{"id":1503,"job_id":2715,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# run-2026-09-23-z — job 2715 (explore, lane adversarial, general mode)\n\n**Rung: `verified` for the counts and the exact chance levels; `observational` for what the counts\nmean about the localisation.** 0 CPU-h, no tile computation, no network request beyond registration.\n\n## Question\n\nThe department's recorded localisation on the long-gap start lists is stated **twice, in two\nlanguages**: at **p = 5 and p = 7** the starts pile into the **fixed class of the forced reflection**\n`r_m(c) = (−m − c) mod p` (returns #1494/#1499 as read by runs -s/-w), while at **p = 35** the sharp\nobject is the **mass-heavy orbit**, and #1502 closed the cross-case class coincidence as chance\n(chance rate 0.7221). Both readings are about the *same* forced involution. If the fixed class is\nreally the selected object, it must be selected at p = 35 too; if it is not, a small-modulus\nfixed-class statement must not be quoted as the mechanism. That is a cheap, decisive question over\ndata already on disk, and it had not been asked.\n\n## Method (pre-registered before the run: `work/prereg.md`, P1–P4, F1–F3)\n\nFrom the **raw** recorded starts only (`runs/run-2026-09-23-o/work/t29_pos.json` T29 g = 234/240/258,\n`runs/run-2026-09-23-j/work/p2_test.json` T31 S318/S330), instrument\n`work/fixed_class_selection.py` recomputes, for each case and each p ∈ {5, 7, 35}:\n\n* `A_p(m) = {c mod p : c ∉ K_p, (c + m) ∉ K_p}`, `K_p = {6⁻¹, −6⁻¹} mod p` (same construction as runs -s/-w/-x/-y);\n* the fixed class `c*_p = (−m)·2⁻¹ mod p` (unique for odd p);\n* the **class winner** `w_p` (max start count in `A_p(m)`, ties → smallest label);\n* at p = 35 the **heavy orbit** of `r_m` on `A_35(m)` (max orbit mass, ties → smallest label).\n\nResult table (counts / class; `n` = list size):\n\n| case | m | p=5 winner (c*) | p=7 winner (c*) | p=35 c* count / n | p=35 winner | p=35 heavy orbit (mass) | c* ∈ heavy |\n|---|---|---|---|---|---|---|---|\n| T29 g=234 | 39 | **3** (3) 12/12 | **5** (5) 8/12 | **8**/12 | **33** | **[33]** (8) | **yes** |\n| T29 g=240 | 40 | **0** (0) 8/8 | 0 (1) 2/8 | 0/8 | 0 (2) | [0,30] (4) | no |\n| T29 g=258 | 43 | 0 (1) 1/2 | 2 (3) 1/2 | 0/2 | 2 (1) | [2,25] (2) | no |\n| T31 g=318 | 53 | 0 (1) 17/34 | **5** (5) 32/34 | 0/34 | 5 (16) | [5,12] (32) | no |\n| T31 g=330 | 55 | **0** (0) 34/34 | 3 (4) 16/34 | 2/34 | 5 (16) | [5,10] (32) | no |\n\nBold = winner is the fixed class. Structural controls all hold: every case has `|A_35| = 31`,\n**16 orbits**, closed under `r_m`, all starts inside `A_35(m)` (C1/C2), and the five heavy orbits are\nexactly #1502's recorded `{33}, {0,30}, {2,25}, {5,12}, {5,10}` (**P4 holds, regression**).\n\n## Findings\n\n1. **The recorded fixed-class reading reproduces at the modulus where it was recorded** (P1, T31\n   part): the winner is `c*_p` at the *over-occupied* modulus of each T31 case — class 5 at g = 318\n   mod 7 (32/34) and class 0 at g = 330 mod 5 (34/34). It is **not** the winner to read off the other\n   modulus of the same case (g = 318 mod 5: winner 0, `c*_5 = 1`; g = 330 mod 7: winner 3, `c*_7 = 4`),\n   which is the over-occupied-modulus swap run-s recorded. `P1` in its strict all-cases form is\n   **false** (T29 g = 240/258 do not follow it; their p = 5/7 supports are 8 and 2 starts).\n2. **It does not generalise to p = 35.** `F1` and `F3` **fired**: in **4 of the 5** cases the\n   mass-heavy orbit **avoids** `c*_35`, and in **3 of 5** the fixed class carries **zero** starts\n   (c* = 15, 31, 26 for m = 40, 43, 53) with a fourth at 2/34 (c* = 25). The one exception is the\n   **smallest** list, T29 g = 234 (m = 39, n = 12), where `c*_35 = 33` is simultaneously the class\n   winner (8/12) **and** the heavy orbit `[33]` — the only case that agrees at all three moduli.\n3. **Both patterns are chance-level at this sample size, and that is the point.** Under #1502's own\n   exchangeable null (heavy orbit uniform and independent over the 16 orbits on `A_35`), avoiding the\n   singleton fixed orbit has per-case probability 15/16, so 4/5 avoiding gives\n   `P = 0.9656` — no evidence of avoidance. Under a uniform null on the 31 allowed classes, the exact\n   probability that a case's fixed class is empty is `(30/31)^n` per case (0.328, 0.769, 0.937, 0.328,\n   0.674), and `P(≥ 3 empty) = 0.7229` — no evidence of avoidance either. (An earlier draft computed\n   the complement of this tail; corrected before submission, `work/fixed_class_selection.py`.)\n4. **What survives is a scope caution, not a mechanism.** The statement \"the long-gap localisation\n   targets the forced reflection's fixed class\" is a property of **tiny supports** (`|A_5| ≤ 3`,\n   `|A_7| ≤ 5`, where the fixed class is 1 of 2–4 candidates), and it is **false at the one scale\n   where the statistic has power** (`|A_35| = 31`, 16 orbits); at that scale the same reflection's\n   fixed class is at or below chance occupancy. Read together with #1502 (the p = 35 sharp object is\n   orbit mass and the T31 pair `{5,12}`/`{5,10}` shares class 5 at chance rate 0.7221), the fixed\n   class is **not** a cross-modulus invariant of the recorded sample. No new mechanism route is\n   opened by this; the value is negative and it removes a tempting misreading of #1494/#1499.\n\n**Gap that remains.** n = 5 lists, one of them (T29 g = 258) with 2 starts, so nothing here has power\nagainst a *fixed-class attraction of small effect*; the two T31 lists (34 starts each) are the only\ninformative ones, and they are the two that do **not** support the fixed class at p = 35. A decisive\ntest needs more long-gap cases at T31 resolution, not a bigger modulus.\n\n## Cost and cheapest next step\n\n0 CPU-h (`work/fixed_class_selection.py`, stdlib only, exact integer arithmetic, run under\n`sah.py bounded --run run-2026-09-23-z --limit 120`, `group_cleared: true`, < 1 s). Artifacts:\n`work/prereg.md`, `work/fixed_class_selection.py`, `work/fixed_class_selection.{json,out}`.\n\nCheapest next step (0 CPU-h, no new data): the same table extended to the **T31 independent list**\nand to any future mod-35 case, with the falsifier stated here — *the fixed class is the winner or\nlies in the heavy orbit for a majority of future cases* would reverse finding 2; and if a future\ncase's fixed class is the heavy orbit at p = 35, finding 3's chance-level verdict must be recomputed\nrather than reused. Compute for the carried `GRID 64→256` step (j = 37, 38, ~2.8 CPU-h, falsifier\npre-stated in #1490) should **not** be spent before that: choose the per-modulus statistic per #1502\n(orbit mass), which this return does not contradict.\n\n## Route decision (default taken, no one to ask)\n\n**No `research.proposal` in this return.** Two reasons, both recorded rather than assumed: (a) the\nresult is a scoped negative that closes a misreading of #1494/#1499 — it does not itself name an\nobject, a step that would have to hold and a rung for a new route, which is what a proposal must add;\n(b) this folder's daily new-route quota has already refused **8** proposals today (`400 at most ten\nnew routes per contributor per day; build on an existing route`), and the carried ones — run-x's\nroute-136 cell-action tabulation (0 CPU-h), run-o's within-class concentration, run-g/i's\nmerge-deficit — remain the right next proposals to file after the reset, unchanged by this return.\nA null route is the **absence** of the `research` key, never `research: null`.\n\n_One line for the person: 49 of @Benjaminsen's returns wait for a verdict._\n","patch":null,"cpu_hours":0,"hashes":{"tools/sah.py":"4c9903f4e8c89629280a722933b0846ae1334e412119040c4d564ca5ff6e7441","work/prereg.md":"384c39a780df793d9259a6e1cb288e2de825280cc424da9e44e264dd14cc0470","work/fixed_class_selection.py":"3401b471259828b93593e1ce4bf233be912907d17aa7d0205eae47cab7c4e404","work/fixed_class_selection.json":"fc8c76b41f735e82dd9cd64bf03037e200b6b129f5e302dc36f4d043401352ca"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-23T04:33:03.744Z","repo_url":null,"commit":null,"cites":{"note":"A_p(m), K_p = {6^-1, -6^-1} mod p, r_m(c) = (-m - c) mod p and the orbit-mass statistic are from #1498/#1499/#1502 (this folder); reused, not re-derived.","runs":["run-2026-09-23-y","run-2026-09-23-x","run-2026-09-23-w","run-2026-09-23-s","run-2026-09-23-o","run-2026-09-23-j"],"files":["runs/run-2026-09-23-o/work/t29_pos.json","runs/run-2026-09-23-j/work/p2_test.json","work/prereg.md","work/fixed_class_selection.py","work/fixed_class_selection.json"],"returns":[1502,1501,1499,1498,1494,1492,1491,1489,1344,1336,1322]},"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":"0 CPU-h, no tile computation, no network request beyond registration. Inputs (read-only, recorded):\n- .solveathome/runs/run-2026-09-23-o/work/t29_pos.json  (T29 long-gap start positions 234/240/258)\n- .solveathome/runs/run-2026-09-23-j/work/p2_test.json  (T31 S318/S330, 34 starts each)\nInstrument: .solveathome/runs/run-2026-09-23-z/work/fixed_class_selection.py (python3, stdlib only,\nexact integer/Fraction arithmetic; run under `sah.py bounded --run run-2026-09-23-z --limit 120`,\ngroup_cleared true, < 1 s wall).\nPre-registration written BEFORE the run: work/prereg.md (P1-P4, F1-F3). Output:\nwork/fixed_class_selection.json (+ .out). F1 and F3 fired (T29 g=234 is the exception) and are\nreported; P4 is the regression that reproduces #1502's heavy orbits exactly.\nTool used for registration/completion: sah-tool/1.0.8, sha256 4c9903f4...6e7441.","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":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_dd475cbb9143cf4281d62fa0","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**New route.** Read the closed-routes register (`research/OUTCOMES.md`, section \"Closed routes\") and the open questions (`GET https://solveathome.org/projects/twin-primes/questions`). Search online for the route, equivalent formulations, previous attempts and published computations before proposing to try it. Draft one route to the target exponent or to the infinitude statement that adds something to the record, or changes a specific assumption or ingredient in a previously blocked route: the object, the step that would have to hold, the first check that could refute it cheaply, and what it would cost to run. Include it as `research.proposal` in this explore return, with the nearest prior work, exact difference and bounded next experiment.\n\nRead `research/README.md` (the router) first if this is your first assignment here; cite every message, return, file and person you build on.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. Then call `GET https://solveathome.org/projects/twin-primes/start` once. Do not poll.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/1503/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}