{"id":5574,"job_id":11286,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route293: the 1/100 lift certificate is evaluated exactly at paired 29# to 31#\n\nJob11286, route293. Outcome: **result**, bounded and exact. The obstruction recorded by return5569 was\na *retention* obstruction: the published producers of return4221 do not retain the residue-masked\nquadratic sums P0, P1 of return5551 or a certified deleted-run bound r, so inequality (17) could not be\nevaluated. It is evaluated here, and it **holds**. The repair is not a new field in the old producer:\nnone of those numbers ever had to be retained. P0, P1, the maximum deleted run r, and the lifted\nmoments F0, F1 themselves are exact functions of (a) the paired29# gap word and (b) the residue classes\nmodulo31, and all of them come out of **one exact pass over one 29# period** in 25 seconds of CPU.\nNothing is fitted, extrapolated or sampled, and no producer of return4221 was rerun.\n\n## 1. What is computed and how\n\nLet P = 29# = 6,469,693,230, A(P) = {n mod P : gcd(n(n+2),P) = 1}, N = |A(P)| = 214,708,725, with\ncyclic gaps g_i and moments M_j = sum_i g_i g_(i+j). The 31-lift is Q = 31P = 200,560,490,130: the\n**31 contiguous copies** of the parent cyclic word, where in block t the copies at\nx ≡ b_t or b_t - 2 (mod 31), b_t = -tP mod 31, are deleted. Its merged cyclic gap word has\nm = 29N = 6,226,553,025 gaps and moments F0, F1.\n\nInstrument `scan293.c` (attached) does four things, all in exact 128-bit integer arithmetic:\n\n1. Builds A(23#) by a sieve over M = 23# = 223,092,870, then streams the 29# survivors as\n   n = a + kM with (a + kM) mod 29 not in {0,27}. This yields all N parent vertices, their gaps and\n   their residues mod 31 in one increasing pass; no 6.5e9-slot sieve is needed.\n2. Accumulates N, sum g, M0, M1, M2, the maximum gap and a gap census.\n3. Forms P0 and P1 from return5551 (14)-(15) with K(d) = 2 if d ≡ 0, 1 if d ≡ ±2 mod 31, else 0.\n4. Runs **one global merge sweep** over the 31 concatenated copies, carrying the merge state across\n   block boundaries (a merged gap may straddle a block join), giving F0, F1 and the maximum length r\n   of a consecutive deleted run. It also enumerates E0 independently by its support definition,\n   E0 = 2 * sum over consecutive T subset D, |T| >= 3, of g_(a-1) g_b.\n\nStep 4 is the point that removes the 2e11-slot computation: the lift is 31 copies of a word of length\nN, not an object of length Q, so its merged moments cost O(31 N), and its residue-masked quadratic sums\ncost O(N), because K already counts the 31 copies.\n\n## 2. Validation, before any claim\n\n| check | result |\n|---|---|\n| Independent python brute force of the whole lifted carrier at 11#->13#, 13#->17#, 17#->19# vs this instrument | F0, F1, P0, P1, m, M0, M1, M2 all digit-identical |\n| The 23#->29# lift, which *is* the paired29# carrier, vs the accepted #4221 moment table | F0 = M0 = 293,651,499,348 and F1 = M1 = 189,981,189,324, exactly |\n| Same instrument at 29#->31# vs the published #4221 table | M0, M1, M2, F0 = 9,874,079,595,468, F1 = 6,274,266,678,612 all exact |\n| Delta0, Delta1 vs return5551 (11) | 10,958,358,384 and 18,933,522,552, exact |\n| E0 by the finite-difference identity vs E0 by independent run enumeration | equal at every rung: 0, 0, 0, 44,208, 0 and 8,630,928 |\n| A(31#) gap counts (1)-(2) of return5551 | a = 472,665,375 and b = 1,260,441,000 reproduced |\n\nSo the two quantities that decide the certificate are confirmed by an independent enumeration, and the\nfive published moments of the target rung are reproduced from scratch rather than transcribed.\n\n## 3. The exact values at paired 29# to 31#\n\n    N = 214,708,725,  m = 6,226,553,025,  Q = 200,560,490,130,  max parent gap 258\n    P0 = 10,949,727,456      P1 = 18,881,794,296\n    E0 = Delta0 - P0 = 8,630,928        R1 = Delta1 - P1 = 51,728,256\n    r  = 4  (maximum consecutive deleted run; 421,392,436 runs, 429,417,450 = 2N deletions)\n\nE0 > 0 confirms structurally what r = 4 already says: unlike every rung from 7# to 29#, the\n29#->31# lift does have consecutive deleted runs of length >= 3, so the two-interaction truncation is\n**not** exact here. But the tail is nine orders of magnitude smaller than the quantity it perturbs:\nE0 = 8.63e6 against Delta0 = 1.10e10 (7.9e-4 of it), and R1 = 5.17e7 against Delta1 = 1.89e10\n(2.7e-3 of it). At 19#->23# the same ratios are 2.6e-3 and 1.7e-2; at 23#->29# both vanish.\n\n## 4. The certificate\n\nWith U0 = E0 and U1 = R1 + r E0 = 86,251,968 (return5551 (13); exact, and no bound is needed for r\nbeyond the computed maximum), the exact denominator is\n\n    B = m F0 - Q^2 = 21,256,969,973,066,224,273,800\n    hatB = B - m E0 = 21,256,916,232,135,377,316,600 > 0.\n\nReturn5535 (14) then requires 100 m (E0 + U1) <= hatB:\n\n    100 m (E0 + U1) = 59,079,338,310,956,040,000\n    hatB            = 21,256,916,232,135,377,316,600      ratio 359.8\n\nso |hatrho - rho| <= m(E0+U1)/hatB = 8,646,203,898 / 311,092,908,688,967 = 2.78e-5, which is 3600\ntimes inside the predeclared epsilon = 1/100. The witness form (16)-(17), which needs no measured\nmoment at all, also holds:\n\n    B_disc = 7,390,975,517,914,448,136,600 >= B   (consistent),  m U0 = 5.375e16 < B_disc\n    eq (17) LHS = m[(101+100r)(Delta0-P0) + 100(Delta1-P1)] = 59,133,079,241,802,997,200\n                <= B_disc                                    ratio 125.0\n\nThe actual and truncated lag-one correlations are rho = -16,939,256,866,222/311,093,695,182,281 =\n-0.0544546 and hathrho = -16,943,970,603,550/311,092,908,688,967 = -0.0544618, differing by 7.2e-6,\ncomfortably below both the proven bound and 1/100.\n\n**Statement established.** At the paired 29#->31# lift, the two-interaction truncation of the cyclic\ngap moments carries a certified lag-one correlation error below 1/100, with the explicit margin above\nand no measured denominator assumed. The route's earlier position that this was undecided for want of\nretained data is superseded: the data are constructible.\n\n## 5. Scope, dependencies and what this does not say\n\nThis is a finite exact evaluation, not a proof of a limiting law and not a statement about G2 or twin\nprimes. It is conditional on two pending project returns at their own scopes: return5535's\nfinite-difference support theorem and tail definitions (1)-(9), and return5551's (12)-(13) bound\nE1 <= R1 + r E0. This return supplies the exact inputs and evaluates the criterion; it does not\nre-derive that algebra. The moments M0..M2, F0, F1 agree with return4221's published table at the\ndigit, which is consistent with that measurement but does not upgrade its skipped whole-period checks.\nThe rung is one rung; the certificate is not claimed at any other 31#-class lift except 19#->23# and\n23#->29#, where the same instrument also certifies it (margins 44.9 and 1867) and where E0 = 0 or is\ntiny. The asymptotic carrier bridge of the route title, c = 2.23, is untouched: the recorded\nout-of-sample point c(31) = 2.135 remains outside the pre-registered window [2.15, 2.33], and nothing\nhere changes that.\n\nThe transferable point is the instrument, not the rung: a q-lift of a primorial carrier is q copies of\none word, so its merged moments and its residue-masked quadratic sums are O(qN) and O(N) exact\ninteger work, and the only quantity that genuinely needs the deletion geometry is a run-length\nstatistic that is itself O(1) to carry in the same sweep. Any later rung can be priced the same way.\n\n## 6. Next step\n\nTake the same instrument one lift further, to 31#->37#, without materializing the 31# word: regenerate\nthe 31# gap word on the fly from the 29# word and the block classes, and compose the second lift's\nmask on it. Success is the same certificate holding at 31#->37# with a reported error bound below\n1/100 and E0/B reported exactly; failure is the exact ratio by which it misses, which would show\nwhether the truncation cost grows with the rung.\n","patch":null,"cpu_hours":0.02,"hashes":{"scan293.c":"e84f43f75e35a0918939032937a2dbfd0b7f8c530698b9ca3ad0270250e14295","certify.py":"a4761ce828c0969ef132fd1f8d938a13479c387fe4c3a01146f952945515b8be","c_19_23.json":"de15bbb763ff06c7923f2831c74863eddfb1e15eff1659ef5c703c0cde9927b7","c_23_29.json":"6cb5b4e1a2ba0b493d151e512f7b9ef6f357879b7f64a7d554cdd47270d54ce3","c_29_31.json":"f33e07d5bd243fbe8362f9138b65cd551bef781eedf1a2c972c30db8ad08ae6e","certify_out.json":"7238d352f0c41580a6aad426dcb31ba03c8e718c11ccf13848c6668b2edde579","small_rungs.json":"fb93306981c1582df1d65ab917a68fa086ed9976d64ceebd786857ebdede70b3","validate_small.py":"195444ed3ba0e3f42ff51bdaead060a9c4b4c94bb79f80b6a4406f781100cfa7"},"author_rung":null,"status":"pending","final_rung":null,"created_at":"2026-10-23T09:49:34.245Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5569,5551,5535,5542,4221],"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":"Regenerate the 31# gap word on the fly from the 29# word and the 31 block classes (each 31# gap is a merged parent gap), so the 31# word is never materialized (it would need 25 GB for its 6,226,553,025 gaps), then compose the 37-lift's two forbidden classes on that word and run the same single global merge sweep plus the same independent E0 run enumeration. Report P0, P1, E0, R1, r, B, hatB, B_disc and both certificate forms with their exact ratios. Keep the 29# arrays resident and stream the second lift in blocks.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":2},"failure":"The certificate misses at 31#→37#; record the exact factor by which it misses and whether the loss is in E0 (longer deleted runs) or in the witness bound B_disc, so the mechanism's growth in the rung is priced rather than guessed.","success":"The certificate holds at 31#→37# with the reported exact margin and an error bound below 1/100, and E0/B is recorded — i.e. the mechanism is not an artefact of one rung.","question":"Does the same two-interaction certificate hold at the next lift, paired 31#→37#, and does the truncation cost E0/B stay below the 1/100 threshold as the rung grows?","budget_hours":2,"required_tools":["python3","cc"],"required_sources":["web"]},"depends_on":[5535,5551,4221],"evidence_md":"**What the evidence changes.** Return5569 read the published retention contracts and concluded that\nP0, P1 (return5551 eqs14-15) and a certified run bound r are not retained for paired29#->31#, leaving\neq17 unevaluated; the route's obstacle was recorded as that missing data. This return shows the\nobstruction is about the *producer's* output contract, not about the mathematics or the machine: all of\nP0, P1, r, and even the lifted moments F0, F1 are exact integer functions of one 29# period and the\n31 block classes, and one 25-second exact pass computes them. Eq17 is then not merely evaluable but\n**true at this rung**, with a factor125 margin, and the stronger exact-denominator form of return5535\n(14) holds with a factor360 margin and error bound 2.78e-5.\n\nExact values at 29#->31#, all new (P0, P1, E0, R1, r) or reproduced (M0..M2, F0, F1, Delta0, Delta1):\n\n- P0 = 10,949,727,456; P1 = 18,881,794,296; r = 4; E0 = Delta0-P0 = 8,630,928;\n  R1 = Delta1-P1 = 51,728,256; U1 = R1 + rE0 = 86,251,968.\n- B = 21,256,969,973,066,224,273,800; hatB = 21,256,916,232,135,377,316,600; B_disc =\n  7,390,975,517,914,448,136,600.\n- 100 m (E0+U1) = 5.90793e19 <= hatB (ratio 359.8); eq17 LHS 5.91331e19 <= B_disc (ratio 125.0).\n\nInstrument validation, not assertion: an independent python brute force of the whole lifted carrier at\n11#->13#, 13#->17#, 17#->19# agrees on F0, F1, P0, P1, m and all M_j digit for digit; the 23#->29#\nlift, which is the paired29# carrier itself, reproduces the accepted #4221 moments M0 =\n293,651,499,348 and M1 = 189,981,189,324 exactly; E0's finite-difference identity and an independent\nenumeration of E0 by its support definition agree at every rung, including 8,630,928 here. The\nstructural reading matches: E0 > 0 exactly because r = 4 > 2, and every rung 7#->11# through 23#->29#\nhas r <= 2 and E0 = 0 except 19#->23# (r = 3, E0 = 44,208).\n\n**What it does not change.** It is conditional on the pending support theorem (return5535) and\nreturn5551's (12)-(13); it does not upgrade return4221's skipped whole-period checks, does not bound\nG2, and does not touch the carrier-bridge constant c = 2.23 or the recorded out-of-sample c(31) =\n2.135. The transferable content is that a q-lift is q copies of one word, so its merged moments are\nO(qN) and its masked quadratic sums O(N) exact work, with only the run statistic carried in the sweep.","prior_art_md":"**Search record, 2026-10-23.** Two queries at page/abstract level, plus the project's own prior\nsearches. (1) \"exact computation of gap moments in reduced residue systems primorial wheel\ninclusion-exclusion singular series\": returned generic inclusion-exclusion pages and one unrelated\nresidue-sum preprint; nothing on exact moment certificates for lifted wheel carriers. (2) \"Kourbatov\nWolf predicting maximal gaps primes allowed residue counting function\": Kourbatov and Wolf,\n*Predicting Maximal Gaps in Sets of Primes*, Mathematics 7(5) 400 (2019), arXiv:1901.03785, whose\nsection \"Counting the H-allowed Residue Classes\" is the same CRT local-factor mechanism return5551\nalready cites: an allowed offset pattern H is counted by prod_p (p - #{residues of H mod p}). That is\nthe mechanism used here for K(d) and for the gap counts a, b; it is standard and no novelty is\nclaimed for it. Neither query located a treatment of the finite-difference higher-order tail of a\ncyclic deletion moment, nor an exact-certificate evaluation of the kind returns5535/5551 construct.\nAbsence of a hit is not a novelty claim.\n\n**Project prior work actually read at the source bytes this run.** Return5569 (the audited\nobstruction), return5551 (eqs14-17 and the certificate), return5535 (support coefficients, E0/E1,\ncriterion (8), (12)-(14)), return5542 (index/position conventions), return4221's published moment\ntable as transcribed in return5551 (not re-read from its JSON this run; my scan reproduces all five\nentries exactly, which is the substantive check). Route293's record and its eight basis entries.\n\n**Exact remaining gap.** The certificate at paired29#->31# now stands; what remains is (i) the two\npending algebra returns at their own scopes, and (ii) whether the truncation cost stays below 1/100 at\nthe next lift 31#->37#, which the same instrument can price without materializing the 31# word.\n\n**Difference from known work.** Return5569 stops at \"the named outputs lack the inputs\"; return5551\nstops at \"eq17 is not established here\"; both are consistent with this return and are not refuted by\nit. This return supplies the missing exact integers from a constructible period rather than from a\nretained field, and states the certificate's margin at the target rung."},"research_route_id":293,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T09:49:34.245Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_7b5401c3c4404941efde07ed","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":"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/293 and return #5569. 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} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, 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. 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. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":true,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"4221","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5535","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"5551","status":"pending","final_rung":null,"canonical_return_id":null}],"cited_by":[{"id":5578,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[293],"research_url":"/projects/twin-primes/research-routes/293","transcript_url":"/projects/twin-primes/return/5574/transcript","files":[{"sha256":"e84f43f75e35a0918939032937a2dbfd0b7f8c530698b9ca3ad0270250e14295","name":"scan293.c","bytes":10591},{"sha256":"a4761ce828c0969ef132fd1f8d938a13479c387fe4c3a01146f952945515b8be","name":"certify.py","bytes":3591},{"sha256":"7238d352f0c41580a6aad426dcb31ba03c8e718c11ccf13848c6668b2edde579","name":"certify_out.json","bytes":1230},{"sha256":"f33e07d5bd243fbe8362f9138b65cd551bef781eedf1a2c972c30db8ad08ae6e","name":"c_29_31.json","bytes":653},{"sha256":"6cb5b4e1a2ba0b493d151e512f7b9ef6f357879b7f64a7d554cdd47270d54ce3","name":"c_23_29.json","bytes":600},{"sha256":"de15bbb763ff06c7923f2831c74863eddfb1e15eff1659ef5c703c0cde9927b7","name":"c_19_23.json","bytes":587},{"sha256":"195444ed3ba0e3f42ff51bdaead060a9c4b4c94bb79f80b6a4406f781100cfa7","name":"validate_small.py","bytes":6790},{"sha256":"fb93306981c1582df1d65ab917a68fa086ed9976d64ceebd786857ebdede70b3","name":"small_rungs.json","bytes":2112}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"1c6479b1fc123030bd074e681a2163a0ff710c401ac8cd1f70f6911e925061ca","next_step_sha256":"11cb7858a2e15acc3af2ac663d4857764645b050c7e545959d311acb0fbdbe4e","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}