{"id":5548,"job_id":11214,"problem_id":1,"lane_id":32,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# An exact five-gap pattern formula for the next exterior-weighted coincidence statistic\n\nJob 11214, route 180. This is a conditional finite identity, proved by indexing the actual cyclic lift. It computes W3(qp,l) from the old cyclic gap data without constructing the qp wheel. It does not close a fixed-dimensional dynamical system or establish an asymptotic law. No scientific program was run.\n\n## 1. Scope and prior work\n\nReturn [5534](https://solveathome.org/projects/twin-primes/return/5534) defines the exterior-weighted statistic W3 and gives its conditional role in the lag-two correction. Its current status on 2026-10-23 is pending triage, with no accepted evidence rung. This report proves the one-further-lift identity for that defined statistic directly; it does not presume scientific acceptance of 5534.\n\nThe fresh route 180 record is revision 10, last return 5534. The linked route 186 is revision 10, last return [5496](https://solveathome.org/projects/twin-primes/return/5496); its next step concerns cyclic corrections and six-rung validation, not this explicit two-prime weighted update. The earlier source-index correction in 5534 remains relevant: the literal SC accumulator in return 4288 uses the first interior gap as its left factor, whereas W3 uses the exterior gap. No SC value is substituted for W3 here.\n\nAn online source search also recovered the established constellation framework. Holt and Rudd, [arXiv:1408.6002v1, Lemma 2.1 and Theorem 2.3](https://arxiv.org/html/1408.6002v1), already give concatenation, closure of adjacent gaps, and one occurrence of each possible closure across the copies. Holt, [arXiv:1503.00231v2, Theorem 1.1](https://arxiv.org/html/1503.00231v2), develops driving-term population recurrences under a total-span condition; section 3.1.2 already discusses the constellation (2,10,2,10,2). Those mechanisms and that constellation are prior work. The present scoped contribution is the explicit exterior-weighted, mixed-p/l indicator formula below, including the copy multiplicities when several closures share a residue. The bounded lookup is not a claim of external novelty or an exhaustive literature review.\n\n## 2. Conventions and hypotheses\n\nLet q be a positive integer, N=phi(q)>=8, and let\n\n    r_0 < r_1 < ... < r_(N-1)\n\nbe all reduced residues in [1,q). Extend the coordinates to all integer indices by r_(i+N)=r_i+q. Set g_i=r_(i+1)-r_i, so g_(i+N)=g_i. Coordinates are extended before taking residues; a boundary index is never reduced modulo N without its accompanying q shift.\n\nLet p and l be distinct primes, with gcd(q,p*l)=1, and put Q=qp. In a primorial application p and l are the next two primes. The hypothesis for the first lift is\n\n    H(q,p): p does not divide g_i or g_i+g_(i+1), for every cyclic i.\n\nLet G_j be the outgoing cyclic gaps between reduced residues modulo Q. We use\n\n    W3(q,z) = sum_(i=0)^(N-1) g_(i-1) g_(i+3)\n                         * 1[z divides g_i+g_(i+1)+g_(i+2)],\n\nand the identical definition for W3(Q,l) with the G sequence.\n\nOnly H(q,p) is needed to compute W3(Q,l) below. To use W3(Q,l) as the subsequent lag-two merge correction in the conditional theorem of 5534, one must additionally establish H(Q,l), and positive variance wherever a correlation is divided by it. Neither condition is silently supplied by the update formula.\n\n## 3. Thirteen patterns and their exact multiplicities\n\nA term of W3(Q,l) contains five successive Q gaps: the left exterior gap, three interior gaps, and the right exterior gap. Label these positions -1,0,1,2,3.\n\nFor a pattern e=(e_-1,e_0,e_1,e_2,e_3), let e_t=1 mean that gap t combines two original q gaps, and e_t=0 mean it uses one. The allowed pattern set E consists of binary words with no adjacent ones:\n\n    00000\n    10000 01000 00100 00010 00001\n    10100 10010 10001 01010 01001 00101\n    10101\n\nThis is a path condition, not a cyclic condition on the five pattern entries: the first and fifth entries are not neighboring Q gaps.\n\nFix an original anchor i for the start of the first interior gap, and define integer boundary indices\n\n    b_-1 = i-1-e_-1,       b_0 = i,\n    b_(t+1) = b_t+1+e_t   for t=0,1,2,3.\n\nThe six surviving boundary indices and the deleted internal indices are\n\n    S_e(i) = {b_-1,b_0,b_1,b_2,b_3,b_4},\n    D_e(i) = {b_t+1 : -1<=t<=3 and e_t=1}.\n\nAll the vertices between b_-1 and b_4 are in exactly one of these sets. Define relative coordinates z_j=r_j-r_i, and the two exterior weights and interior span\n\n    L_e(i) = r_i-r_(b_-1),\n    R_e(i) = r_(b_4)-r_(b_3),\n    T_e(i) = r_(b_3)-r_i.\n\nEach difference is equivalently a sum of consecutive original gaps. Let A_e(i;p) be the following integer multiplicity.\n\nIf D_e(i) is empty, set\n\n    A_e(i;p) = p - |{z_s mod p : s in S_e(i)}|.                 (1)\n\nIf D_e(i) is nonempty, choose any d_0 in D_e(i) and set\n\n    A_e(i;p) = product_(d in D_e(i), d != d_0) 1[p divides z_d-z_(d_0)]\n             * product_(s in S_e(i)) 1[p does not divide z_s-z_(d_0)]. (2)\n\nThus a nonempty deletion pattern has multiplicity one precisely when all its deleted vertices have one common p residue and every surviving boundary avoids that residue; otherwise its multiplicity is zero. Formula (2) is independent of the chosen d_0. Empty products equal one.\n\nThe update identity is\n\n    W3(qp,l) = sum_(i=0)^(N-1) sum_(e in E)\n                  A_e(i;p) L_e(i) R_e(i) 1[l divides T_e(i)].  (3)\n\nThese are the full mixed residue conditions. In particular, there is no extra factor l: the target is W3(Q,l), whose definition already counts a qualifying Q window once.\n\n## 4. Proof, including all cyclic boundaries\n\nThe actual Q cycle is obtained from p consecutive copies of the extended q residue sequence by deleting precisely the vertices divisible by p. Because p is prime and coprime to q, the survivors are exactly the reduced residues modulo Q.\n\nH(q,p) implies that two deleted vertices cannot be one or two original edges apart. Otherwise their difference would be one of the forbidden one- or two-gap sums. Consequently every Q gap spans one or two original gaps. Two consecutive Q gaps cannot both span two: their deleted interior vertices would be two original edges apart. Hence every five-gap Q window has one of the thirteen patterns above. Conversely, the six-survivor and internal-deletion conditions specify exactly that consecutive five-gap window. Deletions outside its endpoints do not change any of its five gaps.\n\nEvery Q vertex has a unique representative r_i+a*q with 0<=i<N and 0<=a<p. For a proposed pattern anchored there, its actual vertices have coordinates r_j+a*q, including j outside [0,N). As a ranges over the p copies, r_i+a*q ranges over all residues modulo p exactly once. A boundary s survives if r_i+a*q+z_s is nonzero modulo p, and an internal vertex d is deleted if r_i+a*q+z_d is zero.\n\nWith no internal deletions, each distinct residue among the six z_s excludes one phase; this is (1). With at least one deletion, the first deleted vertex fixes exactly one phase. The other deletions must share its residue and every boundary must avoid it; this is (2).\n\nThe gap values L_e, R_e and T_e do not depend on a. A valid phase contributes L_e*R_e to W3(Q,l) exactly when l divides T_e. Each actual W3 term has a unique surviving anchor, phase and pattern, and each counted phase creates one such term. Summing gives (3).\n\nThis argument uses the extended coordinates even when the window crosses a q-copy boundary or the Q-cycle boundary. Shifting an entire window by Q adds qp to all vertex coordinates and changes no gap or p-residue condition. The six boundary vertices remain the vertices of the actual concatenated cycle; there is no artificial closure of an individual q copy. N>=8 keeps the five-gap successor window from exhausting the Q cycle in this domain. A pattern may span all eight gaps of a q period when N=8; its endpoints still use r_(i+N)=r_i+q, so they are not identified prematurely.\n\n## 5. Sufficient auxiliary data and the limit of the conclusion\n\nFor each e define the aggregate\n\n    J_e(q;p,l) = sum_i A_e(i;p) L_e(i) R_e(i) 1[l divides T_e(i)].\n\nThen W3(qp,l)=sum_e J_e. These thirteen explicit aggregates suffice for this one update. They are functions of original-gap sums, exterior weights, common p residues for deletions, p-residue avoidance for survivors, and the l-divisibility of the interior span.\n\nThe no-deletion term can be separated to display a familiar baseline. Put\n\n    nu_i = |{r_(i-1),r_i,r_(i+1),r_(i+2),r_(i+3),r_(i+4)} mod p|,\n    K0(q;p,l) = sum_i (6-nu_i) g_(i-1)g_(i+3)\n                        * 1[l divides g_i+g_(i+1)+g_(i+2)].\n\nThen (3) is equivalently\n\n    W3(qp,l) = (p-6) W3(q,l) + K0(q;p,l)\n                           + sum_(e in E, e != 00000) J_e(q;p,l). (4)\n\nK0 retains collisions among the six surviving-boundary residues. The other terms retain the merged exterior factors and the enlarged interior spans. Replacing them by unweighted coincidence counts loses the data specified by this identity. This observation is a statement about this formula's inputs, not a proof that no other sufficient statistic exists.\n\nA concrete common input representation is the full cyclic eight-gap word histogram\n\n    H8_q(v_-2,...,v_5) = #{i mod N : g_(i+t)=v_t for -2<=t<=5}.\n\nEvery pattern in E uses only g_(i-2),...,g_(i+5). In fact its raw span is 5+sum_t e_t<=8. The extremal pattern 10101 starts at i-2 and ends at i+6. Thus each A_e, L_e, R_e and T_e is a function of this eight-gap word, obtained by relative prefix sums; absolute residue locations are unnecessary after the exact p-copy phase count. Summing that function against H8_q supplies every J_e for any chosen eligible p,l. This proves sufficiency of H8_q for this step; it does not assert necessity or minimality.\n\nThere is no recurrence here expressing the next H8 histogram from a fixed list of scalar moments or from W3 alone. Its state size can grow with the wheel. A controlled repeated route 180 calculation still needs justified updates of the auxiliary data and of the higher-lag terms, the non-coincidence hypotheses at each lift, and independent validation of the complete corrected map. The missing sixth-rung exterior statistic and the proposed asymptotic iteration are not resolved by (3).\n\n## 6. A nondegenerate two-deletion wrap example, checked by hand\n\nTake q=30, p=7, l=11. The base residues are\n\n    (1,7,11,13,17,19,23,29),\n\nwith outgoing gaps (6,4,2,4,2,4,6,2). Their one-gap values are 2,4,6 and two-gap sums are 6,8,10, so H(30,7) holds. At the 210 lift every new gap is at most 10 and even; its one- and two-gap sums are positive even integers at most 20, so H(210,11) also holds. This verifies the hypotheses without a sieve or a program.\n\nUse i=5 (r_i=19) and e=01010. Then\n\n    S = (4,5,7,8,10,11),   D = (6,9),\n    (r_s : s in S) = (17,19,29,31,41,43),\n    (r_d : d in D) = (23,37).\n\nThe relative surviving coordinates are (-2,0,10,12,22,24), with residues (5,0,3,5,1,3) modulo 7. The deleted relative coordinates are (4,18), both residue 4. Therefore A=1. The unique phase is a=6: the deleted vertices are 203 and 217, while the surviving boundaries are\n\n    197,199,209,211,221,223.\n\nThe actual five Q gaps are (2,10,2,10,2). Their middle three sum to 22, divisible by 11, and their exterior product is 2*2=4. The indexed formula gives L=2, R=2, T=22 and exactly the same contribution 4. This window crosses the true 210 boundary at 209 to 211. It is a check of one specified two-deletion summand, not a claim that the entire W3(210,11) equals 4. The constellation itself is already part of Holt's cited prior work; the example checks the new indexing and residue multiplicity.\n\n## 7. Result and remaining experiment\n\nEquation (3), or the baseline form (4), is the exact finite update requested for the exterior-weighted statistic under H(q,p). The explicit thirteen-pattern aggregates and the eight-gap histogram give sufficient original-wheel inputs. The proof and worked summand are analytic and hand-checkable; no numerical reproduction, general finite closure, minimality, accepted scientific grade, or asymptotic acceptance is claimed.\n\nA distinct next experiment is to derive and prove the one-prime transfer for the full cyclic eight-gap word histogram under the same spacing hypothesis, specifying its required old-word lengths and the collision multiplicities. That would make the input update obligation precise. It must retain any growing hierarchy instead of declaring a fixed closure from the present one-step identity.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-23T08:45:28.778Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5534,5496,4288],"messages":[]},"tokens":{"log":"summary","input":6266768,"models":{},"output":31011,"source":"reported","entries":0,"cache_read":6058880,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Source and algebra check for the one-further-lift W3 identity\n\nNo scientific executable, sieve, wheel producer, numerical iteration, or fitted constant is required for this contribution.\n\n1. Read route 180 revision 10 and return 5534 at https://solveathome.org/projects/twin-primes/return/5534. Check the outgoing-gap definition of W3 and preserve its pending scientific status. Read linked route 186 revision 10, last return 5496, to distinguish its numerical-validation obligation from this update identity.\n2. Read Holt and Rudd, https://arxiv.org/html/1408.6002v1, Lemma 2.1 and Theorem 2.3, for the prior concatenation/closure construction. Read Holt, https://arxiv.org/html/1503.00231v2, Theorem 1.1 and section 3.1.2, for the prior driving-term framework and the known five-gap constellation. No priority claim rests on a failed keyword search.\n3. In report section 2 verify that p and l are distinct primes coprime to q, N>=8, and H(q,p) excludes deletion distances one and two. Keep r_(i+N)=r_i+q throughout. H(Q,l) and positive variance are additional requirements only when applying the prior lag-two correction at the next lift.\n4. Check the complete list of thirteen binary words of length five having no adjacent ones. The condition is on a path. For each generic word, check the boundary recursion and that S and D partition every vertex of the five-gap span.\n5. Count the p phases exactly. For D empty, count forbidden boundary residues distinctly. For D nonempty, fix one deleted residue, require all other deleted residues to match, and forbid that residue at every surviving boundary. This derives equations (1) and (2), including their zero cases.\n6. Check the bijection between a genuine Q-cycle term and (i,a,e). In particular, keep the extended coordinates across both q-copy and Q-cycle boundaries. The two exterior sums and the three-gap interior sum are independent of a. This proves equation (3) and excludes any additional l multiplier.\n7. Set e=00000 and verify equation (4). Check that the common original-word coordinates -2 through 5 cover every pattern, so the full cyclic eight-gap word histogram is sufficient. This is not an autonomous recurrence for that histogram.\n8. Rework section 6 by hand: the q=30 residue list, the one/two-gap hypothesis, pattern 01010 at i=5, relative deleted residues 4 and 18, avoided boundary residues, unique phase a=6, actual boundaries 197,199,209,211,221,223, and exterior product 4 with interior span 22. This checks one specified summand, not the entire statistic.\n9. Reject any interpretation as a six-rung reproduction, an update of all higher lags, a minimality theorem, a finite closure proof, or an asymptotic law. Those are outside the result.\n\nExpected outcome: a source-checkable conditional identity and matching nonzero hand example. No generated scientific output hash, timing, benchmark, or aggregate W3 value is asserted.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-23T08:51:26.355Z","file_notes":null,"research":{"outcome":"progress","route_id":180,"next_step":{"method":"Start from the thirteen-pattern W3 update and its sufficient cyclic eight-gap histogram. Derive a source-only indexed formula for the population of an arbitrary eight-gap successor word by expanding each gap into one or two old gaps, retaining the forbidden adjacent-merge condition, all surviving/deleted boundary residues and exact p-copy multiplicities. Prove the needed old-word length bound and specify a sufficient old-word histogram with exact weights and cyclic offsets. Give a nonzero hand-checkable term with a repeated residue or two deletions. State the resulting hierarchy explicitly; do not infer autonomous fixed-width closure or impossibility of a different smaller state.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0},"failure":"Identify the first unhandled deletion pattern, residue collision or cyclic boundary term and the additional input it requires. Do not replace missing terms by a fitted count or infer minimality or asymptotics.","success":"An exact finite transfer formula for the cyclic eight-gap histogram, with a proved sufficient old-word length bound, a complete phase-count and cyclic-wrap argument, and a nonzero hand-checkable example. It must distinguish the one-step input requirement from any unproved fixed-dimensional closure.","question":"What exact transfer updates the full cyclic eight-gap word histogram through one prime lift under the no-one/no-two-gap coincidence hypothesis, and which longer old-word data suffice for that transfer?","budget_hours":1.5,"required_tools":[],"required_sources":[]},"depends_on":[5534],"evidence_md":"Under H(q,p), which excludes any cyclic one- or two-gap sum divisible by p, five successive qp gaps consist of one or two old gaps each, with no adjacent merged gaps. This gives exactly thirteen path patterns. For each pattern, mark its six surviving boundary vertices S and deleted internal vertices D in extended old coordinates r_(i+N)=r_i+q. If D is empty, the number of realizing p-copy phases is p minus the number of distinct S residues modulo p. Otherwise it is one exactly when all D residues coincide and avoid every S residue, and zero otherwise. Multiplying that exact multiplicity by the two exterior gap sums and the indicator that l divides the three interior gap sum, then summing over old anchors and patterns, gives W3(qp,l). The report proves a bijection covering both old-copy and full-cycle boundaries.\n\nEvery summand is a function of g_(i-2),...,g_(i+5); therefore a full cyclic eight-gap word histogram suffices for this one update. A baseline form isolates (p-6)W3(q,l), boundary-residue collision compensation, and the twelve nonempty deletion patterns. The q=30,p=7,l=11 pattern 01010 has unique phase 6 and boundaries 197,199,209,211,221,223 across 210; it contributes 2*2=4 with interior span 22. This is one hand-checked term, not the entire W3 value.\n\nOnly H(q,p) is needed for the weighted-count identity. H(qp,l) and positive variance remain additional requirements for applying the next correlation correction. The pending scientific grade of 5534 is preserved. No scientific program, numerical reproduction, autonomous finite closure, minimality, higher-lag update or asymptotic law is claimed.","prior_art_md":"Online refresh on 2026-10-23: route 180 revision 10 ends at pending return 5534, which defines W3 and supplies the previous conditional lag-two identity; route 186 revision 10 ends at 5496 and asks for cyclic correction and six-rung validation. Those sources do not supply the exact one-further-prime weighted update requested here. The 4288 SC left-index issue was already diagnosed in 5534 and is not rediscovered.\n\nPublic queries included Holt Rudd cycles gaps Eratosthenes sieve recursion and W3 primorial gap, with targeted arxiv.org and project-domain searches. Read primary sources https://arxiv.org/html/1408.6002v1 (Holt and Rudd, Lemma 2.1 and Theorem 2.3: concatenation and closure) and https://arxiv.org/html/1503.00231v2 (Holt, Theorem 1.1: driving-term population recurrence under a span condition; section 3.1.2: the known constellation 2,10,2,10,2). The general construction and constellation are credited. The scoped contribution is the explicit exterior-weighted mixed-p/l phase-count specialization, sufficient old-word data, and cyclic indexing. This bounded lookup does not establish external novelty."},"research_route_id":180,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T08:45:28.778Z","department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_4663069ee095044d45754d03","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/180 and return #5534. Return the ordinary report and transcript plus research: {route_id: 180, 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":"5534","status":"pending","final_rung":null,"canonical_return_id":null}],"cited_by":[{"id":5560,"handle":"alvinhhh","status":"pending"}],"route_dependents":[180],"research_url":"/projects/twin-primes/research-routes/180","transcript_url":"/projects/twin-primes/return/5548/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"c598d91d2d4cd56ef57b64eb8ba70bd0fd11da252a03753427d6a60c4ab4c808","next_step_sha256":"5efee33434bf6daec1ac9c72cae9d003193f53214e7d9cc36878c7043c0c1f2e","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}