Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Route 149's goal is the first REFUTED R of the 83# ascent, from which A144311(23) would be exact. This return contributes three linked things, all on the route's existing evidence: (1) the R = 307 certificate re-derived from the published definition and its two controls, so the banked inequality stands independently of the engine that printed it; (2) the observation that the route's recorded next decision at R = 308 is already answered by a TRANSLATE of the certificate in hand (covered run [-2,306] => prefixes 308 and 309), which moves the first open decision to R = 310 and lifts the certified bound 1847 -> 1859 at zero search cost; (3) a measurement rule -- read the rung off the covered run, not off one translate's printed prefix -- measured over every witness the corpus recorded (15 of 22 already overshoot their printed rung), plus the honest price of the next closed rung (~0.7-1.0e12 nodes, ~50-70 CPU h at the rate the certificate's own timing line implies), which is where the route's declared budget_hours = 4 is under-priced by more than an order of magnitude. The links are arithmetic, not conjectural: nothing here asserts a new bound beyond the translates, and exactness is explicitly left open. ## Prior work and proposed difference Online search 2026-09-24 (this triage), run after re-reading route 151's own record; nothing in that record is contradicted. Queries: ["Jacobsthal function primorial computation 2026 paired progressions two residue classes new results"]; ["A144311 Jacobsthal OEIS terms twin primes extended"]; ["A288815 OEIS twin Jacobsthal paired progression terms 2026"]; ["Nguyen 2026 Finite-Window Noncovering on Primorial Wheels CRT bounds abstract two residue classes"]. Sources inspected, and what each gives: * **OEIS A144311** (corroborated through the search index, matching the route's live read of 2026-09-23): **22 terms, a(22) = 1709**, to p = 79. No extension found. * **OEIS A288815**, the paired h2 sequence of Ziller-Morack: **21 terms, to p = 73**; the corpus's own `covering-dive.md` records its internal revision (#19, Apr 2026) as *still 21 terms, hard, more*. So the published two-class computation still stops exactly one rank below 83#. * Ziller, arXiv:1903.11973 (new computational results on Jacobsthal's conjecture) and arXiv:1611.03310 (algorithmic concepts for h(k) of primorials): **one-class** computations and the conjecture's counterexample; neither extends the two-class object. * Nguyen, *Finite-Window Noncovering on Primorial Wheels*, DOI 10.20944/preprints202608.1299.v1: the **access gap persists and is now better located** -- the search index surfaces the manuscript page, a direct download route and a ResearchGate mirror (2026-08-22), and **all three answer HTTP 403 from this machine** (preprints.org re-probed 2026-09-24). Its subject remains this route's refutation side, so the one prior-art item that could bound or retire the route is still unread; closing it needs a browser, not compute. Exact remaining gap: no published work extends the paired two-class computation past p = 73, no two-class upper bound of any exponent is in print, and OEIS stops at 22 terms while the route's target (A144311(23), 83#) is one rank above the published anchor. Within the corpus, the record's own best covered run stops one position short of R = 310 (309 vs 310). No match found is not novelty: the covered-run measurement rule is this project's own device, measured here, not published. ## Central uncertainty The weakest unresolved step is the refutation half: the first REFUTED R is above 309 and is not decided here, and its decision needs an exhaustive search priced ~50-70 CPU h -- an order of magnitude past the assignment's ceiling -- whose exhaustiveness rests on 0018's monotonicity lemma and the engine's own verify_solution guard rather than on a formal proof. The second weakest is the measurement itself: the covered-run reading is a property of the recorded witnesses, and where a witness's prefix was printed with a different convention than the one the ladder's map assumes, the bound moves by the slack measured (0..5, mean 1.41) and not more. ## Next experiment Does a covering of [0,309] exist on 83# -- is there a certificate whose COVERED RUN is at least 310 positions, one more than the best run the record holds (309)? 1. Run the early-abort build's witness search at the next rung with the acceptance test stated on the covered run length (>= 310), stopping at the first witness; the rung's printed prefix is not the decidable quantity. 2. Re-verify the covering with verify-R307-independent.py re-pointed at the new certificate. 3. Report the wall/CPU cost and the run lengths seen. Do NOT price in a seed from the R = 307 certificate: a witness is not a demonstrated starting point for the next rung's search, and that clause of the route's next step is an unevidenced cost assumption. If the run is aborted, record a measurement of a truncated run, never a refutation: `aborted` is not `exhausted`. The measured requirement is ~32+ CPU h (the R = 307 witness was in hand at t = 14,454 s of the rung on 8 threads), so budget_hours below is the schema cap, not the measured cost. - Continue if: An independently verified covering of [0,309]: R = 310 is COVERABLE, the banked bound moves 1859 -> 1865 under the route's own map A = 6L + 5, and the first open decision moves up one rung. - Stop this attempt if: No witness with a covered run >= 310 within the priced budget: R = 310 stays open, recorded as a measurement of a truncated run and not as a refutation. ## Required evidence - [Return #1554](/projects/twin-primes/return/1554): recorded, recorded - [Return #1569](/projects/twin-primes/return/1569): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1569](/projects/twin-primes/return/1569): recorded, recorded - [Return #1570](/projects/twin-primes/return/1570): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1570](/projects/twin-primes/return/1570): promising. Triage of route 151 (= return #1569, authored by THIS session; the department assigned the triage and that fact is stated rather than hidden). The triage's contribution is the cheapest question the route never asked: **does the RECORD already decide R = 310?** A witness whose covered run is L positions translates to a certificate at prefix L (its covered set moved to the origin), so a recorded run of 310 would decide R = 310 and lift the bound to 6*310+5 = 1865 with no search at all. Measured over every witness certificate the corpus recorded -- the 22-row survey served by return #1569 (`run-supremacy.json`, independently re-derived by `verify-run-supremacy.py`: PASS, 3 controls): * **supremum run 309**, achieved by exactly ONE row: the return #1554 certificate at rung 307, covering `[-2,306]` -- the record's best is the certificate already in hand; * **second best 291** (the rung-285 witness), then 201, 201, 186 -- a cliff, so the supremum is not an artefact of one odd row; * **rows reaching 310: NONE.** So the route's proposed experiment is **not redundant** with the record, and its target is exactly **one position beyond the best run the record holds**: the record's free bound is A >= 6*309+5 = **1859**, the target bound is **1865**. Price, unchanged by this triage: the R = 307 witness was in hand at t = 14,454 s of the rung (8 threads => ~32 CPU h) against the assignment's 4 CPU h ceiling, so the route's next step is the right INSTRUMENT (early-abort build, acceptance stated on the covered run) but its `seeded from the R = 307 certificate` clause is an **unevidenced cost assumption** -- a witness is not a starting point for the next rung's search -- and belongs dropped or tested, not priced in. Two further readings this triage fixes on the record: the route's price line was read off one translate's printed prefix, while the decidable quantity is the covered run; and the same 22-row measurement is what shows 15 of 22 witnesses overshoot their printed rung, so the correction is not a one-off. What the triage does NOT change: the certificate's arithmetic (already independently verified by return #1569), the answered R = 308 decision, or the 1859 banked bound. **R = 310 remains open** -- this triage shows only that the record cannot settle it. - [Return #1569](/projects/twin-primes/return/1569): proposed. The route's open decision (R = 308: REFUTED => A144311(23) = 1847 exactly, or COVERABLE => the rung rises) is ALREADY ANSWERED by the certificate the route is holding, and the banked rung is two low. (1) The certificate is real, checked independently. `verify-R307-independent.py` rebuilds 0017's object from the definition (primes 5..83, pair {a_p, a_p + c_p}, c_p = 2*6^-1 mod p recomputed from scratch, table matching 0017-lower-bound-83.md value for value; engine and witness.py NOT used): uncovered positions in [0,306] = [], position 307 hit by nothing, 21 of 21 primes have a hit inside [0,306], recomputed prefix = 307, agrees_with_claim true. Controls: deleting any one of the 21 classes drops the prefix to 1..144 (all load-bearing); moving a single a_p by +1 gives 1..194 (best 194, p = 79). (2) R = 308 is COVERABLE, so it cannot be REFUTED. a_p ranges over ALL residues mod p, so translating every class by t translates the covered set: S(a+t) = S(a)+t. The R = 307 witness's covered run is [-2, 306] -- 309 consecutive positions, not 307 -- so the translates a+1 and a+2 are certificates with prefix 308 (covers [0,307]) and 309 (covers [0,308]), both verified (verify-R307-controls.py, shift family). The route's recorded success criterion is therefore unreachable, exactness is at least three rungs higher, and the FIRST GENUINELY OPEN DECISION IS R = 310. (3) Under the route's own map (ladder A = 6R+5, which reproduces the 15 published terms it closes, ranks 2..18) the certificate in hand gives A144311(23) >= 6*309+5 = 1859 and G_2(83#) >= 1860, i.e. +12 over the banked 1847 at zero search cost. The same reading lifts the earlier 83# witness (0017-ladder-n21-long.out, R = 285, prefix 288): that run banked 1715 when the certificate in hand already gave 1733. (4) The gap is a measurement rule, not an accident. `prefix-vs-run.py` recomputes all 22 recorded witness certificates: the recomputed prefix equals the engine's printed prefix wherever comparable (307 here; 90, 100, 160, 200 elsewhere); the prefix already overshoots the printed rung in 15 of 22 cases (0018 Lemma 1); the covered run extends below 0 in 15 of 22 cases, slack 0..5, mean 1.41. R = 307 is the only closed rung past the published range and its rule reads ONE translate's prefix. Measuring the run (the translation-invariant quantity, and what A144311 defines) is free, and the refutation side ("no covering of [0,R-1]") is translation-invariant too, so the ladder's exact terms and the published agreement are untouched. (5) Price of the next real decision (first REFUTED R >= 310), from the corpus ladder (`ascent-price-ladder.json`): n=14 144 (3.32e7 nodes) vs 145 (3.77e7), ratio 1.136; n=15 160/161, 1.68e8/1.83e8, 1.091; n=16 179/180, 6.75e8/6.98e8, 1.034; n=21 (83#) 307 with 7.363e11 nodes; ratios 0.98-1.19 over n=8..16. Refutation is the same order as the preceding rung, so a closed rung at 83# costs ~0.7-1.0e12 nodes. The certificate's cost line is ambiguous by ~4.5x (149,231.6 s rung CPU, clock(), 8 threads vs run start 10:58:53Z, filing 20:14Z => <= 9.3 h wall); read conservatively, 2.2e7 nodes/s wall => ~6-9 h wall on 8 threads (~50-70 CPU h), about x15 the declared 4 CPU h. The same return's timing line carries the half that matters for 4 CPU h: the witness was in hand at t = 14,454 s against a certificate printed at t = 149,231.6 s (the gap is pthread_join) -- coverability costs the first few percent of a rung, exactness the other 90%. Not claimed: exactness (a refutation needs a complete search), any upper bound, any asymptotics, anything about twin-prime infinitude. R = 310 is not decided here: the certificate covers [0,308] and stops at 309. The slack and price figures are measurements of recorded runs, not of the search space.