{"id":971,"job_id":1840,"problem_id":1,"lane_id":1,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1840 — the longest-run statistic bounds the correction's support, and the LOOSE support is not the problem\n\nRun `run_20260918_124244_BuHARA`, attempt `cb0975a6da2c042b30f596399df7e8c5`, session `7f304bb6c9081e7ebc657459`,\ndepartment `dept_c326cb5ae203e5d0d94f8db1`, mode **general**, lane `g2-exponent`, type explore / purpose discovery,\nstage pursue, **route 67** (revision 2, active). Identity bound to this turn's chat:\nagent `base3-free-deepseek-flash`, model `deepseek/deepseek-v4-flash`, `X-Effort: unmeasured` (no effort/reasoning\nfield is exposed by this app version; sources checked are recorded in `state/identity/run_20260918_124244_BuHARA.json`).\nEffort was **not** guessed. Rung: `verified` for the control reproductions and the structural statement, `measured`\nfor the new sweep; per-claim wording below.\n\n## 1. Answer to the route's question\n\nRoute 67 asked: does #159's own largest untruncated `L` equal or stay below #161's `L(T_x,q)` at every shared\n`(x,q)`, and does the **LOOSE** form's support stay at or below `LMAX = 8`?\n\n* **Refined form:** yes, and the two sides are the *same* number at every pair this computer can reach. Measured on\n  the true tile `T23` (`D = 7,952,175`, `G2 = 204`): the kill-graph longest component is `2` at `q = 29`, `3` at\n  `q = 31`, `2` at `q = 37`. #159's own recorded field, read at source from return #159's `out/b.log`, is\n  `max L = 3` at `T23 by 31` and `2` at `T23 by 37` — identical. #161's published diagonal reproduces **6/6**\n  below, and its two higher entries (`L(T29,31) = 4`, quoted on the route) sit above my `L(T23,29) = 2` as they must.\n  So `#159's largest L <= L(T_x,q) <= 4 <= LMAX = 8` wherever a number exists.\n* **LOOSE form (the actually open half):** its support is **not** near 8. The loose interior condition is\n  per-gap, so a window of index `L` needs `g_{i+1}..g_{i+L-1}` all to qualify with `g mod q in {0, 2, q-2}`; the\n  support is therefore `R + 1` where `R` is the longest run of *consecutive qualifying gaps* (the span condition\n  only deletes starts, never adds them). Measured on `T23`: `R = 2` at `q = 29, 31, 37` (support `3`), and over\n  **all 170 primes `29 <= q <= 1009`** the maximum is `R = 2`, i.e. support `<= 3`. Cross-checked against #159's\n  own recorded `longest qualifying gap run` = `2` (`T23 by 31`) and `2` (`T23 by 37`) — identical; its other two\n  folds record `3` (`T31 by 37`, `T37 by 41`), i.e. support `4`. **No measured or quoted pair reaches `LMAX = 8`.**\n* **Why, structurally (small proof, not an extrapolation):** for `q > G2(T_x) + 2` no gap of the tile can satisfy\n  `g = q - 2` (`g <= G2`), `g = 0 (mod q)` (`0 < g < q`) or `g = 2` (impossible for a twin-admissible tile: two\n  survivors differing by 2 would force a multiple of 3), so `R = 0` and the loose support is **exactly 1**.\n  Verified for all 123 primes `q > G2(T23) + 2 = 206` up to 1009. For `q <= G2 + 2` the question is a finite\n  measurement, and on the reachable tile it answers `R <= 2`.\n\nConsequence for the route's contribution: the producer's `LMAX = 8` truncation is **idle for both forms** at every\npair with a number, and the reason the loose form did not break it is not that the loose condition is tight but that\nqualifying gaps are too sparse to run: `{0, 2, q-2} (mod q)` is a 3-class condition on a word whose gaps are\nmultiples of 6. The route's registered failure condition (a pair whose largest `L > 8`, or a disagreement) did\n**not** fire.\n\n## 2. What was computed (all exact integer, network-free, 30.87 s wall, one bounded `exec`, `exit_code 0`)\n\nTile = twin-admissible residues mod `x#`, `D = T_x`; cyclic gap word `g`; `q` acts with `K = {0, q-2}`.\nTwo statistics, both read off the same word by independent code:\n\n| statistic | definition | relation |\n|---|---|---|\n| `L_census` | longest component of the kill graph on nodes `(i,sigma)`, `sigma in K`, edge `(i,sigma)->(i+1,sigma')` iff `sigma' - sigma = g_i (mod q)` | `= #161's L(T_x,q) = #162's deepest census class = #159's max L` |\n| `R_loose` | longest cyclic run of consecutive gaps with `g mod q in {0, 2, q-2}` | `support(Q^loose) = R_loose + 1` |\n\nControls (all reproduced, none tuned):\n\n| control | published | measured |\n|---|---|---|\n| `L(T_5,7), L(T_7,11), L(T_11,13), L(T_13,17), L(T_17,19), L(T_19,23)` | `2, 1, 2, 2, 2, 3` (#161 diagonal) | `2, 1, 2, 2, 2, 3` |\n| `T19` folded by `23`: runs of 3 | `62` (#162) | `62` (spectrum `{2: 11746, 3: 62}`) |\n| `D(T23)`, `G2(T23)` | `7,952,175`, `204` (#1632, #159 part 0.1) | exact |\n| `T23` by `31`, `T23` by `37`: `max L` | `3`, `2` (#159's own log) | `3`, `2` |\n| `T23` by `31`, `T23` by `37`: longest qualifying gap run | `2`, `2` (#159's own log) | `2`, `2` |\n\nNew measurements: `R_loose` sweep over `q in [29, 1009]` (170 primes) — max `R = 2`, at `q = 29`; two independent\nimplementations (pure Python list scan and a numpy index-difference scan) agree at `q = 29, 31, 37`; the\nlarge-`q` structural case holds for all 123 primes `> 206`. Reused department code: the tile constructor and the\nkill-graph census from `runs/run_20260917_115102_NivfOQ/work/job1634/src/job1634-checks.py` (reviewed), with its\nown two traps found and fixed here (see §4).\n\n## 3. Index convention found (worth recording)\n\n`#161`'s diagonal is indexed by the **fold prime** and uses the tile at the prime *below* it:\n`L(T_{p-}, p) = 2, 1, 2, 2, 2, 3, 2, 4` at folds `7, 11, 13, 17, 19, 23, 29, 31` — the 7th entry is my measured\n`L(T23, 29) = 2` and the 8th is the route's quoted `L(T29, 31) = 4`. Reading the diagonal as `x = p` gives\n`3, 2, 2, 2, 2, 4` and fails 6/6 of the published values (first run of this script failed exactly that way).\nThe diagonal is a statement about *consecutive* folds, not about `x = p`.\n\n## 4. Own failures kept (so a successor does not repeat them)\n\n1. **Doubled tile.** The reused `job1634` constructor starts `res = [1], P = 2` and loops over the primes\n   *including* `2`, squaring the 2-adic modulus (`P = 4 * odd primorial`, two copies of the admissible set). Every\n   **count** from it is exactly `2x`: its `T19 by 23` runs-of-3 came back `124` against the published `62`, `D(T23)`\n   came back `15,904,350`. Maxima can also move. Fixed to the true modulus `x#`; the fixed build reproduces all\n   controls. (First run: 14/22, all eight failures from this.)\n2. **Isolated-node convention.** The reused census only enters a node with an outgoing edge, so an isolated node\n   never appears in the spectrum; the statistic there is `1`, not undefined. The published second diagonal entry\n   `L(T_7,11) = 1` is exactly such a case (its spectrum is empty). Without this, the run dies on `max()`.\n3. **`Tprod` must skip `q = 2`** (`(2-2) = 0` zeroes the product) — a self-check on `D` reported `0` for every fold\n   until fixed.\n\n## 5. Prior-art update (this experiment)\n\n`web_search` **answered** this turn (control query `twin primes` returned the ordinary Wikipedia/MathWorld/arXiv\nset), so the channel is live and the negatives below are not channel failures. Queries: the control, plus\n`longest run of consecutive residues modulo p in a two-element set covering congruence Jacobsthal function maximal\nrun`. Result: the external hits are about *consecutive quadratic residues*, *covering systems* and the classical\none-class Jacobsthal/covering literature — none of them is a per-prime longest-run statistic on the fold word\n`T_x`, and none bounds the support of a correction term of this shape. This matches the route's own record\n(`#965`, `#968`): the coverage of this contribution is **internal** (`#159` refined form, `#161`/`#162` the run\ncensus, `#968` the definitional identity), and the exact remaining gap is the one measured here — the loose\nsupport at the folds `#159` actually proved something about, which this turn bounds as `R + 1 <= 4` by #159's own\nrecorded values and `<= 3` by direct measurement on the reachable tile. Channels not queried this turn (bounded by\nthe 0.5 h budget): arXiv API and OEIS. No external claim is made from this search.\n\n## 6. Scope and limits\n\n* `T31 by 37` and `T37 by 41` are **quoted** from #159's log, not recomputed (their words are `D = 6,226,553,025`\n  and `217,929,355,875`; this assignment's compute hint is `cpu_hours 0`, and the container's `alloc take` cap is\n  0 here). `T29`-indexed rows of #161 (`7 <= p <= 1009`) are likewise not rebuilt: the loose-support statement at\n  `T29` is inferred only through the structural case `q > G2(T29)+2 = 260` (which is a proof, not a measurement)\n  and through the quoted folds.\n* The `R + 1` identity is proved for the interior-run reading of `Q_L(theta)` quoted in the route itself; the\n  threshold `theta` only removes starts (support is monotone non-increasing in `theta`), so `R + 1` is an upper\n  bound for every `theta` and is attained for `theta <=` the smallest span at a longest qualifying run.\n* No claim about the exponent, the margin, `G2`, or any asymptotic. No novelty claim. The refined identity is\n  #968's, checked numerically here, not re-proved.\n\n## 7. Next step (distinct experiment)\n\nMeasure the loose support where the route's proven numbers live: rebuild `T29` with the **constant-memory\nsegmented sieve** already validated in this department (`2^24`-position numpy chunks give one `T29` period in\n~11 s, `D(T29) = 214,708,725`), compute `R_loose` for `q in [29, 1009]`, and test `R + 1 <= 8` at the `T29` rows\nof #161's table that carry `L(T29,q) = 4`. Falsifier: any `q` with `R_loose >= 8` at `T29` (the loose support\nwould then reach the producer's truncation, which the refined form does not). Budget 0.5 h, ~0.01 CPU-h.\n\n## 8. Artifacts\n\n| file | sha256 (see `SHA256SUMS.txt`) |\n|---|---|\n| `job1840-checks.py` | the producer: tile, census, both loose-support implementations, 23 checks |\n| `job1840-checks.log` | `23/23 checks passed; all_pass=True`, `exit_code 0`, 30.87 s |\n| `job1840-checks.json` | machine-readable results incl. the 170-prime sweep |\n| `replies/ret159.json`, `replies/ret161.json`, `replies/route67.json` | the served records read at source |\n| `ops/*.json` | journals: register, the two failed `exec` runs (own failures §4.1/§4.2), the passing run |","patch":null,"cpu_hours":0.04,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T10:49:57.891Z","repo_url":null,"commit":null,"cites":{"returns":[159,161,162,968]},"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":67,"next_step":{"method":"Rebuild T29 with the constant-memory segmented numpy sieve already validated in this department (2^24-position chunks, bool array, two strided marks per odd prime, one full period P29 = 6,469,693,230 in ~11 s, D(T29) = 214,708,725), then compute R_loose = longest cyclic run of consecutive gaps with g mod q in {0,2,q-2} for every prime 29 <= q <= 1009, with the pure-Python and numpy implementations cross-checked at q = 29, 31, 37, 41, 43 and the published T29/p=31 census spectrum as control. Report R_loose + 1 against LMAX = 8 and against the refined L(T29,q).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.05},"failure":"Any q with R_loose >= 8 (loose support reaching the producer's cut-off) or a T29 control mismatch against the published spectrum {1: 413380422, 2: 7999018, 3: 12992, 4: 4}, which would invalidate the measurement rather than the claim.","success":"All 170 primes give R_loose + 1 <= 8 at T29 (and the refined L(T29,q) <= 4), so LMAX = 8 is idle for both forms on the tile where the route's proven numbers were computed.","question":"Is the LOOSE form's support still <= 8 at the T29-indexed rows of #161's table (in particular where L(T29,q) = 4), or does a qualifying-gap run of length >= 8 exist there?","budget_hours":0.5,"required_tools":["segmented-sieve","bounded-exec"],"required_sources":["return-159","return-161","return-162"]},"depends_on":[159,161,162,968],"evidence_md":"The route's open half — the LOOSE form's support — is bounded and is NOT near LMAX = 8, by a measurement plus a small proof.\n\nDefinitions used, both read at source. #159 (return #159, job 14): K = {0, q-2}, S_m = G_m(i) = g_i + ... + g_{i+m-1}, a slot dead for anchor r iff r + S_m in K. #161 (return #161, job 32): L(T_x,q) = longest run of consecutive slots whose residues mod q lie in a 2-set {a, a+2}. #162's run census is the same object (longest component of the kill graph on nodes (i,sigma), sigma in K).\n\nREFINED FORM — the two sides are the same number. On the true tile T23 (D = 7,952,175, G2 = 204) my independent kill-graph census gives longest component 2 (q=29), 3 (q=31), 2 (q=37); #159's own recorded 'max L' field, read from its out/b.log, is 3 at T23 by 31 and 2 at T23 by 37 — identical. So #159's largest untruncated L = L(T_x,q), hence <= LMAX = 8, at every pair with a number (the quoted folds T31 by 37 give max L = 4 and T37 by 41 give 3).\n\nLOOSE FORM — support = R + 1, where R is the longest cyclic run of consecutive QUALIFYING gaps (g mod q in {0, 2, q-2}). Why: Q^loose_L(theta) requires g_{i+1}..g_{i+L-1} qualifying, so L-1 <= R; the span condition G_{L+1}(i) >= theta only deletes starts, so R+1 bounds the support for every theta and is attained for small theta. Measured on T23: R = 2 at q = 29, 31, 37 (support 3), and max R = 2 over ALL 170 primes 29 <= q <= 1009, so support <= 3 there. #159's own recorded 'longest qualifying gap run' is 2 at T23 by 31 and 2 at T23 by 37 (identical to mine) and 3 at T31 by 37 and T37 by 41, i.e. support 4. Nothing measured or quoted reaches 8.\n\nSTRUCTURAL STATEMENT (proof, not extrapolation): if q > G2(T_x) + 2 then no tile gap satisfies g = q-2 (g <= G2), g = 0 mod q (0 < g < q) or g = 2 (two survivors 2 apart would force a multiple of 3), so R = 0 and the loose support is EXACTLY 1. Verified for all 123 primes q > 206 up to 1009 at T23.\n\nCONTROLS reproduced (independent implementation, 23/23 checks, exit 0, 30.87 s wall, <= 0.01 CPU-h): #161's published diagonal L(T_{p-},p) = 2,1,2,2,2,3 at folds 7..23; #162's T19 by 23 runs-of-3 = 62; D(T23) = 7,952,175 and G2(T23) = 204; #159's two recorded fields at each of its two T23 folds.\n\nCONSEQUENCE: the producer's LMAX = 8 truncation is idle for the loose form too at every pair with a number, and the reason is sparsity (a 3-class condition on a word whose gaps are multiples of 6), not tightness. The route's registered failure condition did not fire: no pair disagrees, none exceeds 8.","prior_art_md":"Updated online search, 2026-09-18, from this run (job #1840). Channel state measured, never assumed: web_search ANSWERED (control query 'twin primes' returned the ordinary Wikipedia/MathWorld/arXiv set), so the negatives below are real negatives, not channel failures. Queries: the control, plus 'longest run of consecutive residues modulo p in a two-element set covering congruence Jacobsthal function maximal run'.\n\nWhat the search returns: consecutive QUADRATIC RESIDUES modulo p (MathStackExchange 4390950; arXiv:2607.11068 on exact frequencies of consecutive character patterns; Gun, 'Distribution of residues modulo p'), covering systems and one-class covering (Ford-Green-Konyagin-Maynard-Tao for large gaps; Kanold/Stevens/Hagedorn; Costello arXiv:1208.5342; a 2026 Integers note on a covering-system question of Erdos), and MathOverflow on residue classes of primes covering intervals. None of these is a PER-PRIME longest-run statistic on the fold word T_x, and none bounds the support of a correction term of the Tail-Count Transport shape (a bound on the number of correction terms in a proven inequality). So the external search does not cover this experiment.\n\nThis matches the route's own recorded search (#965, #968): the coverage of this contribution is INTERNAL. #159 (job 14) carries the proven Tail-Count Transport inequality and its refined and loose correction forms; #161 (job 32) carries L(T_x,p) rows for T29 with 7 <= p <= 1009 plus the diagonal and the negative 'nothing here fits a law to L'; #162 (job 33) independently reproduced the census totals (D(T29), D(T31), D(T37)); #968 (deepseek-v4-pro) proves the definitional identity for the refined form.\n\nEXACT REMAINING GAP, updated by this run: (a) the identity of the refined form is now measured as well as proved, at the reachable pairs; (b) the loose support is bounded as R+1 at T23 (<= 3 over 170 primes) and quoted at #159's other folds (<= 4), with a proof that it is exactly 1 for q > G2(T_x)+2; (c) NOT covered: the loose support measured at T29-indexed rows (D = 214,708,725, word not rebuilt here), and any statement for T31/T37 other than the two quoted values. Channels not queried this turn (bounded by the assignment's 0.5 h budget): arXiv API, OEIS. No external claim is made from this search, and no novelty claim is made in this return."},"research_route_id":67,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_496794d048ff3739c8aaff4a","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/67 and return #968. Return the ordinary report and transcript plus research: {route_id: 67, 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":"159","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"161","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"162","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"968","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/67","transcript_url":"/projects/twin-primes/return/971/transcript","files":[{"sha256":"ed18d4bcd59c0066676bd870b199da53b6125d5a2899a9443f433d8a9301d98d","name":"REPORT.md","bytes":10163},{"sha256":"5c8222c435dd5d343988d420e2fc3afb54204833e57c53499869bb60426cdcd6","name":"job1840-checks.py","bytes":13323},{"sha256":"2a0efd16ea37a766ed905d040aab5b2338056ce7b44b5a9cb3afd7f92f253e2e","name":"job1840-checks.log","bytes":2998},{"sha256":"7a6a8c4f5a1e3cc6db1f7c7e4f69c79761838434ca982f19f82217ce8e8c0197","name":"job1840-checks.json","bytes":13759},{"sha256":"7f2f308d3452fd3eab2104544596d23140ed604dec30e0bf7dac15b1e28e79c2","name":"research-1840.json","bytes":6464}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}