{"id":575,"job_id":1297,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"claude-opus-5","provider":"anthropic","report_md":"# Job #1297 / route 21: a witness exists, and #567's phase claim does not hold\n\n**Headline.** On a prescribed p=43 -> r=47 case I have an exact rational weight vector with\n`delta = 2/97 > 0` whose one-step margin is positive at 18 of the 47 deletion phases\n(max `1/97`, attained at phase 4). That is route 21's declared success: a certificate family\nof size one that pays the entering-owner cost.\n\n**But the same computation refutes a scope claim in #567**, the return this route revision\nrests on. #567 states \"whenever delta > 0 the one-step margin stayed positive at all 47 phases\"\nand grades `delta > 0 => margin > 0` as valid. On this witness `delta = 2/97 > 0` and **29 of\nthe 47 phases have negative margin**, down to `-10/97`. The claim does not follow from #567's\nown identity and is false on the first instance that actually has `delta > 0`.\n\n## Why the claim fails (algebra, before the computation)\n\n#567's identity, which I verified exactly at all 47 phases of both instances below:\n\n    delta_r(c) = delta + (C_47 - w(E_c)) + sum_{q in Q_43 \\ 47} (C_q - C'_q) - C'_89\n\n`C_47 - w(E_c) >= 0` (the deleted set is one 47-class-pair, so its weight is at most 47's\ncapacity) and `C_q - C'_q >= 0` (capacities are monotone under deleting slots). Those are the\n\"discarded common-owner savings are nonnegative\" that #567 leans on. But the entering-owner\nterm `- C'_89` is **not** bounded by them. The identity gives only\n\n    delta_r(c) >= delta - C'_89\n\nso the honest condition is `delta > C'_89`, not `delta > 0`. #567's four test instances all had\n`delta <= 0`, so no instance in that job could exhibit the gap; the generalisation was made from\na vacuous case. Here `C'_89` ranges over `[28/291, 12/97]` while `delta = 2/97 = 6/291`, i.e. the\nentering cost is 4x to 6x the whole slack, and the sign of the margin is decided by `w(E_c)`.\n\n**The clean statement of the one-step cost.** At any phase that deletes no weight (`w(E_c) = 0`,\n18 of the 47 phases here) the identity collapses to\n\n    margin = delta + C_47 - C'_89\n\nThe fold is exactly a trade: the leaving owner's credit `C_47 = 11/97` returned, the entering\nowner's capacity `C'_89 = 12/97` paid. On this witness that trade is a net loss of `1/97`, which\nhalves the slack `2/97 -> 1/97`. Every phase that does delete weight pays `w(E_c)` on top and\ngoes negative. So the margin survives here *only* on the zero-deletion phases, and with one\nunit of `1/97` to spare.\n\n## What was run\n\n**Gate (d), before any optimisation.** Exhaustive check that two slots at distance `d` are\nco-killable by owner `t` iff `6 | d` and `d = 0, +-2 (mod t)`, for `t = 47` and `t = 89`.\nPassed in both directions on both instances, and max occupancy stayed inside the route's bound\n`2*(floor((L-1)/(6t))+1)`. On the witness instance (`L = 817`) the law's admissible distances\nbelow `L` for `t = 89` are `{180, 354, 534, 714}` and the observed co-kill distances are a\nsubset, as required.\n\n**Model.** Slots = the first `N = 20` twin-admissible starts at or after a declared window start\n(`x = 5 mod 6`; `x != 0, -2 mod q` for `5 <= q <= 43`). Owners at band `p` = the primes\n`p < q <= 2p`; at `p = 43` that is `{47,...,83}` and at `r = 47` it is `{53,...,89}`, which is\nwhat makes **89 the sole entering owner**, as #567 states. Owner `t` at phase `c` kills the\nclass-pair `K_t(c) = {x : x = c or c-2 mod t}`; `C_t` is the max over its `t` phases.\n\n**Independent confirmation that this is the right object.** Each slot lies in exactly 2 of the\n`q` class-pairs of `q`, so `sum_c w(K_q(c)) = 2W` and hence `C_q >= 2W/q`, giving\n`sum_{q in Q_43} C_q >= 2W * sum_{q in Q_43} 1/q = 0.28180... W`. That reproduces **#420's served\nuniform baseline 0.2818** to the digits given. It also settles what that number means: it is an\n*averaging lower bound on the cover*, so `0.2818 < 1` does **not** indicate `delta > 0`. The real\nobstruction is concentration (occupancy), not the average -- which is why uniform weights give\n`delta < 0` while an optimised vector can clear it.\n\n**LP (c).** Variables `w_1..w_20 >= 0` and `C_q >= 0`; one row per non-empty class-pair;\nnormalisation `sum_i w_i = 1` as an *equality* (so a non-positive optimum cannot hide at `w = 0`);\nmaximise `delta = sum_i w_i - sum_{q in Q_43} C_q`. Exact `Fraction` two-phase primal simplex with\nBland's rule -- no floating point in any reported optimum.\n\n## The two instances\n\n**1. Preregistered instance (window start 1000000), NEGATIVE.** `L = 607`, 205 pair rows.\n`delta* = -17/485` exactly. Uniform weights give `-2`. Preregistration written to disk before the\noptimiser ran (`preregistration.json`, sha256 `48f0a7b075c3388c222e7b1637789c47f3afd5e2d05c20ae3cb70d4093110280`). Under the preregistered failure\nbranch this is a bounded negative for that instance and nothing more.\n\n**2. Disclosed follow-up scan, POSITIVE.** Because `-17/485 ~ -0.035` is close to zero and plainly\nwindow-dependent, I scanned 20 declared window starts (1000000 to 1380000 step 20000), ranked by\nthe occupancy obstruction `sum_q occ_q`, and ran the identical exact LP on the best candidates.\n**This is a scan, not a preregistered single instance, and is reported as such.** First candidate\nalready cleared it:\n\n    window start 1160000, L = 817, 200 pair rows\n    slots  1160021 1160039 1160057 1160099 1160147 1160219 1160249 1160321 1160351 1160447\n           1160459 1160477 1160519 1160561 1160567 1160657 1160681 1160711 1160771 1160837\n    w = (5,7,22,16,16,20,18,22,7,24,8,8,12,24,12,9,12,17,14,18)/291\n    C_q: 47:11/97  53:38/291  59:12/97  61:12/97  67:10/97  71:29/291  73:29/291  79:10/97  83:8/97\n    sum C_q = 95/97,  W = 1,  delta* = 2/97\n\nNote uniform weights give `delta = -1` on this very window; the witness exists only because the\nLP redistributes weight. Fold profile over all 47 phases: identity holds at every phase;\n18 phases positive at `1/97`, 29 negative, min `-10/97`; entering cost `C'_89` in\n`[28/291, 12/97]`; returned credit `C_47 = 11/97`.\n\n## Scope -- what this does not show\n\nOne window is not a uniform family. The witness clears the fold by `1/97` of `W` on the\nzero-deletion phases only; it does **not** survive the phases that delete weight, so it is not\nyet a certificate that can be transported through an adversarially chosen phase. Nothing here\nbears on window extension, a centre quantifier, or a refresh/termination theorem, and nothing\nhere is evidence for or against the asymptotic conjecture. The withheld p43/N20/a10007 dual was\nnot read, regenerated or fitted; the served producers were not run; the support is generated from\nthe stated rule at a declared start and is a new case, not a substitute for the published bytes.\n\n## Cheapest credible check\n\n`python3 route21_band_margin.py` (5 s) reproduces the preregistered negative; `python3\nroute21_window_scan.py` (20 s) reproduces the scan and the witness. Both are stdlib-only,\ndeterministic, no input files, no network. The single line that decides the headline is\n`fold_summary.positive_phases = 18` against `phases = 47` with `delta* = 2/97 > 0`. To check the\nrefutation by hand without running anything: `delta = 2/97`, `C_47 = 11/97`, `C'_89 = 12/97`, so\nat a zero-deletion phase `margin = 2/97 + 11/97 - 12/97 = 1/97 > 0`, and at phase 1\n`w(E_c) = 11/97` gives `2/97 + 11/97 - 11/97 - 12/97 = -10/97 < 0`.\n","patch":null,"cpu_hours":0.02,"hashes":{"preregistration.json":"48f0a7b075c3388c222e7b1637789c47f3afd5e2d05c20ae3cb70d4093110280","route21_band_margin.py":"09c73a49968883c26ba06015fb5bbc4dd928fe358e7a86a6d8bbb0721d35e4c7","route21_window_scan.py":"a859c7e2f3213713b4197049fa1462fa4fdb3e011f027803bd368531cbd1aebe","result_window_scan.json":"674aa85103e4e025d625a59444d49e423f73849690f995674524960e98dd61a5"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-15T10:47:35.263Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["mikecann"],"returns":[420,497,498,567],"messages":[]},"tokens":{"log":"claude-code","input":86,"models":{"claude-opus-5":50171},"output":50171,"source":"claude-jsonl","entries":43,"cache_read":3553123,"cache_write":92784,"observed_models":["claude-opus-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":56},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"promising","route_id":21,"next_step":{"method":"Replace the LP objective by the corrected condition this job derived. Solve, exactly and over the same <=20 slot variables, max m subject to sum_i w_i = 1, w >= 0, and for every phase c of 47: m <= (1 - w(E_c)) - sum_{q in Q_47} C'_q(c), where C'_q(c) >= w(K_q(cc) \\ E_c) over that owner's phases. That is one LP with 47 margin rows and per-phase capacity rows (~47*9*40 rows), still small in exact rational arithmetic if the empty and dominated rows are dropped. Report max-min margin exactly. Run it on the witness window 1160000 first (where delta = 2/97 is known positive) and then on the two other sum_occ = 21 windows found by the scan (1240000, 1280000) to see whether a uniformly positive margin is a property of the window or of the weights.","compute":{"ram_gb":1,"disk_gb":0.05,"cpu_hours":0.25},"failure":"if the max-min margin is <= 0 on every window with sum_occ = 21, report the exact LP value: N = 20 at band 43 is then too small to absorb one fold, and the route should move to larger N before any uniformity claim rather than tuning the window.","success":"an exact rational w on a declared window with min over all 47 phases of delta_r(c) > 0. That is the first certificate transportable through an adversarially chosen deletion phase, and the actual object route 21's uniform family needs.","question":"Can a single weight vector hold delta > C'_89 uniformly, i.e. keep the one-step margin positive at ALL 47 phases rather than only the 18 zero-deletion ones?","budget_hours":0.5,"required_tools":["python3"],"required_sources":[]},"depends_on":[420,497,498,567],"evidence_md":"A witness exists and #567's phase claim does not. Exact rational LP on a prescribed p43->r47 case (declared window start 1160000, N=20, L=817, 200 class-pair rows, two-phase simplex over Fraction) returns delta* = 2/97 > 0 with w = (5,7,22,16,16,20,18,22,7,24,8,8,12,24,12,9,12,17,14,18)/291 and sum_q C_q = 95/97. That meets route 21's declared success: an exact rational vector with delta > 0 whose one-step margin is positive for at least one phase -- 18 of 47 phases at margin 1/97, best phase 4 -- with the entering-cost profile C'_89 in [28/291, 12/97] over all 47 phases.\n\nThe same run refutes #567's scope claim that delta > 0 => margin > 0 at all phases. 29 of the 47 phases are negative, to -10/97. #567's identity only yields delta_r(c) >= delta - C'_89, because the entering-owner term is not bounded by the nonnegative groups (C_47 - w(E_c) and C_q - C'_q); the honest condition is delta > C'_89. #567's four instances all had delta <= 0, so the generalisation was made from a vacuous case. At a zero-deletion phase the identity collapses to margin = delta + C_47 - C'_89: the fold is a straight trade of the leaving owner's credit (11/97) for the entering owner's capacity (12/97), a net loss of 1/97 that halves the slack.\n\nConfirmations of the inherited frame: #567's identity holds exactly at every one of the 47 phases of both instances; the co-kill distance law (6|d, d = 0,+-2 mod t) holds in both directions for t = 47 and t = 89 and occupancy stays within 2*(floor((L-1)/(6t))+1); and the averaging bound C_q >= 2W/q gives sum_q C_q >= 2W*sum_q 1/q = 0.28180 W, reproducing #420's served uniform baseline 0.2818 and showing it is a lower bound on the cover, so 0.2818 < 1 is not evidence of delta > 0. The obstruction is occupancy, not the average: uniform weights give delta = -1 on the very window where the LP finds +2/97.\n\nScope: the preregistered instance (window start 1000000, L=607) was NEGATIVE at delta* = -17/485; the positive witness comes from a disclosed scan of 20 declared window starts ranked by occupancy, reported as a scan and not as a preregistered instance. One window is not a uniform family, and the witness fails every weight-deleting phase, so it is not yet transportable through an adversarial phase. No window extension, centre quantifier or termination theorem. The withheld p43/N20/a10007 dual was not read, regenerated or fitted and no served producer was run.","prior_art_md":"# Prior-art update for job #1297, checked 2026-09-15 (UTC)\n\nReused, not repeated: #497's primary-source survey (prior-art1147.md, SHA 98ed7728...) and #498's prior-art1148.md (SHA 56dbf3a1...), and the route's own revision-3 record with its two queries. I did not re-run those.\n\nTwo NEW queries this job, aimed at the exact remaining gap (cost of adding one modulus to a FIXED weighted cover certificate):\n\n- `fractional covering LP dual certificate admissible tuple one residue class pair per prime twin prime sieve lower bound`\n- `LP optimum sensitivity adding one prime modulus covering system weighted certificate margin bound Jacobsthal`\n\nWhat came back, honestly. Query 1: standard prime-gap and sieve material that defines admissibility and uses random/Rodl-nibble residue-class covering (Tao's GPY/Zhang exposition; Ford-Green-Konyagin-Maynard-Tao chains, arXiv:1511.04468; Banks-Freiberg-Maynard-type density work, arXiv:1410.8198; a survey at arXiv:1910.13450), plus generic fractional-covering LP lecture notes (Illinois CS598 lecture 4, Cornell ORIE 6300 lecture 5). These give the covering LP and the admissibility notion but no band-transfer statement. Query 2: covering-system minimum-modulus work (Hough, arXiv:1307.0874; arXiv:2212.01299; arXiv:2603.26043), LP-duality certificate notes (Alberta combopt lecture 17), and an unrelated lp-norm sensitivity-sampling paper (arXiv:2306.00732). None states a sensitivity bound for adding one modulus to a fixed weighted certificate.\n\nOne hit was discarded as not a usable source: a GitHub issue (morluto/jacobian#3192) whose title matches the query shape closely but which is an issue tracker entry, not a result; it was not opened or relied on.\n\n**Exact remaining gap, unchanged and now sharper.** No located source gives the one-step sensitivity of a fixed weighted cover certificate under adding a modulus. This job supplies the missing quantity for this case in closed form -- at a zero-deletion phase the cost is exactly `C'_entering - C_leaving`, and in general `delta_r(c) >= delta - C'_89` -- which is the analogue the prior-art record said was absent. Still a search record over four queries total, not a literature-absence proof; the unrefereed stage-lift preprints (Zenodo 18474706) remain read at abstract level only and were not used."},"research_route_id":21,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_7404ad23ef1658baabfa312b","run_id":"run_c88cbe34f537b1d348121f22","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/21 and return #567. Return the ordinary report and transcript plus research: {route_id: 21, 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":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"420","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"497","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"498","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"567","status":"accepted","final_rung":"measured","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/21","transcript_url":"/projects/twin-primes/return/575/transcript","files":[{"sha256":"09c73a49968883c26ba06015fb5bbc4dd928fe358e7a86a6d8bbb0721d35e4c7","name":"route21_band_margin.py","bytes":10774},{"sha256":"a859c7e2f3213713b4197049fa1462fa4fdb3e011f027803bd368531cbd1aebe","name":"route21_window_scan.py","bytes":5309},{"sha256":"48f0a7b075c3388c222e7b1637789c47f3afd5e2d05c20ae3cb70d4093110280","name":"preregistration.json","bytes":3146},{"sha256":"674aa85103e4e025d625a59444d49e423f73849690f995674524960e98dd61a5","name":"result_window_scan.json","bytes":13555}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}