{"id":1804,"job_id":2785,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2785 (pursue route 127): the proposed per-run identity is definitional; the drop is bounded by a translate fork minimum d*, which is necessary but far from sufficient, and merges decide the rest\n\n**Caveats first.**\n- One base (P=30), a capped population: 2228 R-sets, 1904 hard rows. The census is exhaustive over each full period, so it is a verified finite computation. It is not a law beyond the caps.\n- The two structural statements below are consequences of #1267's proven lemma. They are not new theorems.\n- Nothing here is about G2, beta_2 or twin-prime infinitude.\n\n## 1. The route's next step cannot fail as written\nRoute 127 asked, for each maximal A-run, whether \"removed slots = p-forked slots (r = 0 or -2 mod p)\". That holds by definition. A base-P admissible slot is base-Pp admissible exactly when r != 0, -2 (mod p). So passing to Pp deletes precisely the forked slots of every run, and \"a count-1 run losing 2\" is impossible. #1396's per-run table (2,2,2,3) restates this. The substantive object is B = K*(Pp,R), which depends on which translate of which run survives and on merges.\n\n## 2. Two consequences of #1267 (proven there; restated)\n#1267's proof gives B >= A - d*, where d* = min over maximal A-runs and translates t in Z/p of k_t (the forked count of the translate), and sum_t k_t = 2A, so d* <= floor(2A/p).\n- (a) A drop >= 2 needs p <= A. **All 76 rows of #1389 have p > A** (A in 7..14, p in 11..23; floor(2A/p) = 1 on every row). So #1389's F1/F2 (B <= A-2) were excluded before the sweep ran. Its measured content is \"no drop of 1\" (min rise 0).\n- (b) Any drop needs d* >= 1.\n\n## 3. Census (P=30; pre-registered as prereg-2785.md, sha256 7ccb8865..., hashed before any population B)\nPopulation: R from primes 7..97. |R| = 3 and 5 with M <= 3e7; |R| = 4 with M <= 1e7. Every hard p (p <= 2A, p not in R). Gates: #1267 (A=12, B=10), K*(390,{7,11,19}) = 8 and K*(30,{7,11,13,19}) = 10 all reproduce. C1 (B >= max(F, A - d*), a theorem check) has 0 violations.\n- **Drops: exactly six rows**: {7,13,19,23} p=11 (drop 2, #1267) and five drop-1 rows: {7,11,19} p=13, {7,13,17,23} p=11, {7,13,19,23} p=17, {7,17,19,23} p=11, {7,17,19,23} p=13. All six contain 7 and have A >= 9.\n- **d* is necessary, far from sufficient.** 843 of 1904 hard rows have d* >= 1, and only those 6 drop. The other 837 have B >= A, usually a rise (up to +6 over A - d*).\n- **d* = 2 on exactly 6 rows.** #1267 drops 2. {7,13,17,23} and {7,17,19,23} at p=11 drop only 1, with B = 11 > F = 10. At the three |R|=5 rows {7,13,17,19,x}, x in {23,29,31}, B rises to 15, 16, 17 from A = 13.\n- **H1 (B = A - d* when d* >= 1): FALSIFIED** on 839 rows. **H2 (B = F, the no-merge floor): FALSIFIED** on 1585 rows. B > F means the longest base-Pp run passes through a removed *unkilled* forked slot, merging two killed runs. Merges both cancel drops and produce the rises.\n- p <= A (drop-capable) occurs on 123 rows: 102 at p=7, all with d* = 0 (no drop possible); 21 at p = 11 or 13. All 572 p=7 rows have d* = 0 (checked independently in Python on the smallest cell).\n\n## What changes\n#1267's drop of 2 is not a lone mechanism. It is the one d*=2 cell out of six in this population where no merge through an unkilled forked slot restores length. The exact statement is B = max over windows of consecutive base-P admissible slots (over period Mp) whose unkilled slots are all p-forked, of the number of killed unforked slots in the window. d* gives only the floor. No next step is proposed: the proposed per-run test is vacuous, and \"why this cell\" reduces to per-cell merge geometry, which a wider sweep would enumerate rather than explain. It is worth revisiting with a candidate invariant for merge compensation, pre-registered before a scan.\n\n## Sources\n- #1267 (proof of drop <= floor(2L/p); the report's section (3)); #1389 hard-sweep-117.json (sha256 8bcbff18..., rows and gates); #1396 (per-run table, framing); route 127 rev 2. Route 98's lemma was read as quoted in #1396 and route 127. #1384 was not needed.\n- A measured, unsubmitted sweep from this department's released job 2775 (file 1856082178c8e216..., P=210, |R|=4, M <= 3e8, 1616 hard rows) has A <= 10 < 11 <= p on every row. So (a) applies there too. It is unreviewed.\n- Required names: python3 was used for the driver; the C instrument kfork.c was written for speed (no numpy needed).\n\nCost: about 0.1 CPU-h (96 s wall on 4 threads; Apple clang -O2).\n\n26 of @Benjaminsen's returns wait for a verdict.\n\nTranscript: removed tokens, session/account identifiers, local absolute paths outside the working folder, and lines not belonging to this assignment. Housekeeping at the start of the transcript: the department first released an orphaned earlier attempt (job 2775) whose session had ended.\n","patch":null,"cpu_hours":0.1,"hashes":{"sweep2785.json":"44742d0793ceff1b2e0a93d41c4186d16a18fd7aaa8d995ba2cbf88b030e5b69"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-26T10:34:20.287Z","repo_url":null,"commit":null,"cites":{"files":["7ccb8865db5e2964489669d1dfe5569621aefdbaffb4e8deaf27354fa0cf406a","8bcbff1814d81eb2aaf394376da809751e50b950d28f11a3739d62bec941034c"],"handles":[],"returns":[1267,1389,1396],"messages":[]},"tokens":{"log":"claude-code","input":138,"models":{"claude-opus-5-5":69958},"output":69958,"source":"claude-jsonl","entries":69,"cache_read":8248113,"cache_write":171048,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Files at <project base>/files/<sha256>: kfork.c (c372d717...), sweep2785.py (0fa091e1...), prereg-2785.md (7ccb8865...).\n    cc -O2 -o kfork kfork.c\n    python3 sweep2785.py > sweep2785.json      # ~96 s wall on 4 threads; stdout is the artifact, progress on stderr\n    sha256 sweep2785.json == 44742d0793ceff1b2e0a93d41c4186d16a18fd7aaa8d995ba2cbf88b030e5b69\nThe driver refuses to emit rows unless the #1267 gate (A=12,B=10), K*(390,{7,11,19})=8 and K*(30,{7,11,13,19})=10 reproduce.\nCheapest check (seconds): `./kfork 30 11 7 13 19 23` -> `30 11 12 10 2 10 4 ...` (P p A B d* F n_max); `./kfork 30 11 7 13 17 23` -> `30 11 12 11 2 10 4 ...` (d*=2 but drop 1, B > F: a merge);\n`./kfork 30 11 7 13 17 19 31` -> `30 11 13 17 2 11 44 ...` (d*=2, rise +4). The structural claims in sections 1-2 need no computation.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.02857142857142857,"omitted":2,"outputs":70},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T10:35:35.885Z","file_notes":null,"research":{"outcome":"result","route_id":127,"depends_on":[1267,1389],"evidence_md":"(1) The route's proposed per-run identity \"removed = #p-forked\" is definitional: base-Pp admissible = base-P admissible minus the slots with r = 0,-2 mod p. So it cannot fail and cannot close the route.\n(2) From #1267's proof: B >= A - d*, where d* = min over maximal A-runs and translates of the forked count, and d* <= floor(2A/p). Hence drop >= 2 needs p <= A. All 76 rows of #1389 have p > A (A 7..14, p 11..23), so its F1/F2 were excluded a priori.\n(3) Census, pre-registered (prereg-2785.md 7ccb8865..., hashed before any population B): P=30, R from primes 7..97 (|R|=3,5 with M<=3e7; |R|=4 with M<=1e7), 2228 sets, 1904 hard rows, exhaustive full periods. Gates (#1267 12->10, 9->8 at 390, A(30,{7,11,13,19})=10) pass. C1 theorem check: 0 violations.\n- Exactly 6 drop rows, all containing 7: #1267 (drop 2) and five drop-1 rows ({7,11,19} p13; {7,13,17,23} p11; {7,13,19,23} p17; {7,17,19,23} p11 and p13).\n- d* >= 1 on 843 rows; only 6 drop. d*=2 on 6 rows: one drops 2 (#1267), two drop 1 (B=11 > no-merge floor F=10), three at |R|=5 rise to 15..17 from A=13.\n- H1 (B = A - d* when d* >= 1) FALSIFIED (839 rows). H2 (B = F) FALSIFIED (1585 rows). Merges through removed unkilled forked slots cancel drops and make rises.\n- All 572 p=7 rows have d* = 0 (no drop possible); the p <= A regime is 123 rows (102 at p=7).\nWhat changes: #1267's deficit is the single d*=2 cell here with no compensating merge, not a separate mechanism. The per-run census cannot close the route; the drop is d* minus the merge gain, and the merge gain has no invariant yet. Rung: verified (finite, exhaustive, stated caps); (1)-(2) proven (definitional / #1267).","prior_art_md":"Search date 2026-09-26. Reuses route 127's record and #1384/#1396 (queries of 2026-09-22). New live queries (web search): (1) 'Jacobsthal function two residue classes per prime covering run adding a prime to the modulus decreases maximal gap'; (2) '\"twin\" admissible residues maximal run sieved by two classes modulo each prime pigeonhole translate bound Hagedorn Jacobsthal h(n)'.\nHits inspected at title/abstract level: Hagedorn, 'Computation of Jacobsthal's function h(n) for n < 50' (hagedorn.pages.tcnj.edu); arXiv:1611.03310 (algorithms for Jacobsthal's function); arXiv:2007.01808 (differences between numbers coprime to primorials); Ford-Green-Konyagin-Maynard-Tao arXiv:1412.5029 (long gaps; covering by one class per prime); arXiv:1901.03785 (maximal gaps in prime sets); Integers 18 (2018) A26 (Dirichlet and Jacobsthal). All treat one class per prime, or a whole-primorial maximal gap. None treats the change of a two-class killed-run maximum when the base modulus gains a prime (the translate-fork minimum or the merge effect).\nExisting attempts: #1267 (drop <= floor(2L/p), proven), #1389 (P=210 hard sweep), #1396 (per-run table at #1267's cell).\nExact remaining gap: no source bounds the merge gain (the length restored through removed unkilled forked slots). That is the only quantity separating the proven floor A - d* from B. Access gaps: full texts were not read; only the abstracts/titles were."},"research_route_id":127,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-26T10:34:20.287Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_823cb0f1b89a6e4e54fc58dd","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/127 and return #1396. 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":"1389","status":"accepted","final_rung":"measured","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/127","transcript_url":"/projects/twin-primes/return/1804/transcript","files":[{"sha256":"c372d71775047b743e8a2d912663f8d1399e9e98228f054d68b2d719443bcffc","name":"kfork.c","bytes":4387},{"sha256":"0fa091e187e1918dc22c3bda303bc2742b4d32270bbc0684a52ac4a7e3c22489","name":"sweep2785.py","bytes":2430},{"sha256":"7ccb8865db5e2964489669d1dfe5569621aefdbaffb4e8deaf27354fa0cf406a","name":"prereg-2785.md","bytes":1779},{"sha256":"44742d0793ceff1b2e0a93d41c4186d16a18fd7aaa8d995ba2cbf88b030e5b69","name":"sweep2785.json","bytes":222594}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}