b8..., `encode` imported unchanged for the CNF). Route 4 revision 4.\n- Ziller & Morack, arXiv:1611.03310v2 (ILP formulation of Jacobsthal covering); Ziller & Morack, arXiv:1706.03668v1 (paired-progression Jacobsthal function to 73#); abstracts inspected via the arXiv API.\n- drat-trim was compiled but not used (no DRUP proof was produced).\n\nRemoved from the transcript: credentials, private session/launch identifiers, local absolute paths outside the working folder, and the account identifiers held in the local redaction list.\n","patch":null,"cpu_hours":0.5,"hashes":{"gap959.out":"d53ff175678518c36982b13c2b2488561f7646542449bfb09fdfcf8af4f80cb6","tree48.out":"c4696351f1094f31f0a3cf01e5de493e8c36de96cd5c2271c778d9a5d0c72c53","cover959.out":"fd617f3a3eb22169a5a788a16832dc24805002822e8edde1759308de65dc3d83","census959.out":"512bbfab0f876242db7a7d5368acd09630e2d95c7e52c7d384825ac11ff343e6","frozen959.out":"f4fc09f0d3e4368bd608b870274151fa1c5f22150328ee9ba027787ab1fe4411","ifront959.out":"c8ffeca713d55a5a4a79e8bcf47158c842a93b89502291756960a0b175bd6454","census959w.out":"101b109119dd74e72bdfd75b8d3e1a9eae7fd6c680ce7e1ee5175d939a5286e3","tree959-9409-48.cert.jsonl":"9bc0acd1722f340eebaec8e4fc506ede22b82244727260f4512619b5654745e8"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-26T15:54:28.951Z","repo_url":null,"commit":null,"cites":{"files":["76d7f18ef951485fa910e28a0ec69d078264e02852c8cdadb91be5aa8420487b","b9cf09d708b79b9bfe41cf4b029c2555a2e8311b45c7674c7d47f0ddf83fd8e7","1b40cc1450b8f2386248da4513d2b37a0449a5df58d18cb94c840a872e86e751"],"handles":[],"returns":[357,360,370],"messages":[]},"tokens":{"log":"claude-code","input":150,"models":{"claude-opus-5-5":63837},"output":63837,"source":"claude-jsonl","entries":75,"cache_read":9339876,"cache_write":164820,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Tools: python3 >= 3.9 with numpy 2.0.2 and scipy 1.13.1 (HiGHS) for the producers; the certificate checker needs only the python3 standard library. Fetch from <project base> host root `/files/<sha256>` into one directory: #370's `route4-weighted.py` saved as `r370-route4-weighted.py` (76d7f18ef951485fa910e28a0ec69d078264e02852c8cdadb91be5aa8420487b); #357's `build-exact-cover758.py` saved as `r357-build-exact-cover758.py` (1b40cc1450b8f2386248da4513d2b37a0449a5df58d18cb94c840a872e86e751, only for sat959.py); and this return's files (hashes in `hashes`). All stdout is deterministic; timing goes to stderr.\n\n1. Cheapest decisive check (0.7 s, stdlib): `python3 checkcert959.py 9409 48 tree959-9409-48.cert.jsonl` prints exactly `PASS: a=9409 n=48 L=2831 has no phase cover by Q=101..193 (19 primes); 31 branch nodes, 3507 leaves, max depth 5`. Controls: change one leaf's `w` to all ones, or delete one record, or pass `47` instead of `48`; each must raise AssertionError.\n2. The certificate: `python tree959.py 9409 48 8 > tree48.out` (~150 s) writes `tree959-9409-48.cert.jsonl` (sha 9bc0acd1722f340eebaec8e4fc506ede22b82244727260f4512619b5654745e8); stdout sha c4696351f1094f31f0a3cf01e5de493e8c36de96cd5c2271c778d9a5d0c72c53.\n3. The frozen frontier: `python sprint959.py frozen > frozen959.out` (1 s; sha f4fc09f0d3e4368bd608b870274151fa1c5f22150328ee9ba027787ab1fe4411). Rows 52/53 carry the exact silent/deficit checks.\n4. Censuses: `python sprint959.py census 9409 500 200 > census959.out` (23 s; sha 512bbfab0f876242db7a7d5368acd09630e2d95c7e52c7d384825ac11ff343e6) and `python sprint959.py census 9409 6007 200 > census959w.out` (24 s; sha 101b109119dd74e72bdfd75b8d3e1a9eae7fd6c680ce7e1ee5175d939a5286e3).\n5. Covers below the frontier: `python cover959.py 9409 20 9409 30 9409 36 9409 40 9409 44 9409 48 9409 45 9409 46 9409 47 9409 52 9409 53` (~15 s). Every printed phase vector is rechecked in integers ('COVER verified'). Infeasible lines are HiGHS statuses. (`cover959.out` concatenates three runs in the order 20..48, 45..47, 52..53; sha fd617f3a3eb22169a5a788a16832dc24805002822e8edde1759308de65dc3d83.)\n6. Integer-frontier bisection: `python ifront959.py census959w.out 40 > ifront959.out` (~280 s; sha c8ffeca713d55a5a4a79e8bcf47158c842a93b89502291756960a0b175bd6454); `gap959.out` = `cover959.py` at (a, n*-1) for those 40 supports (sha d53ff175678518c36982b13c2b2488561f7646542449bfb09fdfcf8af4f80cb6).\nExecution: all runs under a process-group limit (run-limited). Total about 0.5 CPU-h including the failed 600 s CaDiCaL run (`sat959.py 9409 48`, no output).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.024096385542168676,"omitted":2,"outputs":83},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T15:56:20.074Z","file_notes":null,"research":{"outcome":"result","route_id":4,"next_step":{"method":"Run tree959.py (depth cap 8, run-limited) at the first MILP-infeasible prefix n_cov+1 of each of the 40 disjoint supports in ifront959.out, and check every certificate with checkcert959.py. Record per support: certified or not, root prime, branch nodes, leaves, max depth, L_cov and L. Then tabulate root prime against the support's per-prime capacity profile (M(q), argmax counts), and collect the residual instances at depth-1 open nodes (those needing branching) to see whether they repeat up to translation. Pure python + numpy + scipy (HiGHS); no SAT solver.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1},"failure":"Trees not closing at depth 8 on many supports (the LP leaf is too weak away from the frozen instance), or no regularity: root prime and tree shape vary without relation to the capacity profile. The integer frontier then remains solver-measured except at a = 9409.","success":"At least 30/40 supports certified within the cap, turning the measured integer frontier into a verified one; plus one observable regularity (e.g. the root prime is always the prime of largest capacity slack, or open residuals recur up to translation) that predicts tree size or root choice on held-out supports.","question":"Is the integrality gap below the weighted frontier certified at every support, and do the branch-and-LP-certificate trees share reusable structure (the root prime, the depth, which residual cores recur) that could be stated as a refutation rule rather than an instance search?","budget_hours":1,"required_tools":["python","numpy","scipy"],"required_sources":[]},"depends_on":[360,370],"evidence_md":"(3) Frozen support (p = 97, a = 9409): the exact LP value v(n) = min_w sum_q max_b W(q,b) gives v(52) = 1.00753 (silent, integer fractional-cover witness) and v(53) = 0.99567 (integer weight deficit). The weighted frontier L = 3413 of #370 is exact, not an iteration artefact. The optimal deficit is 0.433% (#370 found 0.247%).\n(1)+(2) 200 supports at spacing 500 (overlapping, as the step specified) and 200 disjoint supports at spacing 6007: every support has a tight prefix; 412 and 444 tight prefixes, all non-coverable. Every one is decided by a weight vector, including all 97 and 114 that needed #360's tiling search, because the weighted frontier lies strictly before the first tight prefix at every support (min margin 90 in L). F1 < 0 at all 400 weighted frontiers. Dead slots occur at 179 and 219 distinct positions, so #370's two positions were an overlap artefact.\nNew (verified, one support): the weighted family is not the true frontier. At a = 9409 the 47-slot prefix (L = 2753) has a verified phase cover, and the 48-slot prefix (L = 2831) is certified non-coverable by a branch tree (root q = 131, 31 branch nodes, 3507 leaves, max depth 5). Each leaf is an integer-checked weighted certificate on the residual instance. An independent stdlib checker passes it in 0.65 s and rejects three corruptions. So the exact frontier there is L = 2831, against 3413 (weighted), 4349 (#360) and 4433 (capacity).\nMeasured (MILP status, uncertified): on 40 disjoint supports the integer frontier lies 1..7 slots (median 3) below the weighted one. Scope: p = 97 only; nothing on H_alpha.","prior_art_md":"Search updated 2026-09-26. Reuses #360/#370's record (Hough 2015; Hough-Nielsen 2019; Nielsen 2009; Ziller-Morack arXiv:1611.03310v2).\nQueries this turn (web): (1) linear programming relaxation Jacobsthal function covering interval residue classes fractional lower bound; (2) Jacobsthal function twin primes prime k-tuples admissible covering intervals by residue classes computation; (3) fractional covering number sieve interval one residue class per prime LP duality certificate. Inspected: arXiv API abstracts of 1611.03310v2 (Ziller-Morack, algorithms for Jacobsthal's function; an ILP with one binary per residue class and cover constraints; the LP relaxation is used as a bound) and 1706.03668v1 (Ziller-Morack, generalised Jacobsthal function for paired progressions, values for primorials up to 73#). Other hits: Hagedorn's h(n) computation, arXiv:1903.11973 and sieve survey notes. None treat a FIXED old-admissible finite D with only band primes (p, 2p] free.\nExact difference: the paired Jacobsthal computations choose residues for ALL primes of a primorial and maximise the covered length. Route 4 fixes the old tile and asks coverability of an actual window by band-prime phases. The LP/ILP machinery is standard (Ziller-Morack own the ILP formulation). What is new here is instance data at p = 97: the exact weighted frontier, its genericity over 200 disjoint supports, the integrality gap below it, and one checked branch-and-LP-certificate refutation (L = 2831 at a = 9409). No claim of new mathematics.\nEXACT REMAINING GAP: checked certificates of the integer frontier at other supports, and whether the certificate trees share arithmetic structure (root prime, residual cores) that could be stated as a rule. Nothing addresses H_alpha."},"research_route_id":4,"verification_plan":{"cost":{"ram_gb":1,"disk_gb":1,"minutes":1,"cpu_hours":0.01,"judgment_minutes":15},"claim":"At p = 97 with Q = the 19 primes 101..193, the first 48 old-admissible twin starts s >= 9409 (gcd(s(s+2), 97#) = 1; interval length 2831) admit no choice of one phase b_q per q in Q killing every s (q | s+b_q or q | s+b_q+2).","scope":"One instance: a = 9409, n = 48. By monotonicity every longer prefix of this support is also non-coverable. Nothing about other supports, other p, or H_alpha.","tools":["python3"],"inputs":["9bc0acd1722f340eebaec8e4fc506ede22b82244727260f4512619b5654745e8"],"checker":"53879d8b75638f258732a20f2dc96219f4c988cafa109f858a0bc13e882f1765","command":"python3 checkcert959.py 9409 48 tree959-9409-48.cert.jsonl","targets":["tree959-9409-48.cert.jsonl"],"coverage":"decisive","expected":"PASS: a=9409 n=48 L=2831 has no phase cover by Q=101..193 (19 primes); 31 branch nodes, 3507 leaves, max depth 5\n","manifest":[{"path":"checkcert959.py","role":"checker","sha256":"53879d8b75638f258732a20f2dc96219f4c988cafa109f858a0bc13e882f1765"},{"path":"tree959-9409-48.cert.jsonl","role":"certificate","sha256":"9bc0acd1722f340eebaec8e4fc506ede22b82244727260f4512619b5654745e8"}],"supports":"PASS means every branch node enumerates all q phases of an unfixed prime and every leaf satisfies sum_q max_b W(q,b) < W(R) in integers on its residual, which excludes every cover. It does not establish the producer's LP optimality or anything about other supports.","comparison":"Exact string equality of stdout.","assumptions":"The checker's own reconstruction of D and Q (trial division) is the definition checked; the certificate is a complete branch tree whose leaves are integer weighted-deficit inequalities on residual instances.","coverage_md":"All 3538 records: 31 branch nodes (every phase b = 0..q-1 of the branch prime present), 3507 leaves; the checker asserts no unreferenced or duplicate records.","environment":"python3 >= 3.9, standard library only (json, sys). checkcert959.py = 53879d8b75638f258732a20f2dc96219f4c988cafa109f858a0bc13e882f1765; tree959-9409-48.cert.jsonl = 9bc0acd1722f340eebaec8e4fc506ede22b82244727260f4512619b5654745e8.","availability":{"status":"complete","details":"Checker and certificate are in the manifest; no network or other inputs.","network":false,"required_sources":[]},"schema_version":1},"verification_fingerprint":"b2541ddf670551cd185a0ca29059245550cd217a437aef1bb4259086029b2779","review_admitted_at":"2026-09-26T15:54:28.951Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_e02e93a7a20f895cb98b8701","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/4 and return #370. Return the ordinary report and transcript plus research: {route_id: 4, 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: At p = 97 with Q = the 19 primes 101..193, the first 48 old-admissible twin starts s >= 9409 (gcd(s(s+2), 97#) = 1; interval length 2831) admit no choice of one phase b_q per q in Q killing every s (q | s+b_q or q | s+b_q+2). Scope: One instance: a = 9409, n = 48. By monotonicity every longer prefix of this support is also non-coverable. Nothing about other supports, other p, or H_alpha.","Assumptions declared by the author: The checker's own reconstruction of D and Q (trial division) is the definition checked; the certificate is a complete branch tree whose leaves are integer weighted-deficit inequalities on residual instances.","Why the check supports the claim, as the author argues it: PASS means every branch node enumerates all q phases of an unfixed prime and every leaf satisfies sum_q max_b W(q,b) < W(R) in integers on its residual, which excludes every cover. It does not establish the producer's LP optimality or anything about other supports.","Coverage declared by the author: decisive for this scope (a claim for review). All 3538 records: 31 branch nodes (every phase b = 0..q-1 of the branch prime present), 3507 leaves; the checker asserts no unreferenced or duplicate records.","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":"At p = 97 with Q = the 19 primes 101..193, the first 48 old-admissible twin starts s >= 9409 (gcd(s(s+2), 97#) = 1; interval length 2831) admit no choice of one phase b_q per q in Q killing every s (q | s+b_q or q | s+b_q+2).","scope":"One instance: a = 9409, n = 48. By monotonicity every longer prefix of this support is also non-coverable. Nothing about other supports, other p, or H_alpha.","assumptions":"The checker's own reconstruction of D and Q (trial division) is the definition checked; the certificate is a complete branch tree whose leaves are integer weighted-deficit inequalities on residual instances.","supports":"PASS means every branch node enumerates all q phases of an unfixed prime and every leaf satisfies sum_q max_b W(q,b) < W(R) in integers on its residual, which excludes every cover. It does not establish the producer's LP optimality or anything about other supports.","coverage_md":"All 3538 records: 31 branch nodes (every phase b = 0..q-1 of the branch prime present), 3507 leaves; the checker asserts no unreferenced or duplicate records.","comparison":"Exact string 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":"360","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"370","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/4","transcript_url":"/projects/twin-primes/return/1840/transcript","files":[{"sha256":"c656054c6e0cddf371f87064062eef779fda202cf8604c54f30e15774455162a","name":"sprint959.py","bytes":9157},{"sha256":"b998de6fa3fd60208b5c6c400efdf3354c414ee3c868b3a4a5ae59e50551f6aa","name":"cover959.py","bytes":2868},{"sha256":"7dbfd023e209b4ffe711ee7c67d9ec3b235cd8143190b84a4e0675d2484b1c90","name":"ifront959.py","bytes":2018},{"sha256":"ad236e224d91200b0c27bff8eeae94316e62d0faf22c16e5901442642ee2f106","name":"tree959.py","bytes":5766},{"sha256":"53879d8b75638f258732a20f2dc96219f4c988cafa109f858a0bc13e882f1765","name":"checkcert959.py","bytes":3215},{"sha256":"11d1a1039eeeb213fed42658173009f2d99fdc97d5a65189c9cc4f45b06ec666","name":"sat959.py","bytes":1418},{"sha256":"f4fc09f0d3e4368bd608b870274151fa1c5f22150328ee9ba027787ab1fe4411","name":"frozen959.out","bytes":7326},{"sha256":"512bbfab0f876242db7a7d5368acd09630e2d95c7e52c7d384825ac11ff343e6","name":"census959.out","bytes":28537},{"sha256":"101b109119dd74e72bdfd75b8d3e1a9eae7fd6c680ce7e1ee5175d939a5286e3","name":"census959w.out","bytes":30680},{"sha256":"fd617f3a3eb22169a5a788a16832dc24805002822e8edde1759308de65dc3d83","name":"cover959.out","bytes":2173},{"sha256":"d53ff175678518c36982b13c2b2488561f7646542449bfb09fdfcf8af4f80cb6","name":"gap959.out","bytes":7063},{"sha256":"c8ffeca713d55a5a4a79e8bcf47158c842a93b89502291756960a0b175bd6454","name":"ifront959.out","bytes":2769},{"sha256":"c4696351f1094f31f0a3cf01e5de493e8c36de96cd5c2271c778d9a5d0c72c53","name":"tree48.out","bytes":396},{"sha256":"9bc0acd1722f340eebaec8e4fc506ede22b82244727260f4512619b5654745e8","name":"tree959-9409-48.cert.jsonl","bytes":1170212}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}