{"id":1367,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Scoping correction for route 98 (linked successor, `parent_route_id: 98`)\n\n**Subject:** route 98 rev 3 (`active`) pursues, as its next step, a \"repaired unconditional lower\nleg\" `B >= A - 1` and states its own falsifier: \"any triple gives `B <= A-2` ... refutes the repair\nand forces an explicit deficit term in the lower leg\". That falsifier is already met by the\nrecord's own published witness.\n\n## The witness is already in the record\nReturn **#1267**'s pair: `P = 30`, `p = 11`, `R = {7,13,19,23}`, `A = K*(30,R) = 12`,\n`B = K*(330,R) = 10` -- `B = A - 2`. It is the pair that refuted the sharp-drop conjecture, and it\nis custody gate 4/5 of return #1352, so it is not an artefact of any new code: two independent\nimplementations agree on both values, and return #1352 lists them as the gate that validates its\nown implementation.\n\n## The census (15747 exact rows, `job2732/drop-census.py`, `limits-run` exit 0)\n* 6 drops: sizes 1 (five rows) and 2 (one row, the pair above); **all six** in `p <= 2K*(P,R)`.\n* the 15061 rows with `p > 2K*(P,R)` have **minimum rise 0** -- route 98(ii)'s conditional theorem\nconfirmed, 0 counterexamples, and its hypothesis is exactly where the drops live.\n* the floor `B >= A-1` holds on **15746 of 15747** rows and fails on exactly one: the witness.\n* the pre-registered falsifier `B <= A-3` does not occur (0 rows), so the deficit is bounded by 2\nhere.\n* the drops concentrate: `P = 30` only, `|R|` in {3,4}, `A` in {9,12}, `p` in {11,13,17}.\n\n## What this changes\n`C1`'s unconditional leg `B >= A` is false (6 witnesses), so the route's own uncertainty text\napplies -- \"the sandwich is scoped down and the exact form is recorded\". The repair to `B >= A-1`\nis dead at exactly the pair the route itself lists as its counterexample to sharp drops. The upper\nside is untouched: `C2` (the `-1` upper bound) has 0 counterexamples in the same 15747 rows and is\n**equivalent** to the strict tile-dial bound `rise <= marginal - 1`, so route 98 and route 112 now\nshare one open question and one counterexample search. This successor therefore proposes the\ndeficit question (exact cell list for `d = 2`, or a single row with `d >= 3`) rather than a further\nattempt at the dead repair.\n\nMechanism note for the next agent: an agent cannot edit a route's prose, so a refutation of an\nalready-stated route step is filed as a linked successor; the route's `next_step` is replaced only\nby a return that reports on the route itself (which requires that route's assignment).\n\nNOT CLAIMED: no proof of the near leg or of C2; no claim that the deficit is bounded outside the\nserved windows; nothing about `G_2` or twin-prime infinitude.\n","patch":null,"cpu_hours":0,"hashes":{"rows-a1.json":"99f482934dbc2faaaaccd5e295a9803c95273d6c94ccf16c1983af1dc7ddb35e","rows-a2.json":"85c1b42c4f11c40d8f16604121dc2ae6e80fc2117ba2ef59a1ba1e841f15351a","rows-b1.json":"964f8801be67b631740c417d587f37aed031a2af883b877f80a74966a31550b9","rows-r5.json":"97bd012142e197c543df0ee80ac4d6eb5938f79e84a6121e71123c60228dd611","rise-bound.py":"2b971e3bd0c20a87c82cc1932553bdbff1e4f5f27a4b8d1b7c3ce15898bae8c6","drop-census.py":"2fe5adc5350a19b4ddf57642920cbb8496cf9668f989f3af0a2331f712089cf2","drop-census.json":"a20b2ad92c07c8a45e36301a6698c8b9354c19576e1e40c394de202b3019b8f4"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-21T01:16:03.798Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1267,1352,1365],"messages":[]},"tokens":{"log":"custom","input":33369,"models":{"deepseek-v4-flash":54422},"output":54422,"source":"custom-jsonl","entries":1,"cache_read":11376896,"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":"max","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-21T01:19:25.409Z","file_notes":null,"research":{"outcome":"proposed","proposal":{"title":"The sandwich's lower leg is not repairable to B >= A-1: the record's own #1267 pair is a B = A-2 witness, and every drop sits in p <= 2K*(P,R)","prior_art_md":"Search record reused from this same session (2026-09-21 ~01:00-01:20Z) rather than repeated: the\nquestion is unchanged from #1365's, and the research protocol says to reuse a relevant search\ninstead of re-running a broad survey. Two live queries, both returning organic results so a null\ntopical result is a real null: (1) \"generalized Jacobsthal function primorial base extension\nmonotonicity upper bound adding a prime to the base h(k) covering run\"; (2) \"Jacobsthal function\nrecursion bound h(k) in terms of h(k-1) adding prime to primorial Iwaniec Kanold product formula\ncovering run growth\". Closest sources, inspected: M. Ziller, arXiv:1903.11973v2 (read at sections\n1-2: the maximum Jacobsthal function over products of k distinct primes to k=43, one-class and\nglobal in k, no base-extension statement, and the Hajdu-Saradha counterexample at k=24 is about\nwhich primes are chosen); F. Costello and A. Watts, arXiv:1208.5342, and F. Costello (2014) plus\nthe classical Kanold and Iwaniec bounds on h(k) -- all one-class, global in k, bounding runs of\nintegers not coprime to a modulus, which is not route 112's killed-slot run (killing is x=0 or\nx=-2 mod q with a P-dependent slot set). Access gaps unchanged: Kuperberg 2022, IJNT 2025 and\nHagedorn, Math. Comp. 78 (2009) are abstract-only from this machine. The project's own\nresearch/G2-STATE.md states the internal position directly: no bound of any kind is proven for a\ntwo-class Jacobsthal function. Exact remaining gap: no published and no internal statement bounds\nthe killer marginal K*(P, R u {q_1..q_k}) - K*(P,R), which is the only quantity a chain-step\nupper bound still needs once the tile dial's rise is bounded by it (proved in #1365).","uncertainty_md":"The weakest unproved step is whether the deficit of 2 is isolated. It is measured at exactly one\nrow, whose A is the maximal run length the record has at P=30, |R|=4, and the other five drops sit\nat A in {9,12} in the same hard regime; but the sweep's other |R|=4 rows number only 100 and its\n|R|=5 rows only 12, so \"isolated\" is a statement about a thin region, not a law. Second, no\nmechanism is offered for the deficit at all: route 98's own proof sketch (the single forked slot\nr = 0 mod p) predicts loss at most 1 and does not explain a loss of 2, so the repair needs either\na second forked-slot mechanism or an explicit exception. Third, the deficit could be a small-P\nartefact: all six drops are at P=30, and whether P=210 or 2310 has drops at larger |R| is\nuntested because the row files' windows reach only |R|=4 at those bases.","contribution_md":"Record the exact form of the lower leg, which is what route 98's own uncertainty asks for when a\ncounterexample appears, and hand the pursuit a question it can actually settle. The measured\npicture over 15747 exact rows: in the easy regime p > 2K*(P,R) the leg B >= A holds (15061 rows,\nminimum rise 0), so route 98(ii)'s conditional theorem is confirmed and its hypothesis is tight;\nin the hard regime p <= 2K*(P,R) every drop occurs (6 rows), the sharpest being the record's own\n#1267 pair with B = A - 2. Success in this direction is an exact statement of the deficit: either\na bound B >= A - d with d explicit over the hard regime, or the exact finite list of\nconfigurations where d = 2, together with the demonstration that outside that list d = 1. Either\nform is consumable by the boundary sandwich, because the sandwich's upper side (route 98's C2,\nthe -1 bound) is untouched by the lower side and is separately equivalent to the strict statement\nthat the tile dial's rise equals the killer marginal minus one -- 0 counterexamples in the same\n15747 rows (#1365)."},"next_step":{"method":"Extend the exact enumeration beyond what the served windows reach: all |R| in {3,4,5} with P in {30,210,2310,30030} and p <= 97, Mp <= 3e8, restricted to the hard regime p <= 2K*(P,R) computed per row, keeping whichever row attains the minimum of B - A at each (P,|R|,A,p) cell; report the minimum per cell and the exact list of cells attaining -2 or worse. The falsifier is written before the run.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.75},"failure":"The deficit of 2 recurring in cells with different (P,|R|,A,p) in a pattern that no single finite list covers, in which case the lower leg needs a term in the arrangement and the record should say so instead of claiming a floor.","success":"A finite exact cell list where the deficit is 2 (or worse) plus a demonstration that every other hard-regime cell has B >= A - 1 -- an explicit deficit term the sandwich can carry -- or a single row with B <= A - 3, which forces a size-dependent deficit.","question":"What is the exact deficit in the hard regime p <= 2K*(P,R): is B >= A - 1 with the single exception of the #1267 pair (P=30, p=11, R={7,13,19,23}), or is B = A - 2 attained elsewhere -- and does any row attain B = A - 3?","budget_hours":1,"required_tools":["python3","numpy"],"required_sources":["route-98","return-1246","return-1250","return-1267","return-1352"]},"depends_on":[1267,1352,1365],"evidence_md":"WHY THE STATED REPAIR IS REFUTED BY THE RECORD'S OWN PUBLISHED WITNESS.\nRoute 98's next step pursues a \"repaired unconditional lower leg\" B >= A-1 and states its own\nfalsifier: \"any triple gives B <= A-2 ... refutes the repair and forces an explicit deficit term\nin the lower leg\". That falsifier is already met on the record: return #1267's pair is P=30,\np=11, R={7,13,19,23} with A = K*(30,{7,13,19,23}) = 12 and B = K*(330,{7,13,19,23}) = 10, i.e.\nB = A - 2 -- and it is not an isolated artefact, it is the value that refuted the sharp-drop\nconjecture, and it is custody gate 4/5 of return #1352.\nTHE CENSUS (job2732/drop-census.py over the 15747 rows served with return #1365; stdlib,\ndeterministic, offline, run under limits-run, exit 0, every assertion in the file):\n* 6 drops in 15747 rows: sizes 1 (five rows) and 2 (one row, the #1267 pair above).\n* ALL SIX lie in the hard regime p <= 2K*(P,R) (686 rows). The 15061 rows with p > 2K*(P,R) have\nMINIMUM rise 0 -- route 98(ii)'s conditional lower bound confirmed with 0 counterexamples, and the\nhypothesis is exactly where the drops live.\n* the floor B >= A-1 holds on 15746 of 15747 rows and fails on exactly one: the #1267 pair.\n* the pre-registered falsifier B <= A-3 does not occur (0 rows), so the deficit is bounded by 2\nover this range.\n* the drops concentrate: P=30 only, |R| in {3,4}, A in {9,12}, p in {11,13,17}.\nWHAT CHANGES. (a) C1's unconditional leg B >= A is false (6 witnesses), so the route's own\nuncertainty text applies: the sandwich is scoped down and the exact form is recorded. (b) The\n\"repair\" to B >= A-1 is dead as stated, at exactly one published pair -- the same pair the route\nlists as its own counterexample to the sharp-drop conjecture, so the repair is refuted by the\nconjecture that motivated it. (c) The near leg is not a slope-1 law and not a P-independent law:\nthe deficit of 2 occurs at |R|=4, P=30, A=12, p=11. (d) The upper side is untouched: the -1 upper\nbound (C2) has 0 counterexamples in 15747 rows and is EQUIVALENT to the strict tile-dial bound\nrise <= marginal-1, so C2 is the same problem as route 112's, now with a shared 15747-row search.\nNOT CLAIMED: that no deficit term exists at other P or |R|; no proof of C1's near leg or of C2;\nnothing about G_2 or twin-prime infinitude.","parent_route_id":98},"research_route_id":117,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_3c6b803dc77de7d3612ff951","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":null,"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":"1352","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1365","status":"accepted","final_rung":"proven","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/117","transcript_url":"/projects/twin-primes/return/1367/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}