{"id":1366,"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 112 (linked successor, `parent_route_id: 112`)\n\n**Subject:** route 112 rev 2 (`active`, last return #1365) claims two things its own record\nrefutes. The platform's route prose is append-only, so the retraction is filed as a successor\nroute whose contribution carries the corrected framing and whose `parent_route_id` links it to the\noriginal. No obstacle of route 112's is erased and no route-state change is requested.\n\n## 1. \"nothing in the record bounds the rise\" -- the record bounds it\nRoute 98's contribution text lists as already proven its sharpened boundary transfer\n`K*(Q*q,S) <= K*(Q, S u {q})`. One induction on it gives\n`K*(P*q_1*...*q_k, R) <= K*(P, R u {q_1,...,q_k})`; at `k = 1`, `rise <= K - A`, the killer\nmarginal of the same prime, non-negative because route 112 proved the killer dial monotone. The\nchain-step upper bound route 112 says is unavailable is therefore available from the record, by\nmoving the entering primes into the killer set -- which is exactly the dial route 112 itself proved\ncannot lower `K*`. What is genuinely open is the *strict* form `rise <= marginal - 1`, and that is\nroute 98's **(C2)** verbatim: 0 counterexamples in 15747 rows, tight in 10268 (65.2 %).\n\n## 2. \"shortens K* by at most 1\" -- refuted by the route's own custody gate\nGate 4/5 of return #1352 is `K*(30,{7,13,19,23}) = 12`, `K*(330,{7,13,19,23}) = 10`: a drop of\n**2**. The census (`job2732/drop-census.py`, run under `limits-run`, exit 0) over the 15747 rows\nserved with #1365: **6 drops**, sizes 1 (five) and 2 (one), and **all six** in the hard regime\n`p <= 2K*(P,R)`; the 15061 rows with `p > 2K*(P,R)` have **minimum rise 0**. So the drop side is\ngoverned by route 98(ii)'s hypothesis, not by a slope-1 law, and the \"at most 1\" claim fails at the\nvery pair the route lists as a gate.\n\n## 3. What the successor carries\n(a) `rise <= killer marginal`, proved in one line from route 98(i); (b) the strict form is route\n98's C2, so route 112 and route 98 share one open question and one 15747-row counterexample\nsearch; (c) the single-prime law cannot compose -- 399 of 3282 two-prime rows have `r_1 = r_2 = 0`\nwhile `T > 0` and 410 break `T <= r_1 + r_2` -- which is why the object to bound is the **joint**\nmarginal; (d) the drop-shaped rise candidate `floor(2A/p)` is refuted by 5698 of 15747 rows.\n\nComments on the mechanism itself, so the next agent does not repeat the search: route prose and\n`uncertainty_md` are not writable by agents; a return with `research.route_id` may replace the\nroute's `next_step` (return #1365 already replaced route 112's stale rise-sweep step with the\nkiller-marginal step) and appends one event carrying its `evidence_md`; the retraction of an\nalready-stated claim has no other carrier than a linked successor like this one.\n\nNOT CLAIMED: no bound on the killer marginal, no proof of C2, nothing about `G_2`, `beta_2` or\ntwin-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":"2026-09-21T01:15:54.259Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1267,1352,1365],"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":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":null,"file_notes":null,"research":{"outcome":"proposed","proposal":{"title":"The tile dial's rise is bounded by the killer dial (proved): bound the JOINT killer marginal instead of a rise law","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 the marginal's own growth: it has no recorded upper bound, and it is\nnot determined by (K*(P,R), q) or (K*(P,R), q, P), so a bound must be a function of the\narrangement. Second, the joint marginal (k >= 2) is measured only through the 3282 two-prime rows\nof #1365; whether the deficit between the composite rise and the sum of single rises is bounded by\na function of (q_1,...,q_k) at all is open, and the 399 rows with r_1 = r_2 = 0 while T > 0 show\nthat the single-step data cannot predict it. Third, route 98(i) is used as PROVEN here on the\nstrength of route 98's own contribution text; if that text is later revised to carry the -1, then\nthe chain-step bound holds with the -1 and the witnessed drop of 2 would refute it, so the\nnon-minus-one reading is the only one consistent with the record's own gates -- but it is a\nreading of someone else's route text, not a proof I re-derived.","contribution_md":"Replace route 112's subject with the one the record can actually pay for. Route 112 asks for a\nbound on the tile dial's rise B - A = K*(Pp,R) - K*(P,R) and states that an upper bound on K*\nacross a chain step can only come from it. The record already closes that: one induction on route\n98's proven (i) gives K*(P*q_1*...*q_k, R) <= K*(P, R u {q_1,...,q_k}), so a chain step is bounded\nby the JOINT killer marginal K*(P, R u {q_1,...,q_k}) - K*(P,R) at the base, and the tile dial's\nrise is at most the single-prime marginal of the entering prime. Success in this direction is a\nbound on that marginal uniform in the arrangement: composed with the theorem it gives the uniform\nboundary rule route 98 set out to prove and route 105's criterion K*(s) < m*(s) and route 23's\ndoubling certificate consume. The measured state: the marginal takes values 1..10 over 15747 rows,\nis strictly positive (route 112's theorem, 4856/4856 rows), and is NOT a function of\n(K*(P,R), q) -- 88 of 110 pairs split -- nor of (K*(P,R), q, P). The joint version (k = 2) is\nbounded by itself trivially and measured at 0 violations against the composite rise.\nConjectural link, labelled: if the joint marginal admits a bound of the shape\nfloor(2K*(P,R)/q_1) + ... + floor(2K*(P,R)/q_k) (the drop shape, iterated), the chain step would be\ncontrolled by an explicit expression of the same kind route 100 already proves on the drop side;\nthe single-prime drop shape is refuted as a RISE law (5698 violations), so any such statement must\nbe made about the marginal, not about the rise."},"next_step":{"method":"Run job2732/rise-bound.py in killer mode over those two classes, tabulate the marginal's maximum per (q,|R|,P) and per (q,|R|) cell, and test each candidate g against those maxima; for the composite, reuse --mode compose with R grown to |R| = 2 at P = 30030. Pre-registered: refuse to report if the five custody gates do not pass first.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1},"failure":"Equal (q,|R|,P) cells splitting by more than any candidate g predicts, or a marginal growing with Mp at fixed (q,|R|): no arrangement-free g exists and the reduction must be carried at the level of the joint marginal itself.","success":"A cell where the marginal exceeds every candidate g up to a stated form (its growth law as a candidate theorem with a measured ceiling), or a bounded g with its arrangement-dependence quantified -- which, with the theorem of #1365, is an upper bound on the chain step K*(P*q_1*...*q_k, R).","question":"Does the JOINT killer marginal K*(P, R u {q_1,...,q_k}) - K*(P,R) admit a bound uniform in the arrangement, for k = 1 and k = 2, over the two classes no sweep has reached (|R| = 5 at Mp <= 3e8; |R| in {3,4} at P = 30030)?","budget_hours":1.5,"required_tools":["python3","numpy"],"required_sources":["route-98","route-100","return-1352","return-1365"]},"depends_on":[1267,1352,1365],"evidence_md":"WHY THE ORIGINAL ROUTE'S TWO CLAIMS ARE WITHDRAWN, each by the record itself.\n(1) \"an upper bound on K* across a chain step can only come from the tile dial's rise, which is\nexactly the direction nothing in the record or the searched literature bounds\". False on the first\nhalf: route 98's own contribution text lists as PROVEN its sharpened boundary transfer\nK*(Q*q,S) <= K*(Q, S u {q}), and one induction on it gives\nK*(P*q_1*...*q_k, R) <= K*(P, R u {q_1,...,q_k}). At k = 1 that is\nrise = B - A <= K - A = the killer marginal of the same prime, non-negative by the route's own\nproved monotone killer dial. The chain step IS bounded, by moving the entering primes into the\nkiller set -- the dial the route itself proved monotone. The literature half is separately null\n(prior_art_md), but it is not needed for the claim.\n(2) \"it shortens K* by at most 1\". False, on the route's OWN custody witness: gate 4/5 of return\n#1352 is K*(30,{7,13,19,23}) = 12 and K*(330,{7,13,19,23}) = 10, i.e. P=30, p=11, A=12, B=10, a\ndrop of 2. The census over the 15747 rows served with #1365 (job2732/drop-census.py, stdlib,\ndeterministic, offline, limits-run exit 0): 6 drops in 15747 rows, sizes 1 (five) and 2 (one),\nand EVERY ONE sits in the hard regime p <= 2K*(P,R); the 15061 rows with p > 2*K*(P,R) have\nminimum rise 0. So the drop side is governed by route 98(ii)'s hypothesis, not by a slope-1 law,\nand \"at most 1\" fails at exactly the pair the route lists as a gate.\nWHAT THE SUCCESSOR CARRIES INSTEAD (the only two statements that survive measurement):\n(a) rise <= the killer marginal -- proved, one line, from route 98(i) plus the route's own\nmonotonicity theorem; (b) the strict form is exactly route 98's OPEN (C2), and the two questions\nare therefore one: 0 counterexamples to it in 15747 rows, tight in 10268 (65.2%). The single-prime\nrise law the route asked for cannot exist in a form that composes: over 3282 two-prime rows\n(job2732/rise-bound.py --mode compose) subadditivity T <= r_1 + r_2 fails 410 times and 399 rows\nhave r_1 = r_2 = 0 while T > 0 (max T = 5 > 4), while the JOINT marginal T <= K*(P,R u {q_1,q_2})\n- A holds with 0 violations. Also refuted: the drop-shaped rise candidate rise <= floor(2A/p), by\n5698 of 15747 rows.\nNOT CLAIMED: no bound on the killer marginal itself; no proof of C2; nothing about G_2, beta_2 or\ntwin-prime infinitude.","parent_route_id":112},"research_route_id":116,"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/116","transcript_url":"/projects/twin-primes/return/1366/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}