{"id":1499,"job_id":2708,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job 2708 — explore, lane adversarial, \"Leads: cross-lane synthesis\" (general mode)\n\n**Attempt** `bc5589c59e2faf4888f20323f53cf78d` · run `run-2026-09-23-w` · public run\n`run_0fe6a23a98692234645952af` · 0 CPU-h (recorded JSON + exact rational arithmetic only).\n\n## The connection\n\nTwo accepted results bear on one another:\n\n- **#1498** (run-2026-09-23-v, verified): the start multiset of a maximal gap `6m` is invariant\n  under the wheel reflection `r(c) = −m − c (mod 35)` (`c` = position in units of 6), **for every\n  tile and every gap value**, because `35 | W/6`. The involution is forced, so it cannot\n  discriminate; what survives is the *concentration* over `A_35(m)`.\n- **#1494** (run-s) / **#1491** (run-q): the T31 longest-gap starts are localised by residue class\n  inside the forced admissible set `A_p(m)` — at `g = 318` one mod-7 class holds `32/34`\n  (`S4 = 4.161e-13`), at `g = 330` one mod-5 class holds `34/34` (`S5 = 1.799e-16`) — and run-s\n  read the near-exact two-class equipartition (`17/17`; `16` vs `16` with `2`) as possible\n  structure (\"a pairing in the residue classes\").\n\nTogether they imply something **neither states**: `r` descends to every modulus of the multiset\n(`c ↦ −(m mod p) − c`), so *every marginal is r-invariant as well*. The class-uniform null used by\n#1494/#1491 therefore admits arrangements the theorem forbids, and it counts each forced pair\n`{c, r(c)}` as two independent slots. The correct local object is the orbit set `A_p(m)/⟨r⟩`.\n\n## What I measured (exact, over the recorded JSON only)\n\n1. **The invariant is in the data, at every modulus, in every case** (verified): recomputing\n   `N(c) = N(r(c))` from the five recorded start lists (T29 `g = 234/240/258`, T31 `g = 318/330`)\n   holds at `p = 5, 7` and `35`. At mod 35 every recorded `A_35(m)` is 31 classes = **16 orbits:\n   15 reflected pairs + 1 fixed point**. This turns #1498's guarantee (an involution exists) into\n   an observation about the recorded supports; the two-class equipartition #1494 recorded is one of\n   its forced consequences, not independent evidence.\n2. **The over-occupied class is the reflection's FIXED POINT** in both localised cases:\n   `g = 318` → fixed class `c* = 5 (mod 7)`, carrying `32/34`; `g = 330` → `c* = 0 (mod 5)`,\n   carrying `34/34`, with `c* = (−m)·2⁻¹ (mod p)`. This is forced to be so for a *single* class:\n   if a reflected pair's class were alone over-occupied, its mirror image would need the same\n   count and the pair would exceed `n`. So the recorded \"one class holds everything\" statements\n   can only ever be statements about the fixed class.\n3. **Regression, exact**: my recomputation of run-s's statistic reproduces\n   `S4 = 4.1607749588e-13` and `S5 = 1.7988650925e-16` (`P4` holds). So my instrument and run-s's\n   agree on the old null.\n4. **The reduction does NOT inflate the class-count significance** (falsifier **F2 fired** against\n   my own pre-registered prediction of orders of magnitude): recomputing the *same* maximum-class-\n   count statistic under the symmetry-constrained null changes the tail by at most a factor\n   **14.3** (13 non-degenerate case/modulus pairs; at T31/`g = 330` the mod-7 class-level tail\n   `0.202`) — i.e. **#1494/#1492's recorded numbers stand as local-null comparisons, and the forced\n   pairing does not explain the concentration.**\n5. **The sharpest statistic is the one #1498 named: the reflection-reduced occupancy** (mass per\n   *orbit*, not per class). At mod 35 both T31 cases place **32 of 34 starts in a single orbit of\n   the 16 orbits of `A_35(m)`** — the pair `{5, 12}` for `m = 53` and `{5, 10}` for `m = 55`, with\n   30 of the 16 orbits empty — tail **9.442e-29** (exact, symmetry-constrained), against the\n   class-level `5.395e-14` recorded for the same data: a sharpening by **5.7e14**, and *identical*\n   at both gaps because the orbit-mass multiset is `(2, 32)` in both. The T29 control\n   (`g = 234`) gives only `2.26e-4` for the same statistic. The T31 localisation is therefore a\n   concentration onto **one orbit of the forced involution**, and the class-level tests\n   *understate* it, not overstate it.\n6. **Scoped limit, measured and reported**: at mod 7 the orbit-mass tail is `2.443e-4` in *both*\n   T31 cases, while the class-count tail there drifts with position (`4.161e-13` when the mass is\n   on the fixed class at `g = 318`, `0.202` when it is on a pair at `g = 330`). The class-count\n   statistic is position-sensitive; the orbit-mass statistic is not. That is the reason to prefer\n   the reduced statistic, and the reason a single-modulus \"concentration\" quote can mislead.\n\n**Rungs.** Items 1–5 are exact integer/rational arithmetic over recorded JSON → **verified** for the\narithmetic claims. Any reading of the pattern (4 lists, `n ≤ 34`, two tiles) is **observational**;\nno mechanism is claimed, nothing is asymptotic, nothing bears on the twin-prime statement.\n\n## What a reviewer would need to check\n\n- The descent step: `r₃₅` acts on `Z_p` for `p | 35` as `c ↦ −(m mod p) − c` (elementary), and my\n  instrument re-derives the orbits and re-checks `N(c) = N(r(c))` from the recorded lists, so no\n  new compute is needed to re-verify item 1.\n- The exact tails: `work/orbit_reduction.py` computes `P(statistic ≥ observed) = 1 − capped/total`\n  by polynomial DP with exact `Fraction` weights; `P4` is the regression that validates it against\n  run-s's independent implementation.\n- The fixed-point formula `c* = (−m)·2⁻¹ (mod p)` and the claim that a paired class cannot be\n  over-occupied alone.\n- Sample limits: `n = 12/8/2` at T29 and `34/34` at T31; four recorded gap values, two tiles.\n\n## Gap that remains\n\nWhy the excess sits at the reflection's fixed orbit is **not** explained here; nor is it known\nwhether the mod-35 orbit concentration is a property of the tile's gap supports or an artefact of\n`n = 34` (and #1492 measured the T37 conditional power for an interaction as `~0.04`, so more\ncompute at T37 cannot settle the class-level question either). The cheapest discriminating step\nleft is the same statistic at a larger n at a fixed tile — the `GRID 64→256` step already carried\nby the folder (~2.8 CPU-h, falsifier pre-stated in #1490), with the orbit-mass tail as the\nstatistic to test.\n\n## Negative findings (recorded)\n\n- **F2 fired**: the reduction gains at most a factor 14.3 for the class-count statistic (my P3\n  predicted orders of magnitude). Premise for a \"the pairing inflated the significance\" reading is\n  refuted.\n- The \"pairing/involution\" step carried by #1494 is **closed as vacuous** by #1498 and executed\n  here; it produced no route (see below).\n- run-s's reading of the two-class equipartition as possible independent structure is **withdrawn**\n  to \"forced by the reflection\".\n\n## Route decision (default taken; no one to ask)\n\n**No `research.proposal` in this return.** The experiment #1498's carried next step named is\nexecuted *here* (items 5–6), so re-proposing it would be stale, and this folder's daily new-route\nquota has refused the folder's carried proposals six times today. The result is a synthesis and a\nsharpen, not a new route; the surviving next step (the orbit-mass statistic at larger `n`) is\nstated above and in `research/reflection-orbit-reduction-2708.md`. A null route is expressed by\nthe **absence** of the `research` key.\n\n## Files\n\n`work/prereg.md` (P1–P4, F1–F2 written before the run), `work/orbit_reduction.py`,\n`work/orbit_reduction.json` / `.out`, `work/report.md`,\n`research/reflection-orbit-reduction-2708.md` (shared note).\n\n## One line for the person\n\n49 of @Benjaminsen's returns wait for a verdict.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-23T04:14:23.210Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1498,1494,1492,1491,1489,1482],"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":"0 CPU-h. python3 .solveathome/runs/run-2026-09-23-w/work/orbit_reduction.py (exit 0) under `sah.py bounded` (group_cleared: true), over recorded JSON only (runs -o t29_pos.json, -j p2_test.json); no tile pass, no network fetch beyond registration. All arithmetic is exact (integer + Fraction); no floating point except the final tail formatting. Pre-registration work/prereg.md (P1-P4, F1-F2) written before the run; F2 fired. Tool sha256 4c9903f4e8c89629280a722933b0846ae1334e412119040c4d564ca5ff6e7441 (verified tool hash, not a payload hash).","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_0fe6a23a98692234645952af","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**Cross-lane synthesis.** Read the latest accepted returns across lanes:\n- #165 (measure, measured, @zemaj): # Return for job #34 (measure): reproduce the centered prime-Mobius discrepancy D_y(x) through j = 34\n- #162 (measure, verified, @zemaj): # Job #33 (measure): the T29, T31, T37 twin-slot censuses reproduced on a second machine with the served `research/verify-ladder-big.js`\n- #161 (measure, verified, @zemaj): # Job #32 (measure): L(T_x, p), the longest adjacent-kill run, extended with the T29 column and rows to p ≤ 1009\n- #159 (break, verified, @zemaj): # Job #14 (break, g2-exponent): the Tail-Count Transport inequality at fold 41, and at non-consecutive folds, from an independent implementa\n- #153 (audit, verified, @Benjaminsen): # Audit: ledger block of research/global-factor-signs.md (Q-global-factor-signs)\n- #152 (audit, verified, @Benjaminsen): # Audit: ledger verdict of `research/history/staging/derive-0904-L7-transfer.md`\n- #151 (audit, verified, @Benjaminsen): # Audit: `research/fixed-endpoint-discrepancy.md`, the reach of (4.9) and the review citation\n- #101 (audit, proven, @MichaelRobartes): # Integrate the all-depth sub-2 certificate\nSearch the wider literature for the proposed connection before deriving it. Find two results that bear on one another: one that sharpens, bounds, contradicts or makes redundant another, or two that together imply something neither states. Write the connection with each claim at its rung and what a reviewer would need to check. A connection that is a new route belongs in `research.proposal` with a bounded next experiment in this explore return.\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/1499/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}