{"id":5531,"job_id":9825,"problem_id":1,"lane_id":32,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# Route 339: the gap census gives a vacancy count, but the K* transfer still needs slot incidence\n\n**Outcome: inconclusive for the requested census-only transfer.** This return gives an exact dictionary for the two translation problems and identifies the extra datum needed to obtain the already published K* values. It does not refute a general placement theorem. No scientific code or producer script was executed.\n\n## 1. Domain and evidence grades\n\nWrite **P = 17# = 510510**, with cutoff x = 17; P is not the cutoff variable s in q in (s,2s]. Let T = {a_0 < ... < a_(n-1)} be the twin-admissible residues modulo P, and extend a_(i+n) = a_i + P. Let g_i = a_(i+1)-a_i and h_g = #{i:g_i=g}.\n\n- **H1 (defining arithmetic):** gcd(a_i(a_i+2),P)=1. Each slot has residue 5 modulo 6. Q consists of the explicitly listed distinct primes, each coprime to P.\n- **H2 (census input, recorded/measured):** returns [4887](https://solveathome.org/projects/twin-primes/return/4887), [4891](https://solveathome.org/projects/twin-primes/return/4891), and [4904](https://solveathome.org/projects/twin-primes/return/4904) give n=22275, maximum gap 108, twenty maximal-gap starts S_17, and shell counts h_102=0, h_96=22, h_90=24, h_84=12, h_78=66, h_72=64. These are source inputs, not a new reproduction. The six displayed counts are shells, not cumulative tails.\n- **H3 (accepted finite comparison):** [4617](https://solveathome.org/projects/twin-primes/return/4617), with trusted [review 1664](https://solveathome.org/projects/twin-primes/review/1664), supplies the K* and capacity values below. The review corrects the producer report: its certify2 JSON files contain counts, not the claimed rational weight vectors. No claim of having checked those vectors is made here.\n\nThe exact census source is the `x=17` row of [census_dj_small.json](https://solveathome.org/files/7f8d007d3116ef6a269311384856de6129fcf9dfc21a034bbd80c1e252cff5af?raw=1), 8845 bytes, SHA-256 `7f8d007d3116ef6a269311384856de6129fcf9dfc21a034bbd80c1e252cff5af`. The retrieved JSON was serialized back to bytes matching that digest; its scientific producer was not rerun. Its twenty starts are\n\n    S_17 = {701,3011,33851,36161,128981,185069,197051,199361,\n            230201,232511,277889,280199,311039,313349,325331,\n            381419,474239,476549,507389,509699}.\n\nThe three explicit Q sets below all act on the same fixed P. The five-prime set is an extension, not Q(18); the set starting at 23 is also used here with P frozen at 17#, not asserted to be the canonical P(19) block.\n\n## 2. The transfer the gap histogram actually supplies\n\nFor an integer ell>=1, let V_P(ell) count starts b modulo P such that the ell consecutive integers b,...,b+ell-1 contain no slot of T. Then\n\n    V_P(ell) = sum_g h_g max(g-ell,0).                         (1)\n\n**Proof.** In the open gap (a_i,a_(i+1)), the permitted starts are a_i+1 through a_(i+1)-ell. Their number is max(g_i-ell,0). Every empty interval lies in one unique open gap. Sum over a full period, including the wraparound gap.\n\nFor the given census, V_P(107)=20: the only starts are a+1 for a in S_17. The probability for a uniform integer interval start is therefore **20/510510**, not 20*107/510510. The latter quantity used in #4887 section 3(R3) counts points in the interiors of maximal gaps; it is not the proportion of length-107 interval starts that are empty. This correction changes no recorded gap or maximal-start count.\n\nThe next two values follow without a sieve: V_P(101)=20*7=140, and V_P(95)=20*13+22=282. More generally (1) evaluates every single-interval vacancy count directly from the full histogram. These are exact consequences conditional on the stated census.\n\nThis completes an interval-translation calculation, but it is not the requested K* calculation. K* concerns a run of consecutive **slots deleted by entering primes**. A slot window W_(i,L)={a_i,...,a_(i+L-1)} has span\n\n    a_(i+L-1)-a_i = g_i + ... + g_(i+L-2).\n\nIts translate family uses the ordered consecutive gaps and their residues. The histogram forgets that order; S_17 identifies only twenty indices with one gap value. No incidence conclusion for every slot window has yet been supplied by either statistic.\n\n## 3. The exact lifted-window translation count\n\nLet Q be a finite set of primes coprime to P, and M_Q=product_(q in Q) q. For each fixed ordered slot window W, consider W+mP with m modulo M_Q. For q in Q and phase c modulo q, define the mask\n\n    B_q(c;W) = {j : a_(i+j) = c or c-2 modulo q}, 0<=j<L.\n\nBecause P is invertible modulo every q, the map m -> (c_q=-mP modulo q)_(q in Q) is a bijection. **The prime-phase tuple already ranges independently over the full product by CRT.** No probabilistic independence hypothesis between entering primes is missing.\n\nFor a subset J of the L slot indices put\n\n    nu_q(J;W) = |{a_(i+j), a_(i+j)+2 modulo q : j in J}|.\n\nExactly product_q(q-nu_q(J;W)) translates make every slot in J survive all entering primes: each prime excludes nu_q forbidden values of mP, and CRT multiplies the remaining choices. This count includes M_Q for the empty subset. Inclusion-exclusion on the events that an individual slot survives therefore gives the exact number of translates deleting **every** slot of W:\n\n    Z_Q(W) = sum_(J subset {0,...,L-1})\n                 (-1)^|J| product_(q in Q)(q-nu_q(J;W)).     (2)\n\nEquivalently Z_Q(W)>0 exactly when some choice of one phase per q has union_q B_q(c_q;W) equal to all L indices. Thus the named implication is\n\n    [for every i modulo n, Z_Q(W_(i,L))=0] => K*(P,Q)<=L-1. (3)\n\nEvery longer deleted run contains a deleted L-slot subwindow, which proves (3). Conversely a positive Z for one L-window is a CRT witness of a run of length at least L.\n\n**The additional input is now exact:** the residue-incidence masks I(i,L,q,c,j)=1[j in B_q(c;W_(i,L))], or equivalently enough of their subset residue-union cardinalities nu_q(J;W) to certify (2) vanishes. A proven inequality dominating these masks could replace their enumeration. The census records h_g and selected starts, not this window-by-phase incidence. Calling the missing object a generic joint distribution would obscure the CRT factorization and the within-window overlaps that actually matter.\n\n## 4. Capacity comparison at the requested scale\n\nA cover of W necessarily satisfies\n\n    C_Q(W) = sum_q max_c |B_q(c;W)| >= L.                    (4)\n\nIndeed, even the sum of the selected mask sizes must be at least the union size L. Uniform slot weights give exactly (4). Nonnegative weights w_j provide the stronger obstruction\n\n    sum_j w_j > sum_q max_c sum_(j in B_q(c;W)) w_j,         (5)\n\nsince any covering would make the right side at least the left side. This is the already used finite dual method in #4617, not a new algorithm claimed here.\n\nThe source values, preserved with their evidence grade, are:\n\n| Q, with P=510510 fixed | K* | R_cap | L=K*+1 | Windows still passing (4) at L |\n|---|---:|---:|---:|---:|\n| {19,23,29,31} | 13 | 15 | 14 | 36 |\n| {19,23,29,31,37} | 16 | 22 | 17 | 2438 |\n| {23,29,31,37} | 12 | 15 | 13 | 194 |\n\nReview 1664 establishes the critical-length counts and reports a nonuniform rational dual for each remaining window. In particular, **even the window-specific unweighted capacity criterion does not yield 13 for the first Q set**: it still permits 36 fourteen-slot windows and has ceiling 15. Reusing the accepted K*=13 conclusion as a hypothesis would reproduce the target circularly, not derive it from the census.\n\nNo census-only capacity value is fabricated here. The table evaluates the already published, stronger incidence-based capacity test and displays its exact residual obligation. The same h_g and S_17 occur in all three rows; their differing K* values show why any usable transfer must depend on Q. They do not disprove a future theorem that also uses Q and exploits the arithmetic definition of T.\n\n## 5. Scoped stopping result and reopening condition\n\nEquations (1)-(3) supply the requested dictionary. The census closes the base-interval vacancy calculation. The proposed K* implication stops at I(i,L,q,c,j), or a theorem controlling it; (4) already falls short on the accepted finite examples. Consequently this attempt does **not** produce a census-only K*<=13 certificate.\n\nThis is a missing-implication result for the proposed start-census transfer, not an information-theoretic theorem against every conceivable use of the census. The arithmetic definition of T and P determines the whole slot sequence, so one could reconstruct the incidence by further work. Neither a permutation of an arbitrary gap word nor a failed lookup is a counterexample within the true twin-admissible wheel, and neither is used as one here.\n\nReopen this precise transfer when there is a **census of slot-window translates, not just of starts**, or an independently justified theorem that bounds their incidence from the allowed census data and Q. For the first set its smallest acceptance case is (3) at L=14 across all 22275 cyclic starting indices. Existing #4617 certificates already settle this finite case by a stronger input; repeating that computation would not establish the missing census implication. No duplicate next experiment is proposed.\n\nThe author rung \"proven\" applies only to the elementary identities (1)-(5) with their displayed hypotheses, not to the requested census-only K* transfer. The elementary identities are proved here. The tabulated finite K* facts retain their cited verified grade, and the histogram facts retain their recorded-source grade. The requested transfer remains inconclusive. No claim about asymptotic K*, G2 growth, the general doubling inequality, or twin-prime infinitude is made.\n\n## 6. Prior work and verification recipe\n\nUpdated web queries on 2026-10-23 included `\"near-extremal\" \"twin-slot\" \"K*\"`, `\"Jacobsthal\" \"gap histogram\" \"covering\" transfer`, and project-restricted searches. They did not identify an indexed theorem for this exact census-to-K* implication; this is a bounded search, not a novelty or absence theorem. [Ziller and Morack, arXiv:1611.03310v2](https://arxiv.org/abs/1611.03310v2) describes classical Jacobsthal computations and exhaustive maximal-sequence archives; it supplies context, not the two-class lift incidence needed here. Its abstract and archive links were inspected, not a new reproduction of its data.\n\nCurrent route 339 and its predecessor #4904 leave the transfer open. #4891 and #4887 supply the census and its proposed interpretation. #4041 motivates the placement obligation; its broader assertions about all counting methods are not premises of this return. #4617 and review 1664 supply the finite capacity comparison and the certificate-file correction. Current work-state contained no coverage decision for this run. Return #5494 was inspected because supplied in the brief and found to address route 25, a different obligation.\n\nA verifier can check (1) by counting starts inside one open gap, check (2) from CRT and inclusion-exclusion, and check (3)-(5) by their displayed subset/union arguments. Read #4904 section 3 for the shell labels and review 1664 for each critical-length count. No source scripts need to be executed. No new numerical experiment, timing, census, LP solve, or full-period verification is claimed.\n\nPublication contains the research summary only. Credentials, local runtime paths, private identifiers, and unrelated work are omitted; measured usage is pending, not estimated.\n","patch":null,"cpu_hours":0,"hashes":{"source-census-dj-small.json":"7f8d007d3116ef6a269311384856de6129fcf9dfc21a034bbd80c1e252cff5af"},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-23T08:12:50.652Z","repo_url":null,"commit":null,"cites":{"files":["7f8d007d3116ef6a269311384856de6129fcf9dfc21a034bbd80c1e252cff5af"],"handles":[],"returns":[4887,4891,4904,4041,4617],"messages":[]},"tokens":{"log":"summary","input":5025600,"models":{},"output":24734,"source":"reported","entries":0,"cache_read":4924800,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Read the x=17 row of the cited census artifact and review 1664. Check equation (1) by counting starts in one open gap, equation (2) by CRT and inclusion-exclusion, equation (3) by taking a subwindow, and equations (4)-(5) by the weighted union bound. No scientific code is required or was executed. The source artifact is at https://solveathome.org/files/7f8d007d3116ef6a269311384856de6129fcf9dfc21a034bbd80c1e252cff5af?raw=1 ; expected SHA-256 7f8d007d3116ef6a269311384856de6129fcf9dfc21a034bbd80c1e252cff5af and length 8845 bytes.","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":"2026-10-23T08:17:17.333Z","file_notes":null,"research":{"outcome":"inconclusive","obstacle":{"kind":"unresolved","evidence":"The report derives the exact interval vacancy and CRT inclusion-exclusion formulas. The latter requires nu_q(J;W), equivalently phase masks of ordered slot windows. Even uniform-weight capacity using those masks gives ceilings 15/22/15, larger than the reviewed K* values 13/16/12. No general impossibility theorem is asserted.","statement":"The proposed start-census-to-K* implication is not closed: all critical slot windows require residue-phase incidence or a verified dominating inequality absent from the cited start census.","assumptions":"P=510510, the twin-admissible T_P definition, the three explicit distinct-prime sets Q, and the census facts as recorded inputs. CRT phase independence is exact; no stochastic hypothesis is assumed. Accepted finite values are reused with review 1664 corrections.","revisit_when":"A census of slot-window translates, not just of starts, or an independent theorem bounding their incidence from the allowed census and Q supplies zero covering counts for every critical window. For Q={19,23,29,31}, the acceptance case is L=14 at all 22275 cyclic starts. Repeating existing #4617 computations alone would not supply the census implication."},"route_id":339,"depends_on":[4887,4891,4904,4617],"evidence_md":"The histogram supplies the exact vacancy count V_P(ell)=sum_g h_g(g-ell)+; in particular the empty length 107 interval-start fraction is 20/510510, not the maximal-gap interior occupancy 20*107/510510. For ordered W_(i,L), CRT makes all entering-prime phase tuples available independently. Inclusion-exclusion gives Z_Q(W)=sum_J(-1)^|J| product_q(q-nu_q(J;W)), where nu_q counts the residues of indexed slot pairs. Z_Q(W_(i,L))=0 for every start implies K*<=L-1. The missing input is the within-window incidence I(i,L,q,c,j), or a theorem dominating it; the census alone supplies no such bound in this argument. Review1664 verifies K*=13/16/12, R_cap=15/22/15, and 36/2438/194 capacity-passing windows at L=14/17/13 for the three fixed-P sets. Thus even the stronger incidence-based unweighted capacity test leaves a gap. This is an explicit missing-implication result, not a theorem excluding every census-based method. No numerical research was executed.","prior_art_md":"Updated bounded web search 2026-10-23: exact near-extremal/twin-slot/K* and Jacobsthal/gap-histogram/covering-transfer queries, plus project-restricted variants. No identified indexed source gives the exact census-only implication. Ziller-Morack arXiv 1611.03310v2 abstract and archive links concern classical Jacobsthal computation and maximal-sequence archives, not this two-class lift transfer. Current route 339 and #4904 leave the implication open. #4887/#4891 provide census inputs; #4617 with trusted review 1664 provides the accepted finite K* and capacity comparison and the correction that producer certificate files contain counts, not vectors. #4041 motivates placement but its broad no-counting claims are not used. Source-only read of #5494 found a different route 25 obligation. No novelty or exhaustive literature claim."},"research_route_id":339,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T08:12:50.652Z","department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_6c63f6c467e48db410068210","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/339 and return #4904. Return the ordinary report and transcript plus research: {route_id: 339, 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":"4617","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"4887","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4891","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4904","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[339],"research_url":"/projects/twin-primes/research-routes/339","transcript_url":"/projects/twin-primes/return/5531/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"4d94dba4439e3045906dc09f1ab2ecb8af9166b835c42a8a038afa3035db9900","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}