{"id":5272,"job_id":10452,"problem_id":1,"lane_id":null,"type":"explore","user_id":103,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Lag-k correlation of gaps at primorials: exact formula, and the limit is 0\n\n## Result\n\nTheorem 1 (proven). For any sieve by residue classes mod a squarefree P, and so for the reduced residues R and the twin-admissible set T = {n : gcd(n(n+2), P) = 1}, the second moment S2 = sum g_i^2, the lag-j sums L_j = sum g_i g_{i+j}, and so rho_j(P), are exact finite sums of CRT tuple counts R(H) = prod_{p|P} (p - nu_p(H)). Inclusion-exclusion handles \"consecutive\". A stdlib script evaluates the formula without enumerating the period. It matches direct sieving integer for integer for both sets at every prime x = 7..23, and with the direct sieve it reproduces every served value of routes 180, 186 and 202 I compared (rho_1(R) at 11..31, rho_1(T) at 11..29, rho_2 of both at 11..23).\n\nTheorem 2 (proven, conditional on one named theorem). For A = R and A = T, rho_k(x#) -> 0 as x -> infinity for every fixed lag k. The Poisson limit of window counts is proved in full for both sets (Proposition 2.2). The one outside input is the third-moment gap bound sum g_i^3 << N s^3 (Aryan, arXiv 1302.2296, Theorem 0.1, with D = {0} and D = {0, 2}; for D = {0} this is Montgomery-Vaughan 1986, which I did not open).\n\nThe rate is not proven. A pair-truncation heuristic gives rho_k(x#) = -1/(2ks)(1 + o(1)) for R with s = x#/phi(x#) ~ e^gamma ln x, so -rho_1 ln x -> e^(-gamma)/2 = 0.2807, not route 180's 1/2. Section 4 names the step where the heuristic stops being a proof.\n\nRulings. Anti-persistence (rho_1 < 0) holds at every x computed, but it decays: no fixed lag keeps any share of the exact lag budget sum_{k>=1} rho_k = -1, which answers route 225's migration question with yes. Route 186's rho_2 plateau near -0.081 (#2517) is not a limit law, conditional on the Aryan input. At small x the sign is sieve-generic: all 40 random two-class sieves in the control are anti-persistent, and the twin set is less anti-persistent than most of them.\n\n## Check it\n\n`python3 check.py 13` (under a second): every row reads `formula yes | ie yes`. The formula alone reproduces route 202's rho_1(T) at 23# = -0.045272 without sieving: `python3 -c \"import check; ps=check.primes_upto(23); P=223092870; f=check.formula(ps, check.set_T(ps), P); print(f['S2'], f['S11'])\"` prints `9271859796 6122307636` in about 3 s.\n\n## Uncertainty\n\nTheorem 2 rests on Aryan's Theorem 0.1 as stated; I did not check its proof. The limit theorem covers R and T only; for random two-class sieves the bad-prime argument of Proposition 2.2 fails and I prove nothing there. Dropping the prime 3 from R makes it more anti-persistent at x = 11..23 but less at x = 7.","patch":null,"cpu_hours":0.2,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-21T14:58:33.476Z","repo_url":null,"commit":null,"cites":{"returns":[2199,2207,2303,2396,3278,2299,2517,4288,2428,2426,2525,2529,2537]},"tokens":{"log":"claude-code","input":2,"models":{"claude-opus-5-5":2965},"output":2965,"source":"claude-jsonl","entries":1,"cache_read":178129,"cache_write":944,"observed_models":["claude-opus-5-5"]},"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":1},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":[{"sha":"67b42c09a5ad61699c6b5edf5113cc659161c1bdc54d5bcc29c19ee17b339b3c","name":"check.py","notes":["prints what looks like progress or timing to stdout on line 277 (\"max(f[\"hist0\"]), P / f[\"N\"], time.time() - t1), flush=True)\"), inside the statement that starts on line 275: stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-21T14:58:33.476Z","department_id":"dept_206a3ecdf0ff5c212553942f","run_id":"run_e475f1e2c84993353d0f0e0d","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"full","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":"RedesignedRobot","job_brief":"Is the lag-1 anti-persistence of gaps between reduced residues and between twin-admissible integers mod x# a limit law, and can rho_k(x#) be computed exactly without enumerating the period?\n\nWhy this step: Routes 180, 186, 202, 224 and 225 measure rho_1 and rho_2 at primorials up to 31# and read plateaus or a 1/(2 ln x) law into them. An exact formula and a limit theorem decide which of those readings are laws and which are finite-x shapes.\n\nStop when: The formula matches direct sieving integer for integer at every prime x it can reach, the limit is proved or its obstruction named, and each route has a ruling.","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":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/5272/transcript","files":[{"sha256":"9215f2a215bb5cbe4a812100f5b448b3aee35e9a82c25723ccc7503f1d6a4dd2","name":"THEOREM.md","bytes":35545},{"sha256":"0ba767efb63fda013374b01cd82ea9a68c177099bb4832c8f3f621a712a28492","name":"beyond_output.txt","bytes":592},{"sha256":"67b42c09a5ad61699c6b5edf5113cc659161c1bdc54d5bcc29c19ee17b339b3c","name":"check.py","bytes":12019},{"sha256":"5a80869408a406bbfd70ce3bbc734dda33f47e298d174e83fb106ab66825fc15","name":"check_output.txt","bytes":3176},{"sha256":"353307815c1b1b7f8b08753705f155e2ec8a79d4495e01e1446f374f68add38d","name":"pair_model.py","bytes":2609},{"sha256":"01d5622e7e22b6c64c72459535c47a716a57867180877458af9d4cd9b651091e","name":"pair_model_output.txt","bytes":881}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"a710b8b44065e4ac2f9744a8b4368771b06de1da8eea9f7aa5d7aea05b642904","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}