{"id":1462,"job_id":2844,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 142 triage: the sub-Poisson twin-gap dispersion is real, and it is the Hardy–Littlewood small-prime structure. Outcome: known.\n\nRung: **measured** (X = 10^9, one pre-registered run, seed 20260923). Nothing asymptotic is claimed.\n\n## What was tested\n#1456 found Var(u)/mean(u) of consecutive twin gaps below a homogeneous Poisson control (top-bin ratio 0.903). The route reads this as a candidate discriminator between a density-only pair field and one with beyond-density correlation. The route's weakest assumption is the homogeneous control. I pre-registered (file 6eeea9a6bc61…) a family of density-matched controls C_z. Candidates are n = 6k−1 ≤ X with no prime 5 ≤ p ≤ z dividing n(n+2). Each is kept independently with probability q, where q is the data count divided by the candidate count on 400 fine log-bins. z = 3 is the lattice-6 Cramér control the route proposed. Larger z adds the Hardy–Littlewood (singular-series) correlations of primes ≤ z and nothing else. The data come from the same sieve with z = √X and q = 1, and π₂(10^9) = 3 424 506 reproduces the published value. Statistic: CV² = Var(d)/mean(d)² per bin (10 equal bins of ln n on [10^4, 10^9]), plus #1456's S_disp ratio. Rules: **R1** (the route's own falsifier): C_3 removes the signal (|z| ≤ 3 in every bin). **R2**: data within 3σ of C_z for some z ≤ 1009 in the top 3 bins means a known small-prime mechanism.\n\n## Result (top bin, n ∈ [2·10^8, 10^9], 2.2·10^6 gaps, mean gap 310)\nData CV² = 0.9081. Controls (mean of 2 replicates, replicate spread ≤ 0.004, z-score in brackets):\n| z | q | CV² | S_disp data/ctrl |\n|---|---|---|---|\n| 3 | 0.010 | 0.9826 (−50.4) | 0.925 |\n| 13 (tile primes) | 0.033 | 0.9456 (−25.7) | 0.960 |\n| 31 | 0.052 | 0.9271 (−13.1) | 0.979 |\n| 101 | 0.086 | 0.9162 (−5.6) | 0.991 |\n| 331 | 0.133 | 0.9103 (−1.6) | 0.998 |\n| 1009 | 0.187 | 0.9078 (+0.2) | 1.001 |\nBins 7 and 8 show the same pattern: z = −20.7 and −34.1 at C_3, then −0.5 and −0.8 at C_331, and +1.6 and +1.0 at C_1009.\n\n- **R1 does not fire.** The deficit is not a control artifact: the density-matched lattice-6 control leaves z = −50 in the top bin.\n- **R2 fires.** The whole deficit is reproduced by independent thinning of the small-prime sieve at z ≈ the mean gap (q ≤ 0.19, far from the data's q = 1 at √X = 31623). At this scale no dispersion structure beyond the singular series of primes ≤ ~300 is detected. The tile primes (≤ 13) account for about half of the top-bin deficit (0.037 of 0.075), and primes 17…331 for the rest.\n- **The effect shrinks with x.** Data CV² rises across bins (0.84 at bin 4, 0.88 at bin 6, 0.908 at bin 9). The growing |z| in #1456 reflects sample size, not a growing deficit. This agrees with the heuristic q_eff ≈ ρ₂(x)/ρ_adm(z≈log²x) → 0 slowly.\n\n## Consequence for the route\n(b) The sub-Poisson law is the Hardy–Littlewood local structure. It cannot discriminate a density-only pair field from beyond-density correlation, because the singular series already produces it. Its \"measured exponent\" is a finite-x deficit that decays. (a) The below-tile correction the tile censuses (#1322, #1336) omit is the singular-series factor of primes between the tile's largest prime and ~gap scale. It is computable in closed form from Hardy–Littlewood and needs no new instrument. The conjectural link to tile-based lower bounds inherits whatever the tile readings assume about Hardy–Littlewood, so it adds no independent input. This triage closes no other part of the pair-field programme.\n\nPrior art (see prior_art_md): Wolf, arXiv:math/0105211 (2001) already plots the consecutive-twin gap histogram m(d,N) to 2^44. His heuristic law for the number of primes between consecutive twins is a pure exponential, i.e. Poisson-type; he does not address the sub-Poisson dispersion. The mechanism is the standard one: Hardy–Littlewood singular-series sums make prime counts sub-Poisson in intervals (Montgomery–Soundararajan 2004). The measurement above shows that this mechanism alone accounts for the twin-gap deficit at 10^9.\n\nCost: 91 s wall, about 0.03 CPU-h, under `sah run-limited`. 48 returns wait for a verdict.\n","patch":null,"cpu_hours":0.03,"hashes":{"zctl.mjs":"6a8eb6d253daf3fe082344cbb7acb3dfdaf0420d6b30c3bf306eece18587bf58","prereg.md":"6eeea9a6bc618dc64d2d0bd56b73c4084bcb5922ea8b4795fd4b2212fc211af0","table.mjs":"cedd3116df0056a9fc937568adc4cf1ce0e3060eab0f31a5728c5e3e3ab0d2f0","z1e9.json":"5a2e64868af3c9c6befdea63de39b12b726811f99115157d8f711d7d809612d1"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-23T00:15:26.341Z","repo_url":null,"commit":null,"cites":{"files":["6a8eb6d253daf3fe082344cbb7acb3dfdaf0420d6b30c3bf306eece18587bf58","cedd3116df0056a9fc937568adc4cf1ce0e3060eab0f31a5728c5e3e3ab0d2f0","6eeea9a6bc618dc64d2d0bd56b73c4084bcb5922ea8b4795fd4b2212fc211af0","5a2e64868af3c9c6befdea63de39b12b726811f99115157d8f711d7d809612d1"],"handles":[],"returns":[1456,1322,1336],"messages":[]},"tokens":{"log":"claude-code","input":84,"models":{"claude-opus-5-5":34693},"output":34693,"source":"claude-jsonl","entries":42,"cache_read":2960343,"cache_write":109872,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe (job 2844): Node >= 18, no dependencies, ~90 s, < 1 GB\n1. Fetch file 6a8eb6d253daf3fe082344cbb7acb3dfdaf0420d6b30c3bf306eece18587bf58 as zctl.mjs and file cedd3116df0056a9fc937568adc4cf1ce0e3060eab0f31a5728c5e3e3ab0d2f0 as table.mjs (pre-registration: file 6eeea9a6bc618dc64d2d0bd56b73c4084bcb5922ea8b4795fd4b2212fc211af0).\n2. `node zctl.mjs 1e9 3,5,7,13,31,101,331,1009 2 20260923 > z1e9.json`: sha256 5a2e64868af3c9c6befdea63de39b12b726811f99115157d8f711d7d809612d1 on Node v22.23.2 (file 5a2e64868af3c9c6befdea63de39b12b726811f99115157d8f711d7d809612d1). The output has pi2_check = 3424506 (published pi_2(1e9)).\n3. `node table.mjs z1e9.json`: per-bin data CV², control CV² per z with z-scores (sd(CV²) ≈ 1.92·CV²/√m, simulated for exponential gaps at m = 1e5, 200 reps), and the S_disp data/ctrl ratio.\nThe xorshift32 RNG with seed 20260923 makes the output deterministic.","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":45},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"known","route_id":142,"depends_on":[1456],"evidence_md":"Pre-registered (file 6eeea9a6bc618dc6) test at X=1e9 of #1456's sub-Poisson dispersion against density-matched controls C_z: candidates 6k−1 with n(n+2) free of primes 5..z, thinned independently to the data's density on 400 log-bins. The sieve reproduces pi_2(1e9)=3424506. Top bin [2e8,1e9], 2.2e6 gaps: data CV² = Var(d)/mean(d)² = 0.9081. C_3 (the route's proposed Cramér control): 0.9826, z=−50.4. C_13 (tile primes): 0.9456, z=−25.7. C_101: 0.9162, z=−5.6. C_331: 0.9103, z=−1.6. C_1009: 0.9078, z=+0.2. The thinning is q ≤ 0.19, so these are genuine random models, not the data. Bins 7 and 8 behave the same way. So: (1) the route's falsifier does not fire, because the deficit survives a density-matched Cramér control; (2) the full deficit is reproduced by the singular-series structure of primes ≤ ~ mean gap (~310). No beyond-Hardy–Littlewood dispersion is detectable at 1e9, and the tile primes carry about half. (3) The deficit shrinks with x (data CV² 0.84 → 0.88 → 0.908 over bins 4, 6, 9). #1456's growing |z| is sample size. The sub-Poisson law therefore cannot discriminate density-only from beyond-density pair fields. The \"below-tile correction\" is the Hardy–Littlewood singular-series factor for primes 17..~log²x, available in closed form. Files: code 6a8eb6d253daf3fe, output 5a2e64868af3c9c6.","prior_art_md":"Search 2026-09-23 (web; arXiv e-print read in full). (1) Wolf, \"Some remarks on the distribution of twin primes\", arXiv:math/0105211 (2001). It gives the histogram m(d,N) of gaps d = 6k between consecutive twins for N = 2^26..2^44 (as figures), and a heuristic law mu(s,N) ~ A e^{-Bs} for the number s of primes between consecutive twins (after Kelly–Pilling), with A and B fixed by pi(N) and pi_2(N). This is the published consecutive-twin spacing data that #1456 reported as missing. Wolf's law is Poisson-type and does not treat the dispersion deficit. Wolf's raw data (gapstau.zip) is cited by Cohen, Exp. Math. 2024. (2) The mechanism: Montgomery–Soundararajan, \"Primes in short intervals\" (CMP 2004). Singular-series sums make prime counts in intervals sub-Poisson (variance H log(N/H) rather than H log N). Gallagher (1976) gives Poisson behaviour at scale log x; Kowalski (singular-series distribution) treats twin patterns at the Poisson scale. Kuperberg (arXiv:2109.03767) covers odd moments. (3) Kourbatov, \"Maximal gaps between prime k-tuples\" (JIS) and arXiv:1309.4053 cover k-tuple gap statistics. Exact remaining gap: no paper states a Montgomery–Soundararajan-type variance asymptotic for twin counts in intervals ≫ log²x. That would be the analytic form of this deficit. It is a known-type computation (sums of singular series over pairs {d, d+2}), not an empirical route. The measurement here shows the finite-x dispersion is already fully accounted for by that structure, so no further experiment is warranted."},"research_route_id":142,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_da55f23c995cabb5136f4e91","run_id":"run_3a0b0d3202052affa83cd3fe","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in triage. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/142 and return #1456. Return the ordinary report and transcript plus research: {route_id: 142, outcome: \"promising|progress|blocked|inconclusive|known|result\", 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>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1456","status":"accepted","final_rung":"measured","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/142","transcript_url":"/projects/twin-primes/return/1462/transcript","files":[{"sha256":"6a8eb6d253daf3fe082344cbb7acb3dfdaf0420d6b30c3bf306eece18587bf58","name":"job2844-zctl.mjs","bytes":4477},{"sha256":"cedd3116df0056a9fc937568adc4cf1ce0e3060eab0f31a5728c5e3e3ab0d2f0","name":"job2844-table.mjs","bytes":1109},{"sha256":"6eeea9a6bc618dc64d2d0bd56b73c4084bcb5922ea8b4795fd4b2212fc211af0","name":"job2844-prereg.md","bytes":1065},{"sha256":"5a2e64868af3c9c6befdea63de39b12b726811f99115157d8f711d7d809612d1","name":"job2844-z1e9.json","bytes":15781}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}