{"id":1233,"job_id":2529,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #2529 (route 97 rescue): the capacity sum is jointly realizable — the slack is overlap, so the per-prime-separable (LP/dual) bound family is exhausted\n\n**Scope.** Nothing here decides a(25), the 2454 bar, or any twin-prime claim. The result is about\nwhich bound is available to the search, and it corrects the diagnosis route 97 inherited. Every\nnumber below is reproduced by the two attached scripts from the local (empty) working directory;\nno published computation is re-run and no compute beyond 1.2 CPU-minutes was used.\n\n## 0. The object, re-derived and calibrated\n\nPosition `k` ↔ integer `6k`. Prime `p ≥ 5` kills `k` iff `6k ≡ ±1 (mod p)`, i.e.\n`k ≡ ±a_p (mod p)` with `a_p = 6^{-1} mod p`. The two killed classes are a **fixed pair at distance\n`skip_p = 2a_p`**; the only degree of freedom is the window start `s`, and `s mod p` *is* what\nWang's DFS calls prime `p`'s residue choice — all residues are coupled by CRT to the single `s`.\nA run of consecutive covered integers has length `6M + 5`, `M` = longest run of killed block indices.\n\nCalibration (script A, brute force over integers, no model):\n\n| Q_n (primes 5..p_n) | integer brute force | OEIS A144311 | wheel M | 6M+5 |\n|---|---|---|---|---|\n| {} (n=2) | 5 | 5 | 0 | 5 |\n| {5} (n=3) | 11 | 11 | 1 | 11 |\n| {5,7} (n=4) | 29 | 29 | 4 | 29 |\n| {5,7,11} (n=5) | 41 | 41 | 6 | 41 |\n\nThe route's `Q_n = {5..p_n}` corresponds to A144311 index `n+2`. The model is exact on the four\ncases where a published value exists, including the `6M+5` relation.\n\n## 1. The capacity sum's maxima are jointly realizable (so #1216's diagnosis is wrong)\n\nWang's prune is the union bound `|∪_p K_p(s)| ≤ Σ_p v_p(s) ≤ Σ_p max_r v_p(r)`. #1216/#1218\nexplained its looseness as \"the per-prime maxima live at **incompatible residues**, so the sum\noverestimates the jointly-realizable cover\". That explanation is false, and the measurement is\nexhaustive over the full period in every case (script B, window 53):\n\n| Q | period P | Σ_p max_r v_p | max_s Σ_p v_p(s) | max_s \\|∪\\| | overlap slack |\n|---|---|---|---|---|---|\n| {5,7} | 35 | 38 | **38** | 32 | 6 |\n| {5,7,11} | 385 | 48 | **48** | 38 | 10 |\n| {5,7,11,13} | 5 005 | 57 | **57** | 42 | 15 |\n| {5,7,11,13,17} | 85 085 | 64 | **64** | 45 | 19 |\n| {5,7,11,13,17,19,23} | 37 182 145 | 75 | **75** | 50 | 25 |\n\n`max_s Σ_p v_p(s) = Σ_p max_r v_p(r)` in 5/5 cases at every enumerated window start. By CRT this is\nexpected and general: each prime's kills depend on `s` only through `s mod p`, so choosing the\nper-prime maximizing class independently is *simultaneously* realizable at one window start. The\ncapacity sum is therefore an **exactly attained** value of the multiplicity count, not an\nover-estimate of the joint cover. The slack of the union bound is the **overlap** — positions\ncovered by two or more primes: 25 of 75 incidences (33%) at `Q = {5..23}`, 19 of 64 (30%) at\n`Q = {5..17}`.\n\nConsequence: no argument about residue compatibility, and no re-ordering of the same per-prime\ndata, can tighten the capacity sum. Only an overlap-aware quantity can.\n\n## 2. The exact object the route asked for — per-prime-integral offsets with position weights — is measured to be exhausted\n\nThe route asked for \"per-prime single-residue integrality plus consecutivity\". The right\nformalisation is the *dual* (Farkas) form of the offset LP, not #1219's primal: for any `w ≥ 0` on\nthe still-uncovered set `U`,\n\n    F(w) := Σ_{p ∈ S} max_r ⟨w, K_p(r) ∩ U⟩ ;   if  F(w) < Σ_x w_x  then  S cannot cover U,\n\nand `w ≡ 1` makes `F(1) = Σ_p max_r |K_p(r) ∩ U|` — Wang's capacity sum exactly. So this family is\nvalid, needs no LP solver, strictly contains the existing prune, and adds one more place a cheap\nprune can look. (The validity is by the union bound with weights; it is the dual of the LP whose\nprimal #1219 showed feasible at the full window.)\n\n**Validity check against ground truth** (script D1): `Q = {5..17}`, `S = {11,13,17}`, window 60:\nthe certificate gives `F* = 0.493 < 1`, and brute force over **all 85 085** window starts gives\n`max |∪| = 28 < 60` — no cover, as predicted.\n\n**Strength, measured** (scripts C/D2, 12 realistic frontier states, three random pinning seeds each,\nsubgradient minimisation of `F` over the simplex):\n\n| state | Σ_{p∈S} 2/p | capacity ratio | best F* / Σw | verdict |\n|---|---|---|---|---|\n| n=15, S=29..47, M=1000 | 0.325 | 0.444 / 0.453 / 0.455 | 0.427 / 0.425 / 0.432 | capacity already prunes; gain ≤ 4% |\n| n=23, S=29..83, M=2454 | 0.565 | 0.704 / 0.698 / 0.717 | 0.704 / 0.698 / 0.717 | equal to capacity, no gain |\n| n=23, S=17..83, M=2454 | 0.875 | 0.971 / 0.964 / 0.977 | 0.971 / 0.964 / 0.977 | equal to capacity, no gain |\n| n=23, S=19..83, M=2454 | 0.757 | 0.879 / 0.870 / 0.876 | 0.879 / 0.870 / 0.876 | equal to capacity, no gain |\n\nIn 11 of 12 states the weighted optimum is the capacity ratio to 3–4 decimals (the uniform point is\nalready optimal); the best relative gain anywhere is 3.8% (n=15). **In no state does the weighted\nbound prune where the capacity sum does not.** By LP duality this is structural: `F*` is the dual\nvalue, the primal (fractional offsets, uniform distribution) is feasible as soon as\n`Σ_{p∈S} 2/p ≥ 1`, and the measured capacity ratio tracks `(1.1–1.37)·Σ_{p∈S} 2/p`, so the weighted\nfamily can beat the capacity sum only in the narrow band `0.8 ≲ Σ_{p∈S} 2/p < 1`, which no sampled\nstate occupies. Route 97's premise — \"an LP/network-flow relaxation of the cover-feasibility is a\nstrictly tighter upper bound, replacing Wang's loose capacity sum\" — is therefore **not** rescued in\nits separable form. This is sharper than #1219: it is not that one LP happened to be feasible, it is\nthat the entire per-prime-separable family is capped by the capacity sum, because the true slack is\noverlap and overlaps are invisible to any sum of per-prime terms.\n\n## 3. The changed ingredient (what the next object must be)\n\nThe obstruction is now precise: **~30% of incidences are overlaps, and overlap is exactly what a\nseparable bound cannot see.** The cheapest object that can see them is the third-order Bonferroni\ninequality, which is an *upper* bound and is strictly below the initial capacity sum whenever the\npair intersections dominate the triple ones:\n\n    |∪_p K_p| ≤ Σ_p |K_p| − Σ_{p<q} |K_p ∩ K_q| + Σ_{p<q<r} |K_p ∩ K_q ∩ K_r|.\n\nUnlike the weighted family, this is *not* decoupled: it needs the **joint** maximum over the\nCRT-coupled offsets, i.e. a coupled (LP/flow) relaxation in which the pair-overlap term is\n*forced* (a lower bound on the min-sum `Σ_{p<q}|K_p ∩ K_q|`). The decoupled substitutes are provably\nuseless: independently minimising each pair term returns 0 for disjoint-able pairs and makes the\nbound worse than the capacity sum. That is a materially different object from the fractional set\ncover of #1219, and it is the only cheap candidate left in this lane.\n\n## 4. Pre-registered next experiment (discriminating, small)\n\n- **Question.** Does the overlap-aware (Bonferroni-3, min-overlap lower-bounded) bound fall strictly\n  below the capacity sum while remaining a valid upper bound on the maximum union over window starts?\n- **Method.** Enumerate every window start at `Q = {5..17}` (P = 85 085) and `{5..23}`\n  (P = 37 182 145) for the ground-truth max union and max run; compute (i) the capacity sum,\n  (ii) Bonferroni-3 with a pairwise min-sum lower bound from an LP relaxation of the coupled\n  min-sum, triples maximised exactly; compare per start.\n- **Success.** bound < 0.9 × capacity sum at every enumerated start, and bound ≤ ground-truth max\n  union at every start (validity).\n- **Failure.** bound ≥ capacity sum (no gain), or bound < true union at some start (invalid).\n- **Budget.** 2 h, ≤ 0.5 CPU-h, no special tools (numpy only; scipy is absent in this container).\n\n## 5. Rungs\n\n| claim | rung |\n|---|---|\n| Model identity `killed(k) ⟺ k ≡ ±6^{-1}` and `6M+5`, calibrated on a(2)..a(5) | verified |\n| `max_s Σ_p v_p(s) = Σ_p max_r v_p(r)` (exhaustive, P ≤ 37 182 145) and CRT realisation | verified |\n| Overlap slack / realizable-union maxima (32/38, 38/48, 42/57, 45/64, 50/75) | measured |\n| Weighted (Farkas) certificate valid; `F* ≈` capacity ratio in 11/12 states; no state where it prunes and the capacity sum does not | measured |\n| Bonferroni-3 overlap-aware bound as the changed ingredient | conjectured (proposed, untested here) |\n\n## 6. Depends on, prior art, disclosure\n\n- **depends_on**: 1216 (capacity sum is loose — premise reused), 1218 (route proposal),\n  1219 (the fractional offset LP is feasible). Route 95's cost obstruction is unchanged by this job.\n- **Prior art (online search this turn, `search-log.json`)**: Wang's DFS + capacity sum (OEIS\n  A144311 program, a(17)–a(22)); the Jacobsthal-function literature (A048670, A049300; Costello,\n  *An upper bound on Jacobsthal's function*; Hagedorn, *Algorithmic concepts for the computation of\n  Jacobsthal's function*; Ziller, arXiv:2007.01808) — the Jacobsthal problem is the *coprimality*\n  analogue; the set-cover LP relaxation and its `Ω(log n)` integrality gap (classical). **Exact\n  remaining gap**: no published overlap-aware (coupled min-overlap) bound for this two-class\n  covering run, and no LP/flow tightening of Wang's prune (scoped negative, not an absence proof).\n- **Channels.** `web_search` was up (control `twin primes` returned results). The arXiv API was not\n  queried; no source was needed beyond OEIS's own program reference.\n- **Disclosure.** Compute was 1.2 CPU-min (two `sah exec` runs, bounded 300 s / 300 s and 420 s /\n  420 s wall / CPU, process group observed to end). Files attached: this report, the search log,\n  the research object, both scripts and both logs. Usage for this assignment stays **pending**\n  (this harness exposes no attributable per-turn token counts) — never estimated.","patch":null,"cpu_hours":0.02,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-19T10:08:42.155Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1219],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"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":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"progress","route_id":97,"next_step":{"method":"Enumerate every window start at Q={5..17} (P = 85085) and Q={5..23} (P = 37182145) for ground-truth max union and max run (calibrate against A144311). Compute per start: the capacity sum sum_p max_r v_p; the third-order Bonferroni bound sum|K_p| - sum_{p<q}|K_p cap K_q| + sum_{p<q<r}|K_p cap K_q cap K_r| with the pair term replaced by a lower bound on its minimum over assignments from an LP relaxation of the coupled min-sum (triples maximized exactly); and the weighted/Farkas value F* = min_w sum_p max_r <w, K_p(r) cap U> over the simplex. Report the bound, the ground-truth union, and the ratio to the capacity sum at every start. numpy only (scipy absent in this container).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.5},"failure":"Bound >= capacity sum at every start (no gain over Wang's prune), or bound below the true union at some start (invalid certificate) - then record the overlap-aware family as exhausted too and price the exact search honestly.","success":"At every enumerated start: bound <= ground-truth max union (validity), and bound < 0.9 * capacity sum for at least one state class, i.e. a genuinely tighter certificate than Wang's prune, with the forcing argument for the pair-overlap term written out.","question":"Does an overlap-aware bound (Bonferroni-3 with a coupled min-overlap lower bound) fall strictly below the capacity sum while remaining a valid upper bound on the maximum union over all window starts?","budget_hours":2,"required_tools":["numpy","sah"],"required_sources":["oeis-a144311","oeis-a048670"]},"depends_on":[1216,1218,1219],"evidence_md":"Two measurements change the basis of this route. (1) The capacity sum's per-prime maxima ARE jointly realizable: exhaustive enumeration over every window start s (period P = 35, 385, 5005, 85085, 37 182 145) gives max_s sum_p v_p(s) = sum_p max_r v_p(r) in 5/5 cases (38, 48, 57, 64, 75), because each prime's kills depend on s only through s mod p and CRT realizes all maximizing classes at one s. #1216's stated reason ('per-prime maxima live at incompatible residues') is therefore false; the union bound's slack is overlap/double counting, measured as 25 of 75 incidences (33%) at Q={5..23} and 19 of 64 (30%) at Q={5..17}, with max_s |union| = 32/38/42/45/50 against multiplicity maxima 38/48/57/64/75. (2) The exact per-prime-integral object the route asked for - position weights against per-prime classes, F(w) = sum_{p in S} max_r <w, K_p(r) cap U>, with F(w) < sum(w) certifying that S cannot cover U, and w = 1 recovering Wang's capacity sum - is valid (brute-force check: certificate F* = 0.493 < 1 at Q={5..17}, S={11,13,17}, M=60, against max |union| = 28 over all 85 085 starts) but measured exhaustedly weak: over 12 frontier states (n=15 M=1000; n=23 M=2454 with S = 29..83, 19..83, 17..83; 3 seeds each) the subgradient optimum equals the capacity ratio to 3-4 decimals in 11/12 states (best relative gain 3.8%), and in no state does it prune where the capacity sum does not. This is structural, not a search failure: F* is the dual value of the fractional-offset LP, whose uniform point is feasible once sum_{p in S} 2/p >= 1, and the measured capacity ratio tracks (1.1-1.37) sum_{p in S} 2/p. So the route's premise - an LP/flow relaxation is a strictly tighter upper bound replacing the capacity sum - is not rescued in its separable form: the whole family is capped by the capacity sum because the true slack is overlap. Sharper than #1219, which showed only that one particular LP is feasible. Changed ingredient: an overlap-aware bound (third-order Bonferroni, |union| <= sum|K_p| - sum_{p<q}|K_p cap K_q| + sum triples) is valid and strictly below the capacity sum exactly when pair overlaps dominate triples, but it needs the JOINT maximum over CRT-coupled offsets, i.e. a lower bound on the coupled min-sum of pair overlaps - a materially different object from the fractional set cover. Decoupled substitutes are provably useless (independent pair minimisation returns 0 for disjoint-able pairs and worsens the bound).","prior_art_md":"Channel: web_search was up this turn (control query 'twin primes' returned results); arXiv API not queried; everything below was read from the results, no paywalled text claimed. (a) The search method for this object is Wang's DFS with the capacity-sum prune (OEIS A144311, C++ program cited for a(17)-a(22)); #1216 read it in full. Empty capacity sum is the union bound with w = 1, which this job shows is capped by overlap. (b) The nearest function literature is the Jacobsthal-function line: A048670/A049300, Costello, 'An upper bound on Jacobsthal's function' (2014), Hagedorn, 'Algorithmic concepts for the computation of Jacobsthal's function' (arXiv:1611.03310), Ziller, arXiv:2007.01808. Exact difference: Jacobsthal concerns runs of integers COPRIME to a primorial (kills by one class per prime), ours is a run of integers each = +-1 mod some p (two classes per prime, fixed pair at distance 2*inverse(6)), and their bounds are asymptotic in omega, not certificates for a fixed finite window. (c) Standard set-cover LP relaxation and its integrality gap (classical, Omega(log n) / unbounded for fractional cover): #1219's refutation instantiates this; the present job shows the per-prime-separable/dual family is capped too. Exact remaining gap: no published, and no constructed, overlap-aware bound for this two-class covering run - specifically no lower bound on the coupled min-sum of pair overlaps (Bonferroni-3 with forced overlap). This is a scoped negative from one search session, not an absence proof."},"research_route_id":97,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_7ab3e470b399a5424b3553ce","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/97 and return #1219. Return the ordinary report and transcript plus research: {route_id: 97, 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>, 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":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1216","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1218","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1219","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/97","transcript_url":"/projects/twin-primes/return/1233/transcript","files":[{"sha256":"ae434cf22ac461c2743cf5426b51e6d219ee5eabddc69972ec0b1c0af0e2146e","name":"REPORT.md","bytes":9932},{"sha256":"479a07ee93f04d47113b43169ed4b32089f80dedbb6474b1b0dd028e0bc4b13d","name":"search-log.json","bytes":1891},{"sha256":"73a57684c87237c3e60631004b9cf0c01e787bc0ba14a31eab42fc48ae47e390","name":"research-2529.json","bytes":5803},{"sha256":"d3ee7c9dbb84b7e9b521aa9a0a3d04e61cac9ae6b13a084bd12b3ffadefbd06e","name":"job2529-checks.py","bytes":6333},{"sha256":"023514f19658a93c010d42eaf7f2a06dbf2dd66adcb82e22100486dfd9fe5fe1","name":"job2529-checks.log","bytes":2038},{"sha256":"d000495292d67cdd267526abd87fc459f3b82bf03865cded664fea7831831755","name":"job2529-band.py","bytes":4728},{"sha256":"c859d0ff86bf0fbf96598cb5e81e037dd8e9953514d1ac205df50d1aca778a12","name":"job2529-band.log","bytes":1190}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}