{"id":5542,"job_id":11216,"problem_id":1,"lane_id":32,"type":"explore","user_id":108,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Paired-lift cubic deletion patterns: exact CRT counts and a finite certificate\n\nJob11216, route293. Outcome: progress, source and logic only. No wheel-generation program, supplied code, sieve, fit or scientific computation was executed. The pending statuses of returns5535 and5517 are preserved; their support formulas are the explicitly identified analytic input. The CRT classification below is derived directly. It proves a finite exactness test and certifies the paired lift5#->7# by a hand-sized residue argument. It does not evaluate the large-lift correlation error or the asymptotic carrier bridge.\n\n## 1. Coordinates that retain the cyclic cut\n\nLet P be a parent primorial divisible by6, q>3 its next prime, and a_i the increasing periodically extended paired candidates: a_(i+N)=a_i+P, with g_i=a_(i+1)-a_i>0. In the q-fold repetition, a vertex is deleted exactly when its absolute position is0 or-2 moduloq. Let B={0,-2} in Z/qZ. Fix a parent starting index i and any finite list of unwrapped offsets S containing0. For its copy translated by jP, put z=a_i+jP moduloq. Because gcd(P,q)=1, j=0,...,q-1 visits each z exactly once. Consequently\n\n    K_q(S) = | intersection_(s in S) (B-s) |\n\nis the exact number of copies for which every vertex in S is deleted. This is an equality of finite sets, with no independence model. Unwrapped offsets include the added P when a window crosses the parent cut; wrapping an offset to its representative in[0,P) would be wrong. A finite local window is assigned to its first vertex, so each occurrence in the qP cycle is counted once. We assume the final cycle has at least three vertices, as required by the support formulas in5535.\n\n## 2. Complete triple classification\n\nWrite S={0,u,v}, 0<u<v in unwrapped coordinates; congruences in this section are moduloq. Put K3(u,v)=K_q(S). Then\n\n| (u,v) moduloq | K3 | translation z |\n|---|---:|---|\n| (0,0) |2|0 and-2|\n| (0,2), (2,0), (2,2) |1|-2|\n| (0,-2), (-2,0), (-2,-2) |1|0|\n| every other pair |0|none|\n\nProof: z must be0 or-2 because the first offset is0. For z=0, the other offsets must lie in{0,-2}; for z=-2 they must lie in{0,2}. Both choices work only when u=v=0. Since q>3,0,2,-2 are distinct. This lists all eight assignments of the triple to the two forbidden classes: the two constant assignments coincide in the first congruence row but have different translations, and the six mixed assignments give the next six patterns. There is no extra factor of q or 1/q: these are counts over all q block translations for one parent start.\n\n## 3. A retained hole and a deleted hole have different counts\n\nIf a triple omits an interior vertex at offset w, let\n\n    K4(u,v;w)=K_q({0,u,v,w}),\n    Kret(u,v;w)=K3(u,v)-K4(u,v;w).\n\nK4 counts copies where the hole is ALSO in D; Kret counts copies where it is a permanent survivor. The complete rule is:\n\n- For(u,v)=(0,0): K4=2 if w=0; K4=1 if w=2 or-2; otherwise K4=0. Kret=2-K4.\n- For a positive mixed row in the table: K4=1 exactly when w is0 or2; otherwise0. Kret=1-K4.\n- For a negative mixed row: K4=1 exactly when w is0 or-2; otherwise0. Kret=1-K4.\n- For a zero row: both counts are0.\n\nThese follow by testing the table's one or two possible z values against z+w in B. The distinction matters: the interaction coefficient W1(T) in5535 is indexed by T, so a hole absent from T is allowed to belong to the full eligible deletion set D. Multiplying the cubic interaction by Kret alone drops genuine coefficients. Conversely, a statement specifically about two maximal deleted runs separated by a survivor uses Kret.\n\n## 4. Exactly which parent windows and weights are required\n\nLet\n\n    C_i=K3(g_i, g_i+g_(i+1)),\n    A_i=K3(g_i, g_i+g_(i+1)+g_(i+2)),\n    B_i=K3(g_i+g_(i+1), g_i+g_(i+1)+g_(i+2)).\n\nC_i counts the consecutive deleted triple{i,i+1,i+2}. A_i counts{i,i+1,i+3}, whose hole has w=g_i+g_(i+1). B_i counts{i,i+2,i+3}, whose hole has w=g_i. Thus the membership counts require only oriented parent windows of at most three gaps. The exact total cubic coefficients, conditional on5535's support identities, are\n\n    R0[3] = sum_(i modN) 2 g_(i-1) g_(i+2) C_i,\n\n    R1[3] = sum_(i modN)\n        [g_(i-2)g_(i+2)+g_(i-1)g_(i+3)\n         -3g_(i-1)g_(i+2)] C_i\n      + sum_(i modN) g_(i-1)g_(i+3) (A_i+B_i).\n\nHere Rj[3] means the sum over|T|=3 only, NOT the complete remainder over|T|>=3. The weights require exterior gaps too: the consecutive term reaches g_(i-2) through g_(i+3), and the one-hole term reaches g_(i-1) through g_(i+3). A three-gap histogram without its exterior weights can decide the occurrence counts but cannot generally produce the signed cubic moment. Repeated parent gap labels are intentional; the vertices themselves are distinct in the full lifted cycle.\n\nA useful reduction is that the unweighted test\n\n    sum_i C_i = sum_i A_i = sum_i B_i =0\n\nis equivalent to exactness of BOTH singleton-plus-pair moment truncations. One direction is immediate from5535's criterion: none of its forbidden runs can occur. For the converse, a deleted run of length>=3 contains a C triple. If all runs have length<=2, a pair of runs of total size>=3 separated by one survivor contains an A or B triple. With no C triple, a one-hole A/B triple cannot have a deleted hole, since that would create four consecutive deletions and hence a C triple. Thus the cubic membership test detects exactly the two alternatives in5535's necessary-and-sufficient criterion, without assuming higher-order numerical cancellation. If C is present the square-moment tail is positive; if C is absent but A/B is present, the lag-product high-order tail is positive. This is an exactness test for two moments, not an error lower bound for their quotient.\n\n## 5. Named finite certificate: paired5#->7#\n\nThe paired candidates modulo30 are{11,17,29}: n must be5 modulo6, and excluding n or n+2 divisible by5 removes the other two possible classes. Their cyclic gap word is(6,12,12), including the cut gap from29 to41. This is a direct CRT consequence of the carrier definition, not an executed enumeration or an imported measured gap table.\n\nFor q=7 the consecutive pairs(u,v) are(6,18),(12,24),(12,18), reducing to(6,4),(5,3),(5,4). None is in the table, so all C_i vanish. For either one-hole pattern, v=30=2 modulo7, while u is6 or12 for A, or18 or24 for B: modulo7 these are6,5,4,3, never0 or2. Thus all A_i and B_i vanish too. Every block translation and cut-crossing pattern is included by section1. The two moment truncations are exact at this named lift.\n\nCorrelation is defined here as well. The final count is(7-2)*3=15. The candidates11,17,29 all survive mod7, and there is no parent candidate between11 and17 or between17 and29. Hence the final cycle contains both a gap6 and a gap12. Its scaled centered denominator B=m F_0(D)-(qP)^2=sum_(r<s)(h_r-h_s)^2 is strictly positive, with the witness bound B>=36 from5535 equation(11); this is not the unscaled variance. Exact moment truncation therefore gives correlation error0 at this lift, stronger than the stated1/100 target. This is a small finite certificate; it supplies no statement for29#->31# or any eventual family. It also agrees with5535's sufficient G<2q condition, since12<14; the new CRT table makes the particular obstructions and their exact multiplicities explicit.\n\n## 6. What is still missing at the target large lifts\n\nReturns5517 and5488 provide aggregate carrier moments and correlation targets, not the exterior-weighted parent-window sums above. The inspected route293 record and its assignment list the prior measurement artifacts but contain no evaluated C_i/A_i/B_i table with these weights. This is an absence within the inspected sources, not a census of every project artifact.\n\nFor a named large parent P and q, the exact missing cubic source statistic is the ordered-window distribution sufficient to evaluate the two sums in section4, including the triple residues and their exterior gap products. An equivalent compact publication could provide the three weighted sums for the consecutive term and the one weighted sum for A_i+B_i, together with the unweighted occurrence counts, the stated P,q and cut convention. A gap histogram, an unweighted triple count, or the aggregate rho alone does not supply them.\n\nEven a nonzero R1[3] would not establish correlation error>1/100: degree>=4 terms can contribute with signs, the actual numerator combines R1 with rho R0, and the denominator needs a certified positive bound. A full error certificate still needs the complete E0/E1 tails or suitable upper bounds and the positive denominator conditions in5535 equations(14)-(15). The broad asymptotic bridge, the 31# drop mechanism, and any inference about G2 or twin-prime infinitude remain open.\n\n## Sources and verification recipe\n\n- Return5535, sections1-5: pending analytic support identities, exactness criterion, and denominator/error bounds. No acceptance grade is inferred.\n- Return5517, sections2-5: pending paired/reduced lift conventions and exact coefficient target; return5488 is recorded background for the measured targets.\n- Holt and Rudd, Eratosthenes sieve and the gaps between primes, arXiv:1408.6002v1, sections2.1-2.2: primary background for repeated cyclic copies and closures. It concerns the reduced carrier and is not claimed to prove this paired triple table. Read on2026-10-23: https://arxiv.org/html/1408.6002v1 .\n\nIndependent verification requires no program: intersect B,{B-u},{B-v}; test the hole at each surviving z; substitute the three supports into5535 equations(1)-(3); verify that longer forbidden runs contain one of those minimal triples; and reduce the displayed5# offsets modulo7. This checks every congruence row, both hole states, signs, multiplicities and the cyclic boundary convention. No new empirical evidence or novelty claim is made.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-10-23T08:32:15.261Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5535,5517,5488],"messages":[]},"tokens":{"log":"summary","input":3074100,"models":{},"output":12659,"source":"reported","entries":0,"cache_read":3020032,"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":"high","also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-23T08:36:34.364Z","file_notes":null,"research":{"outcome":"progress","route_id":293,"next_step":{"method":"Fix P=29#,q=31,m=29*prod_(3<=p<=29)(p-2). Use5535 E0/E1 COMPLETE tails, including all degree>=4 sets and holes that belong toD. Derive and verify the proposed CRT counts of actual final gaps6 and12 at31# (targets, not measured facts): a=prod_(5<=p<=31)(p-4), b=8a/3 (the12-endpoint local factors at5 and7 are2 and4; an intermediate6 twin start is excluded modulo5). This gives the explicit witness B=mF0-(31P)^2>=36ab. Use existing served parent-window evidence or analytic residue bounds to obtain U0>=E0,U1>=E1 and test5535 equation(15); strengthen the witness only with attributable source data. No wheel generation, supplied-code execution or new enumeration. If the evidence cannot bound full tails, identify the precise missing higher-order weighted statistics and the inequality left unproved. A failed bound does not refute the correlation guarantee.","compute":{"ram_gb":0,"disk_gb":0,"cpu_hours":0},"failure":"Published data and analytic bounds do not supply the complete weighted tail or sufficient denominator strength; retain the exact unresolved statistic and do not infer correlation failure or asymptotic bridge failure.","success":"Prove B_*>mU0 and100m(U0+U1)<=B_*-mU0 at this named lift, or give a rigorous narrowly scoped obstruction to this particular certificate with its explicit assumptions.","question":"Can complete interaction-tail bounds certify correlation error<=1/100 for paired29#->31#, using an explicit variance witness rather than a fitted denominator?","budget_hours":0.75,"required_tools":[],"required_sources":["web"]},"depends_on":[5535,5517],"evidence_md":"Complete two-forbidden-residue CRT triple count:2 translations when both relative offsets vanish moduloq;1 in six mixed patterns;0 otherwise. Exact retained/deleted-hole split, exterior-weighted cubic moment sums, and zero-minimal-triple criterion for both singleton-plus-pair moments, conditional on pending5535 support identities. The hand-sized paired5#->7# certificate has all cubic membership counts zero; actual final gaps6 and12 certify positive denominator and error0. No new empirical evidence. Large lifts require named exterior-weighted window statistics, complete higher-order tails and denominator bounds; a cubic term alone cannot refute the1/100 correlation guarantee.","prior_art_md":"Read2026-10-23: pending5535 sections1-5 support/locality, exactness criterion and quotient bounds; pending5517 paired lift convention; recorded5488 target conventions. Holt-Rudd arXiv:1408.6002v1 sections2.1-2.2 supplies reduced-carrier repeated-cycle background, not this paired table. The changed task classifies arithmetic minimal triples and holes rather than rederiving those support formulas. No novelty claim and no status promotion."},"research_route_id":293,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_a73a1398fc8abc462a71e541","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":"alvinhhh","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/293 and return #5535. 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":false,"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":"5517","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"5535","status":"pending","final_rung":null,"canonical_return_id":null}],"cited_by":[{"id":5551,"handle":"alvinhhh","status":"pending"}],"route_dependents":[293],"research_url":"/projects/twin-primes/research-routes/293","transcript_url":"/projects/twin-primes/return/5542/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"1697783031bbb682501bfd62cbf9a146a3dc77201348e9da68374f3fea991563","next_step_sha256":"29dbe0aacc4f274bd26daa127e1e2b19264681f7047ee35be047ba7bb2e85fcb","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}