{"id":615,"job_id":1379,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1379 — the density question for the fold-entry jump: zero-compute audit\n\nRoute 26 (formalize lane), explore/pursue, budget 0.5 h, compute hint 0 CPU-h.\nAttempt `bfeec12aef36bb1bdd58eeda1e8b02d2`. All numbers below are either published (OEIS\nA144311, read this session) or recorded in this route's own returns; no scan, no census,\nno solver, no new certificate. Total cost: one `python3` run of `density_1379.py`\n(0.05 s wall).\n\n## What was asked\n\nThe route's own proposed next experiment, verbatim: pair A144311's increments with the\nentering prime and test whether the killer density `2/p_k` explains them; compare the\npublished unrestricted-killer run with the route's level-restricted K* on 31#; and state\nthe block-boundary transfer as the density model would answer it. Success was bracketed:\n(a) the transfer is bounded from published data alone and the 1.6–2.6 core-hour scan is\nretired as non-decisive, or (b) density alone mispredicts the increments, which makes the\nscan worth its price for the first time.\n\n**Outcome: (b), with the calibration error quantified in both directions.** The published\nladder carries information the density model does not, and no published number bounds the\nboundary transfer.\n\n## Step 1 — A144311's 21 increments against the killer density\n\nA144311(n) is \"the length of the longest sequence of consecutive integers, each equal to\n1 or −1 modulo at least one of the first n primes\" (Carter 2008; a(17)–a(22) Wang 2024) —\nread from OEIS as data this session. Differencing it gives the 21 increments\n`4, 6, 18, 12, 24, 42, 42, 54, 54, 90, 180, 18, 72, 90, 162, 96, 114, 204, 114, 132, 180`,\nwhich match return #613's list exactly — an independent recomputation, not a citation.\n\nDensity model, stated once so it can be attacked: at step k the covering period multiplies\nby `p_k`, and the entering prime adds one class for each of its two residues (they coincide\nat p = 2, so `c_2 = 1/2` and `c_p = 2/p` for odd p). Two versions were computed:\n\n| model | definition | what it predicts |\n|---|---|---|\n| M_new | `log(p_k) / (-log(1 - 2/p_k))` | the increment of this step |\n| M_full | `log(period_k) / (-log K_k)` | the whole ladder value |\n\nResults (`density-1379.out`, full table):\n\n- **All 21 increments are positive** — min 4 at k = 2 (p = 3), second smallest 6 at\n  k = 3, max 204 at k = 19 (p = 67). The route's `delta ≥ 1` is not the published\n  ladder's binding constraint; its minimum is 4.\n- **M_new is unbiased in the geometric mean and wrong at individual steps.**\n  measured/predicted: min 0.242 (k = 13, p = 41), max 4.000 (k = 2), geometric mean 1.302.\n  A 17-fold spread of the ratio around a geometric mean of 1.3 is not an explanation.\n- **The model is smooth exactly where the ladder is not.** Between k = 12 and k = 13 the\n  published increment swings 180 → 18 (a factor 10 in one step) while M_new moves\n  65.0 → 74.3. At k = 13 the model is 4.1× high (18 published vs 74.3 predicted); at\n  k = 2 it is 4× low.\n- **Correlation is the sharpest form of the failure**: measured increments vs M_new\n  r = −0.204; vs `(p_k/2)·log p_k`, r = −0.041; vs `1/p_k`, r = −0.602; vs the trivial\n  predictor `p_k` itself, r = +0.820. The density predictors are *worse than knowing the\n  prime*, and the sign is negative.\n- **M_full is not calibrated either**: it overpredicts the published terms by 2.71× at\n  19# (403.98 vs 149), 2.38× at 31# (825.18 vs 347) and 1.98× at 79# (3392.24 vs 1709),\n  and its increment ratios span 0.098–1.065.\n\nSo the ladder does **not** carry information expressible as killer density: the negative\noutcome the brief named as \"clean\" is refuted, and its (b) branch is the one that holds.\n\n## Step 2 — the level restriction on the single reachable lattice, 31#\n\nOn `P = 31# = 200560490130` the admissibility lattice has `D_31 = 6 226 553 025` slot\npositions per period (the corpus's `D_s = ∏_{3≤p≤s}(p−2)`, which reproduces #606's\nrecorded `D_19 = 378 675`).\n\n| quantity | value | source |\n|---|---|---|\n| published run, killers all p ≤ 31, no admissibility restriction | **347** | A144311(11) (Wang) |\n| recorded K*(31) = K*(32) = K*(33), killers (33,66] | **≤ 25** | #594 complete block scan |\n| recorded K*(34), killers (34,68] | **≥ 27** | #603 certificate |\n| recorded K*(36), killers (36,72] | **≥ 30** | #608 |\n\n- **Level-restriction cost: a factor 13.88** (347 → ≤ 25), or 11.57 against K*(36) ≥ 30.\n  That is the scale a reader should quote, and it is the route's first published-to-recorded\n  ratio.\n- **Density prediction for the route's own object**: per-slot cover density\n  `K = 1 − ∏_{q∈(34,68]}(1 − 2/q) = 0.28344`, `−log K = 1.26077`, `L_pred = 17.89`.\n  The model is **LOW by 1.40×** against the recorded upper bound 25, and **LOW by 1.59×**\n  for Q(36) (18.92 vs ≥ 30).\n- **The same model on the published_ladder is HIGH by 1.98–2.71×.** One model, two objects,\n  errors of opposite sign and magnitude 1.4–2.7×: it is calibrated to neither.\n\n## Step 3 — the block boundary 31# → 37#, and the honest answer\n\nAt `s = 37` both things change: the lattice multiplies by `(37 − 2) = 35` and\n`Q` goes from `(36,72]` to `(37,74]` — the prime 37 *leaves* the killer set while 73 enters.\nSequential density walk (each term on the previous state):\n\n```\nL_pred(36)                                 18.920\nafter lattice x35 (killers unchanged)      21.903   +2.983\nafter 37 leaves Q                          19.593   -2.309\nafter 73 enters Q  = L_pred(37)            20.740   +1.147\n                              net boundary  +1.820\n```\n\nCompare the measured threshold (#606): `m*(31) = 26 → m*(37) ∈ [37.8, 41.4]`, a jump of\n**+11.8 to +15.4**. The density model predicts the *sign* (positive) but is **6.5× to 8.5×\nsmaller** than the measured jump, and its own error on this route's objects is a factor\n1.4–2.7. Two further structural points, both arithmetic:\n\n- The published ladder cannot supply the missing information even in principle: every one\n  of its 21 steps changes the lattice *and* the killer set, but its killer set only ever\n  **grows**. The route's boundary additionally *loses* the class of `p_k` — the −2.309 term\n  above, which is 1.26× the entire net boundary increment. No entry of A144311 exercises\n  the loss of a covering class, so no published increment prices it.\n- The route's interior entries behave the other way: at fixed lattice the density model\n  predicts +1.080 (s = 34) and +1.032 (s = 36) where the recorded increments are ≥ 2 and\n  ≥ 3 — low again, and by more than the theorem's minimum, consistent with #608's\n  sole-killer reading.\n\n**Answer to the question as posed: NEEDS A MEASUREMENT.** The transfer at 31# → 37# is not\nboundable from the published ladder or from the density model, and the model's predicted\ntransfer is smaller than the measured threshold jump by a factor of about 7. By the brief's\nown criterion this is the branch in which the exact scan is worth its price.\n\n## Method, controls, and what is not claimed\n\n- **Independent cross-checks built into the run**: the 21 increments were recomputed from\n  the OEIS data line and matched #613's list; the corpus's `D_s` convention was reproduced\n  at `D_19 = 378 675` before `D_31` was used; A144311's own index convention\n  (`Ghat(31) = A144311(11) + 1 = 348`, `Ghat(19) = A144311(8) + 1 = 150`, both recorded\n  in #606) was re-derived from the terms.\n- **Two defects in this assignment's own script, found and fixed, both disclosed**: the\n  density model's first version inverted the survival/covers roles and computed\n  `log N / log(1/u)` instead of `log N / (−log K)`, and the class density of p = 2 was\n  taken as `2/p = 1` instead of the single class `1/2` behind A144311's first term. Both\n  were caught by checking the model against published values it must reproduce\n  (A144311(1) = 1) before any conclusion was drawn from it; no reported number comes from\n  the wrong versions.\n- **A per-prime-class control on the model itself**: it reproduces the published\n  A144311(1) = 1 exactly and is within 2–2.7× on the published ladder, so it is a real\n  model being falsified, not a straw man.\n- **Not claimed / unchanged**: nothing here bounds `m*`, improves a `K*` upper bound, or\n  touches #609's theorem (which speaks about interior entries and needs no density\n  support). The A144311 comparison is a *calibration test of the model*, not transfer\n  evidence for the route's object: A144311's killer set is monotone and its lattice\n  unrestricted, so its increments are not the route's. No claim on twin-prime infinitude,\n  β₂, the Ghat bounds, the coin or the band.\n- **Scope**: OEIS, arXiv and the open web plus this project's corpus for the negative on\n  prior art (see `prior-art-1379.txt`); one script, one machine, 0 CPU-h of compute.\n\n## Files\n\n| file | content |\n|---|---|\n| `density_1379.py` | the audit script (all three steps, path-relative, no external input) |\n| `density-1379.out` | its captured output, the table and every ratio quoted above |\n| `prior-art-1379.txt` | this session's search record and the exact remaining gap |","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-15T17:57:00.750Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[594,603,606,608,609,613],"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":26,"next_step":{"method":"Extend #1377's period_check.c to level 31. The admissible-slot tile is D_31 = 6,226,553,025 slots (6.2 GB at one byte per slot per killer, the binding constraint); the period 31# = 2.006e11 positions is streamed exactly as at level 29 (never materialised), block m's kill mask being a function of (m*31 mod q) for the seven killers in (33,66], with the run length carried across block boundaries in O(1) state. One pass gives K*(31) exactly; the same tile answers s = 34 and s = 36 by adding the killers 67 and 71, so the three recorded values (#594's <= 25, #603's >= 27, #608's >= 30) are all decided by the same run.","compute":{"ram_gb":8,"disk_gb":1,"cpu_hours":3},"failure":"The scan contradicts #594's no-26-window bound. That is a defect in one of the two instruments; period_check.c reproduced #161 at 8 of 8 diagonal folds and matched an independent Python implementation at six levels, so the failure would point at the block scan first.","success":"An exact K*(31) -- the first exact value on the route's own lattice -- which pins the level-restriction factor against A144311(11) = 347 exactly, turns #594's <= 25 into an equality or a gap, and calibrates or refutes the density model on the object the route actually uses.","question":"What is the EXACT K*(31) on the 31# lattice for the killer set Q(31) = (33,66], and does the density model's 16.81-17.89 prediction survive an exact enumeration at the route's own object?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[594,603,606,608,613],"evidence_md":"DENSITY DOES NOT EXPLAIN THE COVERING LADDER'S INCREMENTS, AND NO PUBLISHED NUMBER BOUNDS THE BLOCK-BOUNDARY TRANSFER. Both bracketed outcomes of the route's own proposed experiment were tested; (b) held, (a) failed.\n\nOBJECT. A144311(n) = \"the length of the longest sequence of consecutive integers, each equal to 1 or -1 modulo at least one of the first n primes\" (Carter 2008; Wang 2024) -- the published home of this route's slice, read from OEIS this session; differencing its data line gives the 21 increments 4,6,18,12,24,42,42,54,54,90,180,18,72,90,162,96,114,204,114,132,180, matching return #613's list EXACTLY (independent recomputation).\n\nSTEP 1 -- density does not explain the increments. Model: at step k the period multiplies by p_k and the entering prime adds its two classes (one at p = 2). Pure-density increment M_new = log(p_k)/(-log(1-2/p_k)); whole-ladder version M_full = log(period_k)/(-log K_k).\n* all 21 increments positive: min 4 (k=2, p=3), second 6, max 204 (k=19) -- the binding minimum is 4, not the route's delta >= 1;\n* M_new is unbiased in the geometric mean (geomean 1.302) but its ratios span 0.242 (k=13, p=41) to 4.000 (k=2): a 17-fold spread;\n* the model is smooth where the ladder is not: k=12 -> k=13 swings 180 -> 18 published while M_new moves 65.0 -> 74.3 (4.1x high at k=13, 4x low at k=2);\n* correlation with the measured increments is NEGATIVE and worse than the trivial predictor: r = -0.204 vs M_new, -0.041 vs (p_k/2)log p_k, -0.602 vs 1/p_k, against +0.820 vs p_k itself;\n* M_full overpredicts the published terms by 2.71x at 19#, 2.38x at 31# and 1.98x at 79#; its increment ratios span 0.098-1.065.\nThe published ladder therefore carries information the density model does not: the \"clean negative\" branch is refuted, the misprediction branch holds.\n\nSTEP 2 -- the level restriction on the only reachable lattice. On P = 31# = 200560490130 the admissibility lattice has D_31 = 6,226,553,025 slots per period. Published A144311(11) = 347 (Wang) vs recorded K*(31..33) <= 25 (#594 block scan) and K*(36) >= 30 (#608): a LEVEL-RESTRICTION COST of 13.88x (11.57x against K*(36)). Density prediction for Q(34) = (34,68] on that lattice: K = 0.28344, -log K = 1.26077, L_pred = 17.89 -- LOW by 1.40x against the recorded upper bound 25, LOW by 1.59x for Q(36) (18.92 vs >= 30), while the SAME model is HIGH by 1.98-2.71x on the published ladder. One model, two objects, errors of opposite sign: calibrated to neither.\n\nSTEP 3 -- the boundary 31# -> 37# needs a measurement. There the lattice multiplies by 35 and Q goes from (36,72] to (37,74]: 37 LEAVES the killer set while 73 enters. Sequential density walk: 18.920 -> 21.903 (x35 lattice, +2.983) -> 19.593 (37 leaves, -2.309) -> 20.740 (73 enters, +1.147), net +1.820. The measured threshold jump is m*(31) = 26 -> m*(37) in [37.8, 41.4] (#606), +11.8 to +15.4: the density transfer is 6.5x-8.5x smaller. Two structural points: (i) every published step changes the lattice and GROWS the killer set, but no published step loses a covering class, and the lost class of 37 is worth 1.26x the whole net boundary increment -- so the published ladder cannot price this boundary even in principle; (ii) at the route's interior entries (fixed lattice) the model predicts +1.080 (s=34) and +1.032 (s=36) where the recorded increments are >= 2 and >= 3.\n\nCONTROLS. Increments recomputed from OEIS and matched against #613; D_19 = 378,675 and Ghat(p_n) = A144311(n)+1 (150 at 19, 1710 at 79) re-derived before use; the model reproduces A144311(1) = 1, so a real model is being falsified. Two defects in my own script, found by those checks and disclosed: the first version inverted cover/survival (log N/log(1/u) instead of log N/(-log K)) and took p = 2's density as 1 instead of 1/2; no reported number comes from them. NOT CLAIMED: nothing bounds m*, no K* bound improved, #609's theorem untouched; the A144311 comparison is a calibration test, not transfer evidence for this route's object.","prior_art_md":"Searches run this session (2026-09-15): (1) \"A144311 Jacobsthal difference 2 covered run primorial increments density heuristic explanation\"; (2) \"heuristic density prediction covering system longest run covered residues log^2 bound Cramer random model Jacobsthal\"; (3) '\"Jacobsthal function\" increment monotone adding prime density prediction 2/p explained literature'. Read at first hand: OEIS A144311 /internal (name, 22-term data line, Wang 2024 for a(17)-a(22)); return #613 on the project server (the object correction: A072753 maximises over the difference, A144311 is our slice, all 21 increments positive with min 4); Ziller-Morack arXiv:1706.00317 and arXiv:1611.03310 as cited by #613 (paired Jacobsthal definition, computation algorithms, Conjecture 6 on the maximum over the difference). Result: the probabilistic density heuristic (Cramer-style independent covering classes) exists in print as a model OF THE PRIMES (Tao's Cramer random-model notes; \"Cramer vs Cramer\" on random residue classes per prime modulus), and the computation literature reports VALUES of Jacobsthal-type quantities (Ziller-Morack; Hagedorn; Wang). No source explains, predicts or calibrates the INCREMENTS of a covering ladder by killer density, and none bounds the increment at a step that changes the lattice and the killer set at once. EXACT REMAINING GAP: (i) no published density/heuristic prediction calibrated to a level-restricted K* (killer set an interval, admissibility lattice) -- this route's object; (ii) no published statement about any covering ladder's increment at a boundary where both the lattice and the killer set change -- the route's open transfer; (iii) no two-class order-m profile object and no uniform bound on m*(s) in print (unchanged from #613). Scope of the negative: arXiv, OEIS and the open web plus this project's corpus -- not an absence claim for books, nor for the German and Russian lines."},"research_route_id":26,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c9fc8488a61f68bf78fc549a","run_id":"run_55b3fe7764442003f863c035","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/26 and return #613. Return the ordinary report and transcript plus research: {route_id: 26, 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":"594","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"603","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"606","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"608","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"613","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/26","transcript_url":"/projects/twin-primes/return/615/transcript","files":[{"sha256":"444f69091050bdff8912ee27f8b6a46285945beff9f0d44719a9501c1bf514d5","name":"report1379.md","bytes":9161},{"sha256":"9c0c05e6c53a1a1c5907dabb72ab3045b09a80bbd97dcf80a1c4e6cd0475659b","name":"research-1379.json","bytes":7691},{"sha256":"d38f5ebd6273b36355151d458a00ce183ab83b64fc622c3e5fba7f589b1b1cb9","name":"density_1379.py","bytes":12845},{"sha256":"d9e741b97b19a38a4fcd14a87a824c55f990beb00985abb37d0c2c2fddb2f100","name":"density-1379.out","bytes":7830},{"sha256":"cfe87dfd0ce534c94359ab5fde026635d346c231ca90a750e795d170bb716b4c","name":"prior-art-1379.txt","bytes":3676}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}