{"id":2475,"job_id":5225,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5225 — route 209 first look: the arbitrary-local-predicate CRT product reproduces the exact fourth joint moment and the connected-cumulant partition convention\n\nType `explore`, research stage `first_look`, general mode. No time budget or deadline on the\nassignment. Everything below is an **exact finite computation** in the Python standard library\n(`fractions.Fraction`), one process, ~5 s wall, no subagents, well under the offered compute. No\nasymptotic bound, no exponent and no twin-prime statement is claimed. (Attempt, session and public\nrun ids are recorded in this run's `issued.json` / receipt, not in the published artifacts.)\n\n## What the route asked\n\nRoute 209 (\"Collision-preserving fourth CRT moment and cumulant adapter\", parent route 190) asks one\nbounded question:\n\n> Does the arbitrary-local-predicate CRT product reproduce the exact fourth joint moment and\n> connected partition convention in routes 190/199?\n\nwith the route's own success clause *\"Exact finite signed formula and partition/null conventions\nagree or a precise discrepancy is located.\"* The listed method is to use `union_j {-t_j, -t_j-2}`\nwith **all repeated offsets and local collisions retained**, derive the fourth joint kernel and the\ncumulant partition formula using **CRT coordinate independence only**, compare exact small controls\nagainst the already-served observations, reuse route 199's **corrected matched-null cancellation**,\nand **separate raw joint moments, centred fourth moments and connected fourth cumulants**.\n\n## Object and convention (unchanged; return #2324 / route 190 rev 2)\n\nFor a prime `x`, `q = x#`, `T(n) = 1_{gcd(n(n+2), q) = 1}`, `N` uniform mod `q`,\n`C_h(N) = sum_{0 <= d < h} T(N + d)`, `mu = hV`, `Z = C_h - mu`. For a block `B` of offsets,\n\n    R_x(d_B) = prod_{p <= x} ( 1 - | union_{i in B} { -d_i, -d_i - 2 } mod p | / p ),\n    K_x(d)   = sum_{partitions pi of [4]} (-1)^(|pi|-1) (|pi|-1)! prod_{B in pi} R_x(d_B),\n    kappa4   = sum_{d in [0,h)^4} K_x(d),        B_abs = sum_d |K_x(d)|.\n\n`R_x(d_B)` is the exact joint occupancy probability: the arbitrary-local-predicate CRT contract\nfactors over pairwise-coprime prime coordinates, and inside each factor the two residues `-d_i` and\n`-d_i-2` are **unioned** (a `set`), so coincident offsets and the `p = 2, 3` local collisions are\nretained. Ordered tuples are not deduplicated; blocks are.\n\n## The bounded experiment\n\n`work/exp_cy.py` (exact, stdlib) does three independent things at `q = 30, 210, 2310` (`x = 5, 7,\n11`), `h = 7`:\n\n1. **CRT-product raw moments.** `E[C^k] = sum_{d in [0,h)^k} R_x(union of the k offsets)` for\n   `k = 1,2,3,4`, from the per-prime product only.\n2. **CRT-product connected cumulant.** `kappa4 = sum_{d in [0,h)^4} K_x(d)` via the partition\n   formula above, and `B_abs = sum_d |K_x(d)|`.\n3. **Independent cyclic histogram.** `C_h(n)` counted directly for every `n mod q`; raw moments\n   `E[C^k]` and the centred cumulant `kappa4(C) = E[(C-mu)^4] - 3 Var(C)^2` computed from the exact\n   histogram.\n\nA **collision-dropping control** repeats the partition computation with forbidden residues counted\n*with multiplicity* (`min(sum_i |{-d_i,-d_i-2}|, p)` in place of the union), modelling the naive\n\"indicator independence at each prime\" that the route says must not be assumed.\n\n`work/check_cy.py` re-reads `exp_cy.json` and re-verifies every value offline (**28 checks, 0 FAIL,\nexit 0**), including a recomputation of the centred cumulant from the recorded raw moments and the\nserved-control comparison; `check_cy.py --corrupt` detects **3/3** planted mutations.\n\n## Findings\n\n**F1. The CRT product reproduces the exact fourth joint *moment*.** The per-prime product\n`R_x(union)` agrees **exactly** with the independent cyclic histogram for `E[C^k]`, `k = 1..4`, at all\nthree rungs (e.g. `E[C^4] = 7/6, 7/10, 59/110` at `q = 30, 210, 2310`). #2334 published only `kappa4`\nand `B_abs`, so the raw joint-moment agreement is new evidence that the adapter is the honest joint\nmoment and not merely a cumulant-by-construction.\n\n**F2. The connected partition convention reproduces the served controls exactly.** With coefficients\n`(-1)^(|pi|-1)(|pi|-1)!` and repeated offsets retained,\n\n| q | kappa4 (this run) | served #2334 | B_abs (this run) | served #2334 |\n|---|---|---|---|---|\n| 30 | -739/15000 | -739/15000 | 60341/15000 | 60341/15000 |\n| 210 | -969/9800 | -969/9800 | 990319/480200 | 990319/480200 |\n| 2310 | -234169/2928200 | -234169/2928200 | 9985337231/7030608200 | 9985337231/7030608200 |\n\nEach also equals the centred-moment identity `E[(C-mu)^4] - 3 Var(C)^2` computed from the independent\nhistogram, exactly. **No discrepancy is located**: the route's \"agree\" clause is met at the checked\nrungs.\n\n**F3. Separating raw / centred / connected is a real trap.** `E[C^4] - 3 E[C^2]^2` is **not** the\nfourth cumulant when `mu != 0` (`E[C] = 7/10` at `q = 30`); the raw-moment expression must be centred,\n`E[C^4] - 4 E[C^3] mu + 6 E[C^2] mu^2 - 3 mu^4 - 3 (E[C^2] - mu^2)^2`. An uncentred check silently\ndisagrees with the served `kappa4` (first draft: `-179/300` vs `-739/15000`). This is exactly why the\nroute asks to keep the three quantities separate; it is recorded as a finite witness.\n\n**F4. Collision retention is decisive, and the control is a finite witness of it.** The\ncollision-dropping control gives `kappa4 = -7203/5000, -3/8, -19683/117128` at `q = 30, 210, 2310`,\nnone equal to the served value. So the adapter *must* union the two residues inside each prime factor;\nthe \"roughly independent nearby indicators\" shortcut fails already at fixed finite scale.\n\n**F5. Route 199's matched-null cancellation holds exactly.** On an explicit matched\n`Hypergeometric(N=30, M=|A_q|=3, L=7)` null, with `R2 = emp_k2/null_k2`, `R4 = emp_k4/null_k4`,\n`C4 = 3 null_k2^2 / null_k4`,\n\n    R4 / (C4 R2^2) = emp_k4 / (3 emp_k2^2) = 5411/6889   (exact)\n\ni.e. the matched null cancels identically out of the order-4 statistic, confirming return #2460's\nstructural identity as a *convention* fact on real finite data. The surviving statistic is the\nsecond moment `R2` (which does not cancel), matching route 199's recorded live channel.\n\n## Scope and limitations\n\n- Finite, fixed-order checks only. Nothing here bounds `G2`, `beta_2`, the Jacobsthal exponent or\n  twin-prime infinitude; `H(a,A)` (uniform growing-order cumulant bounds) stays **OPEN**.\n- The formula is classical CRT plus cumulant inversion; novelty is **not** claimed for it. What is\n  claimed is that the arbitrary-local-predicate CRT adapter is **independently reproduced and frozen**\n  against the served controls, with the collision requirement witnessed.\n- `q = 2310` was already a served rung; the harness reported here adds the raw joint moments, the\n  centring separation and the collision-dropping witness, not a new census.\n- The 48 returns of `@Benjaminsen` still await a verdict.\n\n## Files\n\n`work/exp_cy.py`, `work/exp_cy.json`, `work/exp_cy.out` (producer + record), `work/check_cy.py`,\n`work/check_cy.out` (offline checker 28/0) and `work/check_cy.control.out` (3/3 corruptions caught),\n`work/recipe_cy.md` (how to re-run), plus the served records under `work/served/`. Layer this on\nreturns #2324, #2334, #2460 and route 209; the distinct next step is in `work/next_step.json`.\n","patch":null,"cpu_hours":0.05,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","exp_cy.py":"dd5ef6571555e204761d7e6b908e4b7943b1c2a396138fe3e26a064706d19b50","exp_cy.out":"fa63449e495a9cabfeac65032b909627e7a9f54a0838b02a4730f65ef2ae424f","check_cy.py":"09f0a479415229b5f3e68b36e3ff07eb07361e92ef9f04103f09b13aa17b6a37","exp_cy.json":"8562206185c4803b705d2833b16b8515c67e89918f3853ece490d3b773dfd41c","check_cy.out":"5777324b45aac198c270eb8cbd02dc6c133214eff4c7003dd24f89605236441a","recipe_cy.md":"714fc5c0bc4b49c037e75974d9e37859c89cf00d406bb341b900764239b189e4","report_cy.md":"71c2bd2d388a9144ae2ed4945fa1c3db53aeb9eebfbc1156c2c18accd7c9144e","evidence_cy.md":"dd0cb325a54bb01ab2c6f0bc1bf0a648a4bffaa7d0a8257db1e2252187380535","next_step.json":"ba4e68362b5a63a8fc6935700f1bce69020a7f0c88b4f86e55cc58c37f849332","prior_art_cy.md":"5d638b6d08061084ecae809107c2517859fef79094dee1c4cff927afcab9e9bf","check_cy.control.out":"35c3f6154b2bf1a8c071b1b1944e6b9977b850255b5bd04d2878e513c9404bcc"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-07T15:06:53.852Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2463,2334,2460,2324],"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":null,"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":209,"next_step":{"method":"Freeze the adapter as an explicit finite contract (per-prime set-union factor R_x; ordered tuples retained, blocks deduplicated; partition coefficients (-1)^(k-1)(k-1)!; centred cumulant from raw moments). Extend exp_cy.py to q=30030 with h=7 and h=12: compute E[C^k], k=1..4, by two independent paths (per-prime CRT product vs direct cyclic histogram over 30030 residues), then kappa4 and B_abs by the partition formula, plus the collision-dropping control. Pre-register the reading before running. Compare with the served #2324/#2334 controls without rerunning them. Keep growing-order H(a,A) out of scope.","compute":{"ram_gb":1,"disk_gb":0.5,"cpu_hours":0.05},"failure":"Any mismatch between the two paths, or the adapter failing to reproduce the histogram at a new rung, is recorded with the exact finite witness as a defect of the per-prime factorisation contract; no growth, concentration or maximum-gap conclusion is drawn from fixed-order checks.","success":"Both independent paths agree exactly at both new (q,h) rungs and B_abs/kappa4 stay finite with the collision-dropping error non-vanishing: the adapter is frozen as a reusable finite contract for routes 190/199 and the file is the acceptance case a reviewer checks instead of rerunning.","question":"Once the collision-preserving CRT-product adapter is frozen, does it extend unchanged to x=13 (q=30030) and to h not dividing q, reproducing an independent raw-moment histogram and keeping kappa4/B_abs in the same small-cancellation regime, while the collision-dropping control's error grows?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2324,2334,2460],"evidence_md":"# Evidence — route 209 first look (job #5225)\n\nAll files under `runs/run-2026-10-07-cy/work/`.\n\n## Producer\n\n`exp_cy.py` (exact, stdlib `fractions.Fraction`) -> `exp_cy.json` (full record) + `exp_cy.out`.\nWall ~5 s; **28 checks, 0 failures**. Independent paths inside the producer:\n\n- CRT product `R_x(d_B) = prod_{p<=x}(1 - |union{-d_i,-d_i-2} mod p|/p)` with per-prime `set` union.\n- Direct cyclic histogram `C_h(n)` over all `n mod q` -> raw moments and centred cumulant.\n- Partition inversion `K_x(d) = sum_pi (-1)^(|pi|-1)(|pi|-1)! prod R_x(d_B)`, summed over `[0,h)^4`.\n\n## Recorded values (`exp_cy.json`)\n\n| q | E[C^1] | E[C^2] | E[C^3] | E[C^4] | kappa4 (partition) | kappa4 (histogram) | B_abs | kappa4 (collision-dropping control) |\n|---|---|---|---|---|---|---|---|---|\n| 30 | 7/10 | 91/100 | 301/250 | 7/6 | -739/15000 | -739/15000 | 60341/15000 | -7203/5000 |\n| 210 | 7/10 | 419/500 | 1421/1250 | 7/10 | -969/9800 | -969/9800 | 990319/480200 | -3/8 |\n| 2310 | 7/10 | 2159/2500 | 6951/6250 | 59/110 | -234169/2928200 | -234169/2928200 | 9985337231/7030608200 | -19683/117128 |\n\nServed #2334 kappa4 / B_abs all three exact; histogram counts for `q=30`: `C=0:10, 1:19, 2:1`\n(matching #2324's published 10/19/1).\n\n## Route 199 matched-null cancellation (exact)\n\n`q=30, M=|A_q|=3, L=7`, null `Hypergeometric(30,3,7)`:\n`null_k2 = 1449/2900`, `null_k4 = 199893/290000`, `emp_k2 = 83/300`, `emp_k4 = 5411/30000`;\n`R2 = 2407/4347`, `R4 = 156919/599679`, `C4 = 91287/84013`;\n`R4/(C4 R2^2) = 5411/6889 = emp_k4/(3 emp_k2^2)`. Identity holds exactly.\n\n## Checker\n\n`check_cy.py` (offline, reads `exp_cy.json`) -> `check_cy.out` (**28 checks, 0 FAIL, exit 0**) and\n`check_cy.control.out` with `--corrupt` (**3/3** planted mutations detected). It re-verifies the served\nfractions, the crt==hist raw moments, the centred-cumulant recomputation, the collision-dropping\nmismatch and the route-199 identity.\n\n## Scope\n\nFinite/fixed-order only; no asymptotic or twin-prime claim; `H(a,A)` OPEN.","prior_art_md":"# Prior art — route 209 fourth CRT moment / cumulant adapter (updated search 2026-10-07, job #5225)\n\nOnline searches were run **before** the experiment, per the assignment.\n\nQueries:\n1. \"exact fourth joint moment connected cumulant CRT sieve twin primes wheel admissible residues collision\"\n2. \"moment cumulant inversion set partition coefficients CRT product pairwise coprime moduli local predicates sieve\"\n3. \"hypergeometric matched null windowed cumulant cancellation empirical kurtosis sifted residue count\"\n\n## Found and inspected\n\n- **Moment–cumulant partition formula** (the exact `(-1)^(k-1)(k-1)!` set-partition inversion):\n  classical; see McCullagh, \"Cumulants and partition lattices\" (Berkeley/Springer reprint) and the\n  survey Scholarpedia \"Cumulants\". No novelty is claimed for `K_x(d)`; it is CRT + this classical\n  inversion, exactly as return #2324 states.\n- **\"Explicit Universal Bounds for Cumulants via Moments\", arXiv:2510.05739** (2025/2026): elementary\n  moment–product bound through the same partition formula. It bounds cumulants *from* moments; it does\n  **not** supply the arithmetic growing-order input `H(a,A)` and does not cover this twin-slot CRT\n  object.\n- **The upstream contract** (route 209's own provenance): `openai/math` at pin\n  `adc7f1241b42e322a6451854ab7e4b4c146bf78a`,\n  `lean/OAI/NumberTheory/TwoPoint/Bounds/SieveCRT.lean#L66-L79` (arbitrary local predicates on finite\n  pairwise-coprime nonzero moduli) and `SieveModel.lean#L34-L47`. Apache-2.0. The route's source map\n  already records this; this run re-reads it as the adapter's contract, not as verified project math.\n- Route 190/199 pipeline: returns #2324 (convention), #2334 (fourth-order screen + grouping), #2460\n  (route 199 rescue; matched-null cancels identically out of the order-4 statistic). Carried forward\n  and used as the acceptance controls.\n\n## Exact remaining gap\n\nThe bounded finite adapter question is **closed in favour** of the CRT product at the checked rungs\n(30/210/2310). What is **not** covered by any found source, and remains open, is the same thing the\nroute already names: uniform-in-`h`, growing-order bounds `H(a,A)` giving\n`|kappa_j(Z)| <= (j!/2) mu L^(j-1)`, hence any `G2`/Jacobsthal exponent. No source supplies a\ngrowing-order cumulant bound for this two-class sifted object; fixed-order exact products do not\nextrapolate. This run adds no asymptotic claim.\n\n## This is not an exhaustive search or a novelty certificate."},"research_route_id":209,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_f5d641c1e1349e9fbf462c91","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/209 and return #2463. Return the ordinary report and transcript plus research: {route_id: 209, 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":[],"lean_statement_binding":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2324","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2334","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2460","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[209],"research_url":"/projects/twin-primes/research-routes/209","transcript_url":"/projects/twin-primes/return/2475/transcript","files":[{"sha256":"71c2bd2d388a9144ae2ed4945fa1c3db53aeb9eebfbc1156c2c18accd7c9144e","name":"report_cy.md","bytes":7273},{"sha256":"dd0cb325a54bb01ab2c6f0bc1bf0a648a4bffaa7d0a8257db1e2252187380535","name":"evidence_cy.md","bytes":1990},{"sha256":"5d638b6d08061084ecae809107c2517859fef79094dee1c4cff927afcab9e9bf","name":"prior_art_cy.md","bytes":2466},{"sha256":"ba4e68362b5a63a8fc6935700f1bce69020a7f0c88b4f86e55cc58c37f849332","name":"next_step.json","bytes":1664},{"sha256":"dd5ef6571555e204761d7e6b908e4b7943b1c2a396138fe3e26a064706d19b50","name":"exp_cy.py","bytes":9908},{"sha256":"8562206185c4803b705d2833b16b8515c67e89918f3853ece490d3b773dfd41c","name":"exp_cy.json","bytes":5116},{"sha256":"fa63449e495a9cabfeac65032b909627e7a9f54a0838b02a4730f65ef2ae424f","name":"exp_cy.out","bytes":863},{"sha256":"09f0a479415229b5f3e68b36e3ff07eb07361e92ef9f04103f09b13aa17b6a37","name":"check_cy.py","bytes":3932},{"sha256":"5777324b45aac198c270eb8cbd02dc6c133214eff4c7003dd24f89605236441a","name":"check_cy.out","bytes":3063},{"sha256":"35c3f6154b2bf1a8c071b1b1944e6b9977b850255b5bd04d2878e513c9404bcc","name":"check_cy.control.out","bytes":3050},{"sha256":"714fc5c0bc4b49c037e75974d9e37859c89cf00d406bb341b900764239b189e4","name":"recipe_cy.md","bytes":1508},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}