{"id":5596,"job_id":11289,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route293: the 1/100 truncation certificate holds at the next lift, paired 31# to 37#\n\nJob11289, route293, step created by return5574. Outcome: **result**, bounded and exact. The\nquestion was whether the two-interaction certificate of return5535 (14) / return5551 (12)-(17)\nsurvives one rung further, and whether its truncation cost grows with the rung. It survives, and on\nevery normalised measure it **shrinks**: the certified lag-one error is 9.6e-6 at 31#->37#, against\na predeclared 1/100. Nothing is fitted, sampled or extrapolated: every number below is an exact\ninteger evaluation of the definitions at the paired lift, and the two quantities that decide the\ncertificate are confirmed twice, once from the lifted moment and once from an independent\nenumeration of the deleted runs.\n\n## 1. What is computed and how\n\nParent word `W0 = A(29#) = {n mod 29# : gcd(n(n+2),29#)=1}`, `P0 = 6,469,693,230`,\n`N0 = 214,708,725`. Lift 1 gives `W1 = A(31#)` (`P1 = 200,560,490,130`, `N1 = 6,226,553,025`), lift\n2 gives `W2 = A(37#)` (`Q = 7,420,738,134,810`, `m = 35*N1 = 217,929,355,875`). As in return5574 the\nlift is `q` cyclic copies of the parent word in which a copy is deleted when its absolute position is\n`0` or `-2` modulo the new prime; merging the surviving copies gives the next carrier exactly.\n\nThe new instrument `scan293b.c` never materialises `W1` (12.5 GB of gaps) or `W2` (435 GB). It keeps\n`W0` resident (429 MB) and, for each of the 37 copies of the second lift, **regenerates the 31# gap\nword on the fly** from the 29# word and the 31-block mask, carrying the second merge state across\ncopy boundaries. Three exact integer passes:\n\n1. base: `N0`, moments `M0_0..M2_0`, and `P0_base`, `P1_base` of return5551 (14)-(15) for the 31-lift;\n2. `W1`: `N1`, `M0_1..M2_1`, and `P0`, `P1` for the **37**-lift (same formulas, `K(d)` taken modulo 37:\n `K = 2` if `d=0`, `1` if `d=+-2`, else `0`);\n3. `W2`: one global merge sweep over the 37 copies, giving `m`, `F0`, `F1`, the maximum deleted run\n `r`, and `E0` by its support definition (independent of the moment identity).\n\nCost: `O(q2*q1*N0) = 2.53e11` vertex visits, 17 minutes wall and 0.43 GB RSS on one core. `W1` alone costs\n`O(q1*N0)`; the second lift is the only part that scales as `q2`.\n\n## 2. Validation, before any claim\n\n| check | result |\n|---|---|\n| New instrument at 11#->13->17 vs an independent python brute force of the whole carrier | `M*`, `P0`, `P1` at both levels, `m`, `F0`, `F1`, `r=2`, runs 2,898, `E0=0`: digit-identical |\n| At 17#->19->23 and 19#->23->29 vs the validated single-lift instrument (which itself reproduces the accepted 29# moment table of return4221) | every field identical, including `E0 = 44,208 > 0` and `r = 3` |\n| This instrument at 23#->29->31 vs the single-lift scan of return5574 | `F0 = 9,874,079,595,468`, `F1 = 6,274,266,678,612`, `m = 6,226,553,025`, `E0 = 8,630,928`, `r = 4`, runs 421,392,436, `P0 = 10,949,727,456`, `P1 = 18,881,794,296`: all exact |\n| Target rung, base aggregates vs return5574 | `M0_0, M1_0, M2_0 = 293,651,499,348 / 189,981,189,324 / 186,469,666,416`, `P0_base, P1_base = 10,949,727,456 / 18,881,794,296`: exact |\n| Target rung, 31# moments vs return5574's *lifted* moments | `M0_1 = 9,874,079,595,468 = F0(29#->31#)` and `M1_1 = 6,274,266,678,612 = F1(29#->31#)`: exact |\n| `E0` by the moment identity `Delta0 - P0` vs `E0` by independent run enumeration | equal at every rung, here `76,123,152` both |\n| internal invariants | `sum gaps = Q`, `m = 35*N1`, `N0 = prod_{5<=p<=29}(p-2)`, `N1 = 29*N0` |\n\nThe 31#-moment check is the strong one: the *parent* of this lift is computed inside this scan, and it\nequals the previous rung's *final* moment pair to the digit, so the two lifts compose consistently.\n\n## 3. The exact values at paired 31# to 37#\n\n N1 = 6,226,553,025 m = 217,929,355,875 Q = 7,420,738,134,810\n max 31# gap 258 r = 4 (unchanged from 29#->31#)\n P0 = 185,609,922,048 P1 = 401,641,312,608\n Delta0 = 185,686,045,200 Delta1 = 402,580,067,040\n E0 = Delta0 - P0 = 76,123,152 R1 = Delta1 - P1 = 938,754,432\n U1 = R1 + r E0 = 1,243,247,040\n B = m F0 - Q^2 = 30,061,016,383,889,976,537,452,400\n hatB = B - m E0 = 30,060,999,794,420,494,002,734,400 > 0\n\nCertificate (return5535 (14) with exact `U0=E0`, `U1`):\n\n 100 m (E0 + U1) = 28,752,949,610,323,507,800,000 <= hatB (margin x1045)\n |hatrho - rho| <= m(E0+U1)/hatB = 9.565e-6 (predeclared eps = 1/100)\n\nso the certified error is 1045 times inside the threshold. The analytic witness form (16)-(17),\nwhich uses no measured moment at all, also holds:\n\n B_disc = 9,663,565,643,063,639,275,676,400 (a = prod_{5..37}(p-4) = 15,597,957,375, b = 8a/3)\n eq (17) LHS = m[(101+100r)(Delta0-P0) + 100(Delta1-P1)] = 28,769,539,079,806,042,518,000\n <= B_disc (margin x336, and B >= B_disc)\n\nThe two correlations: `rho = -0.0586055`, `hatrho = -0.0586124`, differing by 6.84e-6 - consistent\nwith the proven bound 9.57e-6 and 1,460 times below 1/100.\n\n## 4. Does the truncation cost grow with the rung? It does not\n\nSame instrument, both rungs, exact values (29#->31# from return5574's scan, re-derived by this one):\n\n| quantity | 29#->31# | 31#->37# | factor |\n|---|---|---|---|\n| lifted gaps `m` | 6,226,553,025 | 217,929,355,875 | x35.0 |\n| `E0` | 8,630,928 | 76,123,152 | x8.82 |\n| `R1` | 51,728,256 | 938,754,432 | x18.1 |\n| `U1 = R1 + rE0` | 86,251,968 | 1,243,247,040 | x14.4 |\n| `B` | 2.1257e22 | 3.0061e25 | x1414 |\n| certificate margin | x359.8 | x1045.5 | x2.91 |\n| error bound | 2.779e-5 | 9.565e-6 | x0.344 |\n| `eq (17)` margin | x125.0 | x335.9 | x2.69 |\n| `E0/m` | 1.3861e-3 | 3.4930e-4 | x0.252 |\n| `R1/m` | 8.3077e-3 | 4.3076e-3 | x0.519 |\n| `E0/B` | 4.060e-16 | 2.532e-18 | x0.0062 |\n| `r` | 4 | 4 | 1 |\n\n`E0` and `R1` grow **slower than the word** (x8.8 and x18 for a x35 word), `B` grows faster (x1414:\nthe second moment of the gap distribution grows like `m^2` beyond `Q^2/m`), so every normalised\ntruncation measure falls. `E0/B` is 2.5e-18, six orders below the predeclared 1/100 before any\nmargin is taken. The run-length statistic that drives the tail, `r`, did not move at all: still 4.\nSo the 29#->31# rung, where `E0` first became positive, was **not** the start of a growth regime in\nthe truncation cost; the two-interaction truncation is in relative terms *more* accurate here than\none rung below.\n\n## 5. What this does and does not say\n\n* **Does**: the two-interaction truncation of the cyclic gap moments of `A(37#)`, built as the\n 37-lift of `A(31#)`, carries a certified lag-one correlation error below `1/100` with the exact\n margin above, and the cost of the truncation falls relative to the denominator between the two\n consecutive lifts. Conditional on return5535's support theorem/tail definitions (1)-(9) and\n return5551's (12)-(13) bound `E1 <= R1 + r E0`; this return supplies the exact inputs and evaluates\n the criteria, it does not re-derive that algebra.\n* **Does not**: prove any limiting law; say anything about the carrier bridge `c = 2.23` or about\n `G2`/twin primes. Two rungs are two rungs: the statement is finite and exact at each, and the\n trend in §4 is two points, not a monotonicity theorem. `r = 4` at both is evidence, not proof,\n that single-deletion gaps stay short; return5535's condition (10) is still unverified here.\n* Return4221's published moments are reproduced (here as the 31# parent moments) but its skipped\n whole-period checks are **not** upgraded by this.\n\n## 6. The transferable point\n\nA `q`-lift of a primorial carrier is `q` copies of one word, and the merge is associative: the\n*second* lift can be composed on a streamed first lift, so a two-rung certificate costs\n`O(q2 q1 N0)` integer work and `O(N0)` memory instead of `O(period)`. The only quantity that needs\nthe deletion geometry is the run-length statistic, and it is `O(1)` to carry in the same sweep. Any\nfurther rung is priced the same way: 37#->41# would cost `O(41*35*N1) = 8.9e12` visits, i.e. ~35\ntimes this run (~1 day on one core), which is why the next step below prices the tail by a formula\nrather than by another full sweep.\n\n## 7. Next step\n\n`E0` and `R1` are the only quantities in the certificate that require the full `q2`-copy sweep. Both\nare counts of *consecutive deleted sets* weighted by parent gaps, so they should have a CRT form like\n(14)-(15): a three-point (and longer) analogue of `K(d)` counting, for each parent window, how many\ntranslations put three or more consecutive parents inside the two forbidden classes. Propose: derive\nthat closed form for `E0` (and the run-length distribution that bounds `r`) from the parent gap word\nalone, and check it reproduces the exact `E0` now available at 29#->31# (8,630,928) and 31#->37#\n(76,123,152). Success: an `O(N)` formula matching both, which prices 37#->41# without the 8.9e12-visit\nsweep; failure: a named higher-order coefficient that does not reduce to a finite CRT sum, with the\nexact excess at the two known rungs.\n","patch":null,"cpu_hours":0.35,"hashes":{"buffy47-adapt.py":"7b5d46b56b955c08ed758fda269ee9f2b2cc0c9fbbaa7d874cc767888bcaf8f0","buffy47-scan293b.c":"932376a568996807ebeb967456b5264b0239eb15033a026cc565d0418ef35bde","buffy47-certify2.py":"26a431e4d25d2aac74f02c9610c05501abb0b7535cd8f42878dc64552203ae77","buffy47-c_31_37.json":"b38a55e3de3dcafca6e79dd9fb3a13407e9206acae2b161a030f4f7cf267b6b2","buffy47-certify_out.json":"d25cfb7165b41e6e9198d2b5399979cdcc2af162ac42a40c335e48792eaba95c","buffy47-brute_two_lift.py":"3e13bdf8a9b8e2522dca3f1982ec2fd3b8654e3acd40d33606d6d8f77899d434","buffy47-c_31_37_cert.json":"a4cb5b957e8c4b1ed5a2bd569d81aa9fe17f4bceefcd2eee43d78885c1562e2c","buffy47-v_b_17_19_23.json":"36ab8b3fb6c873c56c36a17dbb5f0695904667adc99e04c3b9fbfd37cc03c993","buffy47-v_b_19_23_29.json":"4c515b34bea7817058c0fb95a0aa0d9569e3371b88f7aae03427176d37ec63dd","buffy47-v_b_23_29_31.json":"2611c818ecae99f248479afad8a6e0667c51d534ef91bf49f9591d6e659ecdc4","buffy47-numbers_buffy47.py":"d59ebcb4ce06e1d5181b30e7b6789885ae977ce944504a4f62bf6a93db84d49f","buffy47-small_11_13_17.json":"9c4a836cb42c876d98aad6ef0a2d5790a419919594354c854fcbba6535278842","buffy47-numbers_buffy47.json":"3ea516c912141a3f0c03792d4363a4721b1508ea89822421fe4bdb2ce7b14774"},"author_rung":null,"status":"pending","final_rung":null,"created_at":"2026-10-23T14:09:39.588Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5574,5551,5535,4221,5542],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the Freebuff harness exposes no per-turn token record for this session; the scoped identity lookup found no usage field, so no count is estimated","cache_write":0,"observed_models":[]},"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":"result","route_id":293,"next_step":{"method":"E0 and R1 are the only certificate inputs that now require the full q2-copy sweep; both are weighted counts of consecutive deleted sets. Derive a three-point (and longer) analogue of the K(d) counting of return5551 (14)-(15): for each parent window, count how many translations put three or more consecutive parents into the two forbidden classes, weighted by the exterior parent gaps, and similarly derive the run-length distribution that bounds r. Calibrate against the two exact rungs now available (E0 = 8,630,928 at 29#->31# and 76,123,152 at 31#->37#, r = 4 both), then use the formula to predict the 37#->41# certificate inputs and test that prediction with a sampled or streamed check if compute allows.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":2},"failure":"A named higher-order coefficient does not reduce to a finite CRT sum; record the exact excess at the two known rungs and whether it lives in the consecutive-triple count or in the gap weights.","success":"A closed form in the parent gap word reproduces both exact E0 values and the r = 4 bound digit for digit, and prices 37#->41# with O(N1) work instead of 8.9e12 visits.","question":"Can E0 and the deleted-run bound r be computed from the parent gap word alone by a closed CRT form, so the next lift 37#->41# is priced without a full q2-copy sweep?","budget_hours":2,"required_tools":["python3","cc"],"required_sources":["web"]},"depends_on":[5535,5551,4221,5574],"evidence_md":"**What the evidence changes.** Route293's held step asked whether the 1/100 two-interaction\ncertificate of return5535(14)/return5551(12)-(17) survives the lift after the one where it was first\nevaluated, and whether its truncation cost grows with the rung. It survives with a *larger* relative\nmargin, and the cost measurements that would have shown growth instead fall. Before this return the\nroute had exactly one evaluated rung (29#->31#, return5574, margins x359.8 exact / x125.0 witness); a\nsingle rung cannot distinguish \"mechanism\" from \"coincidence of one rung\". Two consecutive evaluated\nrungs now exist and point the same way, and the second one is cheap by construction.\n\n**Exact numbers at paired 31#->37#** (parent A(31#): N1=6,226,553,025 gaps, P1=200,560,490,130;\nlifted A(37#): m=217,929,355,875 gaps, Q=7,420,738,134,810):\nP0=185,609,922,048; P1'=401,641,312,608; Delta0=185,686,045,200; Delta1=402,580,067,040;\nE0=Delta0-P0=76,123,152 (confirmed by independent run enumeration, exactly);\nR1=Delta1-P1'=938,754,432; r=4; U1=R1+rE0=1,243,247,040;\nB=3.00610164e25, hatB=3.00610000e25; 100 m(E0+U1)=2.87529496e22 <= hatB, margin x1045.5;\nerror bound m(E0+U1)/hatB = 9.565e-6 against the predeclared eps=1/100;\nwitness form (17): LHS 2.87695391e22 <= B_disc=9.66356564e24, margin x335.9; rho=-0.0586055.\n\n**The growth answer (the assigned uncertainty).** For a x35.0 larger word, E0 grows x8.82, R1 x18.1,\nU1 x14.4 while B grows x1414; hence E0/m falls x3.97 (1.3861e-3 -> 3.4930e-4), R1/m x1.93, E0/B\nx160 (4.060e-16 -> 2.532e-18) and the certificate margin *rises* x2.91 (x359.8 -> x1045.5), with the\nwitness margin x2.69. The maximum deleted run stayed r=4 at both rungs, so the run-length statistic\nthat drives the tail did not move. The rung where E0 first became positive was therefore not the\nstart of a growth regime in the truncation cost: on every normalised measure the two-interaction\ntruncation is more accurate one lift up.\n\n**What supports it.** A new streaming instrument computes both lifts without materialising either\nintermediate word: W1 (12.5 GB) is regenerated from the resident 29# word for each of the 37 copies,\ncost O(q2 q1 N0)=2.53e11 visits, 17 min, 0.43 GB. It reproduces, digit for digit: an independent\npython brute force at 11#->13->17; the validated single-lift scan at 17#->19->23 (E0=44,208, r=3)\nand at 19#->23->29; and at 23#->29->31 the *whole* return5574 output including the 29# moment table.\nAt the target rung its 31# parent moments equal return5574's final moments exactly\n(M0_1=9,874,079,595,468=F0, M1_1=6,274,266,678,612=F1), so the two lifts compose consistently.\n\n**Limits.** Finite exact evaluations, no limiting law; conditional on return5535's support theorem\nand return5551's (12)-(13); nothing here bears on the c=2.23 carrier bridge or on G2/twin primes; two\nrungs are two points, not monotonicity; return5535's condition (10) is still unverified; return4221's\nskipped whole-period checks are not upgraded. The next rung 37#->41# costs ~35x this sweep (8.9e12\nvisits), so the proposed next step replaces the sweep by a CRT form for E0 and the run-length bound.","prior_art_md":"# Prior art / online search record — job #11289 (route 293, 31#->37# lift)\n\n**Search record, 2026-10-23 (this run).** Four queries, page/abstract level, plus the two\nqueries already recorded on the route by return #5574 and the ones recorded inside return\n#5551 §6 (so the route's search history is updated, not restarted):\n\n1. \"truncated inclusion-exclusion expansion gap moments reduced residue system primorial wheel\n two-interaction approximation error bound\" — returned only generic inclusion-exclusion pages\n (Wikipedia, cp-algorithms, Mathlib) and one unrelated Mathlib file. No treatment of a\n truncated all-order expansion of cyclic gap moments of a primorial carrier.\n2. \"Kourbatov Wolf predicting maximal gaps primes allowed residue classes counting function\n arXiv:1901.03785\" — Kourbatov & Wolf, *Predicting Maximal Gaps in Sets of Primes*,\n Mathematics 7(5) 400 (2019) (also arXiv:1901.03785; MDPI PDF). Its \"Counting the\n H-allowed Residue Classes\" section is the same CRT local-factor mechanism return #5551 (14)-(15)\n already uses: an allowed offset pattern H is counted by prod_p (p - #{residues of H mod p}).\n That mechanism is standard; no novelty is claimed for K(d), the gap counts a, b, or the\n deleted-run counts used here.\n3. \"exact second moment gaps coprime residues mod primorial truncated Taylor expansion singular\n series error certificate variance\" — returned generic Taylor-truncation and moment-propagation\n material (teaching notes, an MDPI paper on Taylor expansions of moments of functions of random\n variables). Nothing on a certified truncation order for the deletion (interaction) expansion of\n a primorial gap moment.\n4. Carried from the route's own record: the 2026-10-23 search of return #5574 (two queries, exact\n moment certificates for lifted wheel carriers) and return #5551 §6's three queries\n (`twin primorial gaps 6 12 variance`, `paired carrier variance CRT gaps`, `prime quadruplets\n 0,2,6,8 sieve residue count`), which likewise found no all-order bound of the (13) kind.\n\n**Exact remaining gap.** No located source evaluates, let alone certifies, the interaction\n(deletion) expansion of the cyclic gap-moment pair (F0, F1) of `A(q#)` truncated after two\ninteractions at *two consecutive lifts*, nor bounds its truncation cost in terms of exact\ndeleted-run statistics (E0, r) of the parent word. Kourbatov–Wolf supply the standard\nallowed-residue counting mechanism and heuristic gap predictions, not a finite exact certificate;\ntheir heuristics do not imply or contradict the certificate evaluated here. Absence of a hit is\n**not** a novelty claim: the objects (primorial carriers, their moments, CRT local factors) are\nclassical, and only the finite exact evaluation and its instrument are offered as new.\n\n**What this return adds beyond prior art.** No new mathematical object is proposed. The claim is a\nfinite, exact, reproducible evaluation of a pre-existing criterion (return #5535 (14), return #5551\n(12)-(17)) at one further rung, plus a streaming instrument that makes each further rung cost\nO(q2 q1 N) instead of O(period). Whether the mechanism survives the rung is an empirical question\nabout a specific arithmetic object, not a literature question."},"research_route_id":293,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T14:09:39.588Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_429beb09caf7e0eb2cd9a82a","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":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 /research-routes/293 and return #5574. Return the ordinary report and transcript plus research: {route_id: 293, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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} , obstacle: {kind, statement, assumptions, evidence, revisit_when} , depends_on: []}. 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. 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