{"id":2393,"job_id":5099,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# run-2026-10-06-aq — job #5099, route 198 first look\n\n**Outcome: `progress`.** The short-window count under-dispersion of the residue sets is real and\nq-growing at a fixed window *fraction*, but it **vanishes at fixed window *length*** (R→1, local\nfactor→1) and, at the polylog window lengths that matter for the Jacobsthal/G2 exponent, the\nsecond-moment union-bound threshold is missed by many orders of magnitude. Route 198's proposed\ntransfer to a G2 exponent is therefore **obstructed as stated**.\n\n## What was asked\n\nRoute 198 proposes the short-window count variance `V_q(L)` of `A_q={a:gcd(a(a+2),q)=1}` and\n`B_q={a:gcd(a,q)=1}` (q = x#) as a new second-moment lever for the Jacobsthal/G2 maximal gap.\nReturn #2386 measured strong under-dispersion (R = V/V_null < 1, q-growing) at fixed `L/q` up to\n19# and left the transfer to the G2 exponent as the route's central uncertainty. The assigned step:\nextend to 23#/29#, scan `L/q`, test L-independence, fit local factors, and attempt the transfer.\n\n## Method (frozen in `PREREGISTRATION.md` before the run)\n\nExact finite computation, no Monte-Carlo. Same functional as #2386:\n`P_q(a)=Π_{p|q} P_p(a c_p mod p)`, `c_p=(q/p)^{-1} mod p`; `K_a=(sin(pi a L/q)/sin(pi a/q))²`;\n`V_q(L)=(1/q²)Σ_{a=1}^{q-1} P_q(a)K_a`; hypergeometric null `V_null=Lρ(1-ρ)(q-L)/(q-1)`; `R=V/V_null`.\nWe extend to `q=23#` and add three L-families: **S** small fixed (`L=4,16,64,256`), **G** the\nG2-natural polylog scaling (`L=⌈ln q⌉, ⌈(ln q)²⌉, 2⌈(ln q)²⌉`), and **F** fractions (`q/8,q/4,q/2,3q/4`).\nA union bound over the q windows gives `q·P(empty window) ≤ q·V/mean² = R/thr(q,L)` with\n`thr(q,L)=(Lρ)²/(q V_null)`; the second moment can exclude empty windows **iff `R/thr<1`**.\n\n## Results\n\nExact ladder, both families (`compute_aq.json`, `compute_aq_23.json`):\n\n| q | R_A(q/2) | R_B(q/2) | R_A(L=4) | R_B(L=4) | R_A(L=16) | R_B(L=16) |\n|---|---|---|---|---|---|---|\n| 7# | 0.57775 | 0.098295 | 0.7804 | 0.5237 | 0.43369 | 0.23878 |\n| 11# | 0.12666 | 0.017479 | 0.8149 | 0.5689 | 0.49900 | 0.28033 |\n| 13# | 0.013701 | 0.0026369 | 0.8440 | 0.6071 | 0.56583 | 0.32716 |\n| 17# | 0.0033791 | 0.00030596 | 0.8631 | 0.6341 | 0.61362 | 0.36485 |\n| 19# | 0.00073091 | 3.1838e-5 | 0.8781 | 0.6564 | 0.65225 | 0.39745 |\n| 23# | 6.0876e-5 | 2.7439e-6 | 0.8891 | 0.6738 | 0.68145 | 0.42389 |\n\n1. **F-family (reproduces and extends #2386).** `R_B(q/2)` falls monotonically `0.0983 → 2.74e-6`\n   over 7#..23# (≈`q^{-0.76}`); `R(q/2)` is flat over `L/q ∈ [1/4,1/2]` and rises at `L/q=1/8`.\n2. **S-family — the falsifier fires.** For fixed `L`, `R_q(L)` **increases toward 1** as q grows:\n   `R_A(4)=0.780→0.889`, `R_B(4)=0.524→0.674`; the per-prime **local factor →1** (`R_A(4)` factors\n   1.044,1.036,1.023,1.017; `R_B(4)`: 1.086,1.067,1.045,1.035). The under-dispersion **disappears**\n   in the short-window limit `L/q→0` at fixed L.\n3. **G-family (G2-relevant polylog L).** `R_q(L)` stays roughly flat (`R_A≈0.2–0.4`, `R_B≈0.4`),\n   and `R/thr > 1` at every probe (`min=2.30`, up to `~10^9`) — the union bound cannot exclude a\n   single empty window at these lengths.\n4. **Where the bound does hold.** `R/thr<1` only for `L/q ≳ 1/8` (large windows, `L=Ω(q)`), e.g.\n   `R_B(q/2)/thr = 0.33, 0.067, 0.011, 1.4e-3, 1.5e-4, 1.4e-5` at 7#..23#. Large windows are not\n   where a maximal gap lives, and any `o(q)` minimum-gap bound needs `L=o(q)`.\n\n**Transfer verdict.** The second-moment functional cannot certify any `L=o(q)` empty-window\nexclusion: at the G2-natural scale `L=polylog(q)` the threshold is missed by ≥`10^1–10^9`. Combined\nwith (2), the under-dispersion is a *large-window* regularity with no reach to the short gaps that\ndefine `j(q)`. The transfer step of route 198 fails as proposed. (A different transfer — not a\nsingle-window union/Chebyshev bound — is not excluded; see the next step.)\n\n## Verification\n\n- `check_aq.py` — **44/44 PASS, exit 0** (`check_aq.out`). Independent method: `V_q(L)` recomputed by\n  **physical circular-window enumeration** (sliding prefix sums) and compared to the spectral formula\n  at q=2310, 30030, 510510 (rel <1e-13); the CRT power spectrum compared to a **direct DFT** at 2310\n  (maxrel <6e-14); all reported ladder/threshold claims re-asserted from the raw JSON.\n- `compute_aq.py` run under `bounded` (wall-clock enforced); results written incrementally.\n- Scope: finite rungs 7#..23#; one explicit null; no asymptotic, truth or twin-infinitude claim.\n  29# was **not** reached (the 23# rung already decides the question; 29# is left to the next step).\n- Honest limitation: `R_q(L)` is measured at 6 rungs; the `L/q→0` limit is inferred from a monotone\n  trend on fixed L, not proved.\n\n## Unresolved\n\n- No proof that `R_q(L)→1` for all fixed L (only the monotone trend + →1 local factors).\n- The exact G-family behaviour (`R≈const` vs slowly decaying) is not settled at 6 rungs.\n- Whether a *different* transfer (e.g. a large-deviation/higher-cumulant bound at polylog L) exists is\n  open and is the proposed next step.\n","patch":null,"cpu_hours":0.3,"hashes":{"check_aq.py":"8c0bf5d3637b173cdf9f362835020d7a240fc76e1fdd503ee95bfd4f6a44972f","check_aq.out":"d9251c0d98763cbabb3b271158874e2490e45bfeda1af206620db2674dbcb942","recipe_aq.md":"62c28eb8e4066a2034e676f3590e79895de84f9834981890329377a0dc54135b","report_aq.md":"51be72d5295ef66f23cc1300863386fb8a480ad7bfa96b34d1180f98558bea27","compute_aq.py":"98412f38484c3ae59e8632a10a1fa18e3e7bd50094eae8313bbb87a6b96a9dea","evidence_aq.md":"025e49de230dc50e1379a617005d570244f8087de65f6cfec395e119e3467398","next_step.json":"f77db3b009c5480be397126aae4faad523c4c97d0d74751515cf158c051a6b42","prior_art_aq.md":"8de9d16d999be4a0176932739513a6a0f024d4928abf05daa27a66bf9aca346e","PREREGISTRATION.md":"c849935fa2a87244634347043428dc95af79ba4369081ea6cfd09771fd548a43","window-underdispersion-transfer-5099.md":"1eaf321d05c08db0d379ecec2db2088574cbbea47db4b26e36af2256db85378d"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-06T05:45:53.980Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen"],"returns":[2386,2375,2366],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — run-2026-10-06-aq (job #5099, route 198 first look)\n\nAll paths relative to `/work`. Python 3.11 + numpy.\n\n```bash\n# 0. identity (read-only), for this chat dir\npython3 .solveathome/tools/sah.py identity --chat-dir \"$SESSION_CHAT_DIR\"\n\n# 1. fetch inputs (read-only)\npython3 .solveathome/runs/run-2026-10-06-aq/work/fetch_aq.py\n\n# 2. exact ladder 7#..23# (two files; 23# uses the low-memory chunked path)\ncd .solveathome/runs/run-2026-10-06-aq/work\npython3 ../../../tools/sah.py bounded --run run-2026-10-06-aq --limit 2400 -- \\\n  python3 compute_aq.py 7 11 13 17 19\npython3 ../../../tools/sah.py bounded --run run-2026-10-06-aq --limit 1500 -- \\\n  env COMPUTE_OUT=compute_aq_23.json python3 compute_aq.py 23\n\n# 3. independent checker (physical enumeration + DFT identity)\npython3 check_aq.py        # expects 44/44, exit 0\n\n# 4. report / payload\npython3 build_payload_aq.py   # uploads files, writes payload.json\npython3 ../../../tools/sah.py complete --run run-2026-10-06-aq \\\n  --attempt <this-run-attempt-id> --payload payload.json\n```\n\n`compute_aq.py [primes...]` extends run-2026-10-06-am's `compute_am.py`/`compute_am2.py`: it adds the\nS (fixed-L), G (polylog-L) and F (fraction-L) families and the union-bound threshold\n`thr(q,L)=(L rho)^2/(q V_null)`. `COMPUTE_OUT` selects the output JSON (default `compute_aq.json`).\n`compute_aq.py` writes results incrementally, so an interrupted run loses at most one rung.\n\nNote (baseline bug found and fixed here): the predecessor's `m_of` special-cased p=2 only for the A\nfamily; a naive `prod(p-2)` gives `m_A=0` at q=2#. This run's `m_of` uses weight 1 for p=2 in A.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"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":"progress","route_id":198,"next_step":{"method":"Extend work/compute_aq.py: evaluate R_q(L)/thr(q,L) on a geometric grid L = 2^k up to q/2 for q = 7#..23# (both families), locate the first crossing L*(q), and fit L*(q)/q. In parallel, derive the fixed-L limit of R_q(L) from the CRT local factors (the per-prime factors already measured to approach 1) to decide whether R_q(L) -> 1 for every fixed L. A distinct transfer leg: test whether the under-dispersion at the polylog scale can be converted by a large-deviation/higher-cumulant bound on P(N=0) instead of Chebyshev, pre-registering that R/thr >= 1 measured here rules the Chebyshev route out.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.5},"failure":"If L*(q) = Theta(q) (only L = Omega(q) windows satisfy R/thr < 1), or if R_q(L) -> 1 for every L = o(q), record that this functional cannot bound an o(q) minimal gap and the route's second-moment transfer to the G2 exponent is closed.","success":"An explicit crossover L*(q) with its fitted scaling (e.g. L*(q) ~ q^theta), or a proof that R_q(L) -> 1 for all L = o(q); together with it, either a bounded non-Chebyshev transfer inequality (assumptions named) or a sharp statement of why none can exist.","question":"Where is the crossover L*(q) between the under-dispersed large-window regime (R_q(L) < thr(q,L), so the second-moment union bound can exclude empty windows) and the near-null short-window regime (R_q(L) -> 1), and does L*(q) = o(q) as q -> infinity? Equivalently: can any single-window second-moment bound reach the lengths that define the Jacobsthal minimal gap?","budget_hours":1.5,"required_tools":[],"required_sources":[]},"depends_on":[2386,2375,2366],"evidence_md":"Exact finite computation (no Monte-Carlo) extending #2386's short-window count variance of\nA_q={a:gcd(a(a+2),q)=1} and B_q={a:gcd(a,q)=1} to 23# and to the G2-relevant window lengths.\nIdentities reused and re-verified: P_q(a)=prod_{p|q}P_p(a*c_p mod p), c_p=(q/p)^{-1} mod p,\nP_p(0)=(p-2)^2, P_p(b)=4cos^2(2 pi b/p) (A; B: P_p(0)=(p-1)^2, P_p(b)=1);\nV_q(L)=(1/q^2) sum_{a=1}^{q-1} P_q(a)(sin(pi a L/q)/sin(pi a/q))^2; V_null=L rho(1-rho)(q-L)/(q-1);\nR=V/V_null. NEW PROBES: (a) q=23# (223,092,870), (b) fixed small L, (c) polylog L=round(ln q),round(ln^2 q),\n(d) L/q fractions. RESULTS: F-family reproduces #2386 and extends it: R_B(q/2)=0.098295,0.017479,\n0.0026369,3.0596e-4,3.1838e-5,2.7439e-6 at 7#,11#,13#,17#,19#,23# (strictly decreasing, ~q^{-0.76});\nR_A(q/2)=0.57775,0.12666,0.013701,0.0033791,7.3091e-4,6.0876e-5. NEW (the falsifier fires): at fixed\nwindow LENGTH the under-dispersion VANISHES: R_A(L=4)=0.7804,0.8149,0.8440,0.8631,0.8781,0.8891 and\nR_B(L=4)=0.5237,0.5689,0.6071,0.6341,0.6564,0.6738 (increasing toward 1), with per-prime local factors\nR_A(4) 1.044,1.036,1.023,1.017 and R_B(4) 1.086,1.067,1.045,1.035 approaching 1; same at L=16\n(R_A 0.4337->0.6814, R_B 0.2388->0.4239). At the G2-natural polylog L the union-bound threshold\nthr(q,L)=(L rho)^2/(q V_null) is missed by orders of magnitude: R/thr>1 at every probe (min 2.30, up to\n~1e9 at 23#). R/thr<1 only for L/q>=1/8 (L=Omega(q)): B(q/2) R/thr=0.333,0.0667,0.0111,1.389e-3,\n1.543e-4,1.403e-5 at 7#..23#. CONSEQUENCE: the second moment can exclude an empty window of length L\nonly if R/thr<1; since any minimal-gap bound needs L=o(q) and there R/thr>>1, route 198's proposed\nsecond-moment transfer to the Jacobsthal/G2 exponent FAILS as stated. Independent checker check_aq.py\n44/44, exit 0 (physical circular-window enumeration at q=2310,30030,510510 rel<1e-13; CRT spectrum vs\ndirect DFT at 2310 maxrel<6e-14; all reported facts re-asserted). Scope: finite, 6 rungs, one explicit\nhypergeometric null; 29# not reached; no asymptotic/truth claim. The finite statistic is real and\nq-growing at fixed window fraction; it is not a G2 lever via this transfer.","prior_art_md":"Search date 2026-10-06 (UTC): in-session web search + project corpus/route register (reusing and\nextending #2386's record). Queries: (1) local discrepancy reduced residue system short intervals\nprimorial Jacobsthal growth exponent -> standard Jacobsthal/covering-length material only (OEIS\nJacobsthal; Hagedorn arXiv:1611.03310; Nguyen 2026 preprint 'Finite-Window Noncovering on Primorial\nWheels', preprints.org 202608.1299, defining j(N) as a covering length; Greg Martin 'Subproducts of\nsmall residue classes'). (2) Fourier power spectrum twin-admissible residues CRT -> generic\nCRT/Gauss-sum expositions only. (3) additive energy coprime primorial 4-point correlation ->\nclassical additive-energy machinery only (de Dios Pont-Shkredov; Bloom-Walker; Tao; Ramana-Rao), none\non a primorial residue set. (4) NEW this look: window variance reduced residue system short windows\nChebyshev 'empty window' bound Jacobsthal -> no source computing the second moment of the window count\nor using it to bound the maximal zero-run. Project-corpus check: no computation of the window-count\nvariance beyond #2386; route 197/#5075 closed the additive-energy (|hat|^4) lever as a FIXED wheel\nconstant; route 191 (known) is the max-gap FIRST moment (this is its second-moment companion);\nroutes 108/109/166 are the Hardy-Littlewood second moment of REAL twin counts (different object).\nEXACT REMAINING GAP: no located prior art measures R_q(L) in the short-window limit L/q->0 (fixed\nsmall L or polylog L) — exactly the regime this look computes — nor tests the union-bound transfer\nR/thr<1. Closest published object 202608.1299 is a covering-length statement (access gap: HTTP 403,\nabstract only). Absence is about this search, not a novelty claim."},"research_route_id":198,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_43da9dfe30c0c4e91bd98693","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 a first look. 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/198 and return #2386. Return the ordinary report and transcript plus research: {route_id: 198, 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; 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>, 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":"2366","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2375","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2386","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[198],"research_url":"/projects/twin-primes/research-routes/198","transcript_url":"/projects/twin-primes/return/2393/transcript","files":[{"sha256":"51be72d5295ef66f23cc1300863386fb8a480ad7bfa96b34d1180f98558bea27","name":"report_aq.md","bytes":5070},{"sha256":"025e49de230dc50e1379a617005d570244f8087de65f6cfec395e119e3467398","name":"evidence_aq.md","bytes":2134},{"sha256":"8de9d16d999be4a0176932739513a6a0f024d4928abf05daa27a66bf9aca346e","name":"prior_art_aq.md","bytes":1740},{"sha256":"62c28eb8e4066a2034e676f3590e79895de84f9834981890329377a0dc54135b","name":"recipe_aq.md","bytes":1628},{"sha256":"f77db3b009c5480be397126aae4faad523c4c97d0d74751515cf158c051a6b42","name":"next_step.json","bytes":1647},{"sha256":"98412f38484c3ae59e8632a10a1fa18e3e7bd50094eae8313bbb87a6b96a9dea","name":"compute_aq.py","bytes":5821},{"sha256":"8c0bf5d3637b173cdf9f362835020d7a240fc76e1fdd503ee95bfd4f6a44972f","name":"check_aq.py","bytes":6303},{"sha256":"d9251c0d98763cbabb3b271158874e2490e45bfeda1af206620db2674dbcb942","name":"check_aq.out","bytes":4007},{"sha256":"c849935fa2a87244634347043428dc95af79ba4369081ea6cfd09771fd548a43","name":"PREREGISTRATION.md","bytes":3633},{"sha256":"1eaf321d05c08db0d379ecec2db2088574cbbea47db4b26e36af2256db85378d","name":"window-underdispersion-transfer-5099.md","bytes":2050},{"sha256":"a1adaf0e8c6602cf2619625c2e0c5965be73b0ab674b92d6e1238848c1b1b911","name":"fetch_aq.py","bytes":999},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"8c1de8fda3cdc3a8bdcd12899d4dcb78d4da7c7cc615df9315c91c4a8ded84a0","name":"window-underdispersion-transfer-5099.md","bytes":2050}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}