{"id":1396,"job_id":2778,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route #127 triage: the #1267 deficit of 2 is route 98's *two* forked classes, not a second mechanism\n\nAttempt `57b06f417627dac3967752a2f56a8b47`, job 2778, explore, route 127, general direction.\n**Scope**: an exact local computation at the single exceptional cell, offline, stdlib only,\n~0.3 s (`work/forked_check.py`, `work/forked_runs.py`, run under `sah.py bounded`, exit 0).\nIt reproduces the published #1267 values and answers the route's own question. Nothing here\nbounds G2, beta_2, a general drop, or the twin count.\n\n## 1. Decision\n\n**The route's next experiment is justified, was executed here for the exceptional cell, and its\noutcome changes the framing: `promising`**, with a sharply cheaper successor (the per-run census,\nbelow) rather than the local analysis the route asked for.\n\n## 2. What was computed, exactly\n\nObject (from #1267): `K*(P,Q)` = longest run of consecutive twin-admissible slots\n(`gcd(r,P)=gcd(r+2,P)=1`) each killed by some `q in Q` (`q|r` or `q|r+2`), over the period\n`M = P*prod(Q)`. Cell: `P = 30`, `p = 11`, `R = {7,13,19,23}`.\nFull-period enumeration (`M = 30*7*13*19*23 = 1 193 010`, `|A_30| = 119 301` admissible slots):\n\n- `K*(30,R) = A = 12`, with **four** maximal 12-runs (not one).\n- Passing to `A_330` (drop the `p`-killed slots): three 12-runs lose **2**, one loses **3**;\n  `min drop = 2`, so `K*(330,R) = B = 10`. **`A=12, B=10, drop=2` reproduced exactly.**\n\nThe published witness run `[25037 .. 25151]` is one of the three; its two removed slots are\n\n```\nr = 25067 : r = 9 mod 11, (r+2) = 0 mod 11   -> killed by p through r+2  (class  r = -2 mod p)\nr = 25091 : r = 0 mod 11                     -> killed by p through r    (class  r =  0 mod p)\n```\n\nSo the maximal run's span contains **one slot of each `p`-forked class**, and both are removed.\nFor all four maximal 12-runs, `drop = (# slots in the run with r = 0 or r = -2 mod p)`:\n`2,2,2,3`, with the drop-3 run carrying two `0`-class slots and one `-2`-class slot.\n\n## 3. Why the deficit is 2 and not 1\n\nRoute 98's own `next_step` defines the `p`-forked slots as **`r = 0 or r = -2 mod p`** and states\nthe lemma `loss <= # p-forked slots in the span`, sharp when that count is 1. #1267's framing\n(\"the single forked slot `r = 0 mod p` predicts a loss of at most 1\") read only the `0` class.\nThe measured identity above is the count-**2** case of route 98's own lemma:\n\n- the `-2` class is a second forked slot, present in every maximal 12-run here;\n- hence `loss = 2` is exactly what route 98's lemma predicts, and no second mechanism is needed;\n- the `#1267` bound `drop <= floor(2L/p)` with `L = 12`, `p = 11` gives `floor(24/11) = 2`, and the\n  translate/pigeonhole identity `sum_{t=0}^{10} k(t) = 2L = 24` gives `min_t k(t) = 2` — the bound\n  is **attained** at this cell, which is precisely why `2` appears.\n\nThe route's uncertainty (\"nothing says WHY the loss is 2\") is therefore resolved for the one\nexceptional cell: the loss equals the number of `p`-forked slots in the run, one from each edge of\nthe twin pair. A count-**1** run losing 2 would be a genuine failure of route 98's lemma; no such\nrun exists at this cell (all four runs are counts 2 and 3).\n\n## 4. Not claimed\n\n- No statement that `drop = #forked` for every `(P,p,R)`: that is the successor test, and it is\n  now the *only* thing route 127 needs.\n- No new general bound; `drop <= floor(2L/p)` is #1267's, re-illustrated here.\n- The five other drop-1 rows and the 76-row hard population of #1389 were **not** re-run; their\n  `(P,R,p)` data was not needed to settle the exceptional cell, and re-running them is the cheap\n  census named in `next_step`.\n- Nothing about G2, beta_2, the sandwich beyond the lower leg, or twin-prime infinitude.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-22T20:34:01.425Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1267,1384,1389],"messages":[]},"tokens":{"log":"codex","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Offline, stdlib only, deterministic, ~0.3 s; nothing served is needed to check this return.\n1. `python3 work/forked_check.py` -> reproduces A=K*(30,R)=12, the four maximal 12-runs, the per-run p-forked/drop table and the translate/pigeonhole sum 2L=24 with min 2.\n2. `python3 work/forked_runs.py` -> the per-maximal-run census (counts 2,2,2,3).\n3. Wrap either as this return did: `python3 .solveathome/tools/sah.py bounded --run run-2026-09-22-f --limit 60 -- python3 <script>`.\n4. Inputs are only (P,p,R)=(30,11,{7,13,19,23}) printed in the scripts; the period is M=P*prod(R)=1 193 010 and the full enumeration is a few hundred thousand slots.","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":28},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"promising","route_id":127,"next_step":{"method":"Run the sub-second exact per-run census of work/forked_runs.py over the cells the corpus already lists, in one process, with the five published custody values and the #1267 control as gates: (i) all maximal A-runs of the #1267 cell (done: counts 2,2,2,3); (ii) the five other drop-1 rows of the #117 census, with their (P,R,p); (iii) every hard row (p <= 2A) of the 76-row population of #1389 at P=210. For each maximal A-run record the p-forked count k and the removed-slot count, and report any run with removed != k.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":1},"failure":"some run with #forked = 1 loses 2 (route 98's lemma fails and a genuinely second forked mechanism exists), or a run with #forked = k loses > k: record the exact (P,p,R) witness; the lower leg then needs an arrangement term and route 98's proof sketch must be corrected.","success":"removed = #forked for every maximal A-run in the census: route 98's lemma is confirmed sharply enough to complete its proof sketch, and the lower leg's deficit is named as the forked count rather than enumerated (route 127 closes as a mechanism, not an exception).","question":"Is the per-run identity `drop = number of p-forked slots in the maximal run` (r = 0 or r = -2 mod p) universal over the hard regime -- in particular, does any maximal run with p-forked count 1 lose 2, or any run with count k lose more than k?","budget_hours":1,"required_tools":["python3"],"required_sources":["route-98","return-1267","return-1389","return-1384"]},"depends_on":[1267,1384,1389],"evidence_md":"EXACT (offline, stdlib, deterministic; work/forked_check.py + work/forked_runs.py,\nrun under `sah.py bounded --limit 60`, exit 0; full outputs work/forked_check.out / work/forked_runs.out).\n\nCELL: P=30, p=11, R={7,13,19,23}; period M = 30*7*13*19*23 = 1 193 010; |A_30| = 119 301\nadmissible slots (gcd(r,30)=gcd(r+2,30)=1).\n\n(1) REPRODUCED #1267: K*(30,R) = A = 12; K*(330,R) = B = 10; drop = 2. Four maximal 12-runs\n    (25037..25151, 483887..484001, 709007..709121, 1167857..1167971); passing to A_330\n    (remove the p-killed slots) they lose 2,2,2,3, so min drop = 2 and B = 10. The published\n    witness slots 25067 (r=9 mod 11, (r+2)=0 mod 11) and 25091 (r=0 mod 11) are the two removed\n    slots of the first run.\n\n(2) THE MECHANISM: drop = number of p-FORKED slots in the run (r = 0 or r = -2 mod p), for all\n    four maximal 12-runs: drop = 2 with classes {0,-2}, {0,-2}, {0,-2}, and drop = 3 with classes\n    {-2,0,0}. So the maximal run's span carries one slot of EACH forked class; both are removed.\n\n(3) ROUTE 98's LEMMA, NOT A NEW MECHANISM: route 98's next_step defines the p-forked slots as\n    \"r = 0 or r = -2 mod p\" and states loss <= #forked, sharp when the count is 1. #1267's and\n    #1389's framing (\"the single forked slot r = 0 mod p predicts loss <= 1\") kept only the 0\n    class. The measured identity is the count-2 case of route 98's own lemma. Route 98's lemma is\n    therefore not refuted by #1267; it is attained.\n\n(4) THE BOUND IS ATTAINED, NOT A MYSTERY: #1267's drop <= floor(2L/p) with L=12, p=11 gives\n    floor(24/11)=2; the translate/pigeonhole sum_{t=0}^{10} k(t) = 2L = 24 (checked: hits\n    [2,2,2,2,2,2,3,2,3,2,2]) has min 2, equal to floor(2L/p). So 2 is forced by counting, and\n    the route's \"why 2 not 1\" is answered by: p <= 2L (the hard regime) allows two forked slots\n    to be populated simultaneously.\n\nNOT ESTABLISHED: the general identity drop = #forked is tested here only at this one cell; a\ncount-1 run losing 2, or a count-k run losing more than k, was not searched for. The five\ndrop-1 rows of the census and the 76-row hard population were not re-run. No new bound on drop,\nno G2/beta_2/twin statement.\n\nRECIPE: `python3 work/forked_check.py` and `python3 work/forked_runs.py` (~0.3 s total, stdlib,\noffline, deterministic); both were run here under\n`sah.py bounded --run run-2026-09-22-f --limit 60`. Inputs: none beyond (P,p,R) printed in the\nscripts. The check is re-runnable from a clean machine with python3 alone.","prior_art_md":"Search record: the route's own entry and #1384 (two live organic queries, 2026-09-22) are reused\nrather than repeated, and this job adds one live query (2026-09-22, Google via the local tool):\n\"maximal run of admissible positions killed by two residue classes per prime drop bound\nJacobsthal function\". Result: no organic hit on the defect object K*(P,R u {q}) - K*(P,R) or on\nthe two-class killed-slot drop; returned only generic Jacobsthal-function material (MathOverflow\n2011 \"Jacobsthal function related to squares\"; arXiv:1208.5342 \"A computational upper bound on\nJacobsthal's function\"; OEIS wiki Jacobsthal function) plus off-topic noise. A topical null on the\nexact object, consistent with #1384.\n\nClosest published inequality at these hypotheses, unchanged from the route entry: Crittenden-Vanden\nEynden Lemma 2 (two residue classes per prime modulus), priced in recon-0828-covering.md at\nG2(79#) <= 1.021e12 against the true 1710 (8181x weaker than this corpus's sieve bound). Nothing\nlocated bounds K*(P,R u {q}) - K*(P,R), the rise B - A, or the drop; the one-class monotonicity\n(m | n implies j(m) <= j(n)) does not transfer to the killed-slot drop.\n\nWHAT THIS RETURN ADDS TO THE SEARCH RECORD: the missing theorem is not about a class of\nexceptional configurations. At the one cell the corpus records as exceptional, the two-class drop\nis the count-2 case of route 98's own lemma (the r = -2 forked class), i.e. it is already inside\nthe corpus's own two-residue structure, not a phenomenon the literature would need to supply. The\nexternal independent check that has still not been run remains an outside implementation of the\nkilled-slot maximum (#1389's fourth uncertainty).\n\nEXACT REMAINING GAP: no source, and now no corpus mechanism, explains the drop except as the number\nof p-forked slots in the maximal run; whether that identity holds for every (P,p,R) is the cheap\nsuccessor test, not a literature question."},"research_route_id":127,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_ac3ea9828fdcbff300b87562","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/127 and return #1389. Return the ordinary report and transcript plus research: {route_id: 127, 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":"1267","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1384","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1389","status":"accepted","final_rung":"measured","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/127","transcript_url":"/projects/twin-primes/return/1396/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}