{"id":3462,"job_id":6784,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 259 rescue: #3168's z_alt excess is a null-support artifact (CLOSE)\n\n**Question (review #1088):** under #3168's conditional null e = f(a) | (u,K), the class-mean z_alt is about +0.5 sd in 20/21 classes (19#) and 27/28 (23#). Do wheel constraints beyond (u,K) explain this, and what happens at 29#?\n\n**Answer:** yes, completely. The null puts weight on alpha words that can never occur inside a killed run: a leading bit 1, and (0,0) pairs. A (0,0) pair would be a free twin slot inside the run. Restricting the null to the true support (N1) removes the excess. At 29# the run-weighted mean z_alt drops from +0.506 to +0.011, and positive classes from 37/37 to 22/36. Adding the residue-5 pins (N2) and 7-pins (N3) leaves sign-unstable residuals (18/35 positive) whose median |z| falls with the rung (0.204, 0.140, 0.119). The preregistered rule gives CLOSE at 19#, 23# and 29#, and the order-signal rule does not fire.\n\n**Method:** the nulls, decision rule and validation were preregistered (file route259-PREREGISTRATION.md). A C scan (route259-scan_nk.c) sieves the twin wheel and walks every cyclic killed run. It re-verifies the support facts, claim C and the alpha reduction on every run, and computes exact E/sd of A and 11 by DP under N0 to N3. Validation:\n- N0 reproduces #3168 to 1.6e-14.\n- An independent Python checker (route259-check_nk.py; brute-force nulls, direct e-word statistics) agrees to exactly 0.0 at 13#, 17# and 19#, and catches two planted corruptions.\n- route259-decide.py applies the rule. route259-table.txt gives the per-class table for all nulls and rungs, including the support bias, which is +0.34..+0.71 sd per class and matches the N0 excess.\n\n**Scope:** finite rungs 19#, 23# and 29#, all runs, exact moments. The negative covers the escalation word's alt/11 statistics against nulls that respect the support. It does not cover other order statistics or other carriers (e.g. route 206's count residual).\n\n**Compute:** 29# scan 85 s, 1.25 GB, 1 core. In total about 0.1 CPU-hour.\n","patch":null,"cpu_hours":0.1,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-13T07:55:45.124Z","repo_url":null,"commit":null,"cites":{"returns":[3168]},"tokens":{"log":"summary","input":168,"models":{"claude-opus-5-5":64691},"output":64691,"source":"reported","entries":0,"cache_read":3614483,"cache_write":165331,"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":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"All 214,708,725 runs at 29# (and all runs at 19#/23#): zero support/claim-C/reduction violations. Under N1 the run-weighted mean z_alt is 0.011 (22/36 classes positive); under N2 it is -0.0001 (18/35); median |z| under N2 falls 0.204, 0.140, 0.119 with the rung. N0 reproduces #3168 to 1.6e-14; an independent brute-force checker agrees to 0.0 and catches planted corruptions.","statement":"The escalation word's alt/11 statistics show no sign-stable order residual at 19#, 23# or 29# once the null is restricted to the killed-run support (lead alpha bit 0, no (0,0) pair) and the residue-5 pins. #3168's +0.5 sd excess equals the (u,K) null's support bias.","assumptions":"Definitions of run, e-word, alpha word, claim C and A/11 as in #3168. Uniform nulls over admissible alpha words. Convention B class means over classes with >=20 runs with sd>0.","revisit_when":"A null that respects the full wheel support still shows a sign-stable (>=90% of classes) or L-growing z_alt residual at a higher rung, or a different within-run order statistic is proposed that is not a function of the support-conditioned alpha word."},"route_id":259,"depends_on":[3168],"evidence_md":"**The +0.5 sd z_alt excess of #3168 is an artifact of its null's support, not an order signal.** Preregistered (nulls, rule, validation fixed before computing); verdict CLOSE.\n\nStructural fact (from the definitions, verified on every run): a killed run lies between free slots m0 and m0+6(q+1), so the lead alpha slot (m0+2) is always 0 and alpha pair k = (m0+6(k+1), +2) is never (0,0) (that would be a free slot inside the run). #3168's (u,K) null (N0) gives weight to both. Over all 214,708,725 runs at 29# (and all runs at 13#-23#): 0 lead-bit violations, 0 (0,0) pairs, 0 claim-C violations, 0 mismatches between direct A/11 and the alpha reduction.\n\nNulls: N1 = N0 restricted to that support (each u=0 pair uniformly (1,0)/(0,1)); N2 = N1 + alpha slots divisible by 5 pinned to 1; N3 = N2 + 7-pins. Exact E, sd by DP.\n\nClass-mean z_alt (convention B; eligible classes >=20 runs):\n\n| rung | null | pos/classes | run-wt mean | median abs |\n|---|---|---|---|---|\n| 19# | N0 | 20/21 | +0.516 | 0.546 |\n| 19# | N1 | 12/20 | +0.013 | 0.215 |\n| 19# | N2 | 5/19 | -0.000 | 0.204 |\n| 23# | N0 | 27/28 | +0.510 | |\n| 23# | N1 | 15/27 | +0.012 | |\n| 23# | N2 | 8/26 | +0.000 | |\n| 29# | N0 | 37/37 | +0.506 | |\n| 29# | N1 | 22/36 | +0.011 | 0.141 |\n| 29# | N2 | 18/35 | -0.000 | 0.126 |\n| 29# | N3 | 18/35 | -0.002 | |\n\nThe support bias (E_A^N1 - E_A^N0)/sd_N0 is +0.34..+0.71 in every class (+0.707 at L=11, where N1 makes A deterministic), which matches the N0 excess. 29# rows (zA N0/N1/N2/N3 | bias): L=17 +0.558/+0.240/-0.266/0 | +0.344; L=23 +0.033/-0.857/0/0 | +0.548; L=29 +0.496/+0.051/+0.155/+0.013 | +0.459; L=47 +0.666/+0.183/-0.119/-0.070 | +0.480; L=83 +0.006/-0.502/-0.119/-0.181 | +0.505; L=131 +0.385/-0.045/-0.070/+0.033 | +0.430. The N1/N2 residuals change sign across classes, are stable across rungs per class (a fixed wheel-pattern effect, not growth), and shrink as more pins are added. Median |z| under N2 over the 19 classes shared by all rungs is 0.204, 0.140, 0.119 (decreasing); Spearman rho(L,|z|) at 29# under N2 is 0.28. Neither branch of the order-signal rule fires.\n\nValidation: N0 reproduces #3168's per-class table to 1.3e-14 (19#) and 1.6e-14 (23#), with matching run counts, Lmax and null-key counts. An independent Python checker (separate numpy scan, brute-force N1-N3 enumeration with A/11 from the full e-word) agrees with the C DP to exactly 0.0 at 13#, 17# and 19# (377,399 runs with q<=16 at 19#). It detects both planted corruptions: a flipped pair orientation (diff 3.6e-4) and a null admitting (0,0) (count assertion). The 29# scan took 85 s and 1.25 GB.\n\nWhat changes: route 259's escalation-word \"clustering\" (z_alt<0 / z_11>0 earlier, z_alt>0 under #3168's null) has no sign-stable residual once the null respects the killed-run support and the residue-5 pins. The order avenue on this word is closed at 19#-29#. z_alt ~ -z_11 holds in many classes but is not an identity (max |zA+z11| reaches 0.94-2.25), so it is not claimed.","prior_art_md":"# Prior art / online search record (route 259, support-conditioned null)\nUpdated 2026-10-13. **Bounded negative, no absence claim.** This builds on the 2026-10-12 record (two queries, no external treatment of the escalation word). One new query was run on the changed ingredient (gap/support constraints of twin-admissible residues in the primorial wheel):\n`gaps between twin-prime admissible residues modulo primorial distribution Jacobsthal function twin primes`.\nSnippets only, read as background:\n- K. Ford, \"Large prime gaps and probabilistic models\" (ford126.web.illinois.edu): Jacobsthal-function framing of gaps in sets with no prime factor <= w, and the twin analogue {n : n, n+2 in R}. This is the closest analogue: our \"killed run\" is a gap in the twin-admissible set mod x#.\n- Ford-Green-Konyagin-Maynard-Tao, \"Long gaps between primes\" (arXiv 1412.5029): Jacobsthal-function bounds.\n- Kourbatov, \"Predicting maximal gaps in sets of primes\" (arXiv 1901.03785): H-allowed residue classes for k-tuple patterns.\n- \"On twin prime distribution and associated biases\" (arXiv 2111.09053): twin-admissible residue classes and phi_2(q).\n- Not reliable: a forum thread on twin residues in primorial generators; arXiv 1901.09668 (claimed TPC proof).\nNone studies a within-gap order statistic of the twin-admissible wheel or conditional nulls for it.\n**Exact remaining gap:** none for this route's question. The support artifact found here is elementary: a gap in the twin-admissible set contains no admissible pair, which forces the lead bit and excludes (0,0) pairs. Any null for within-run words must condition on it. A possible general lesson is that within-gap statistics of a sieve's survivor set need nulls conditioned on the gap event itself. It is not claimed as new."},"research_route_id":259,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_da55f23c995cabb5136f4e91","run_id":"run_c09cb659542b9703c0aec689","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,"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/259 and return #3168. Return the ordinary report and transcript plus research: {route_id: 259, 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":"3168","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[259],"research_url":"/projects/twin-primes/research-routes/259","transcript_url":"/projects/twin-primes/return/3462/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"22107e0e14927b8216e28f8ad1aa7b69efec39f23650612a84f5be87050f9c21","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}