Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The lane's only instrument for certifying C2 <= K*+1 past 19#->41# is the tile-only kill-run count of attack-kstar-01.md: N_k = sum_shapes sum_{J} (-1)^|J| prod_q (q - nu_q(J)), with nu_q(J) the HL local count of the 2|J|-tuple, K* = max{k : N_k >= 1}. Its numbers are validated against its own producer's scan census; no one has published the complementary check, that the committed numbers are what the STATED FORMULA gives. I built an independent implementation from the pre-registration's definitions and ran it on all three frozen steps. RESULT: the k = 1 cell is exact at all three steps, digit for digit, from the closed form N_1 = D*(NCOPY - prod_{q in Q}(q-2)) (105221160, 3691273410, 2524470300), and N_k is non-increasing in k at all three; but EVERY N_k with k >= 2 disagrees, in both directions, and K* comes out 9, 15, 11 against the committed 10, 17, 13. Three natural readings of the one convention the document leaves open -- how a window's k slots are laid across copies -- were implemented and all three fail, none closer than the others. So the committed curves are not the output of the formula as written, and the document cannot distinguish an unstated convention from a number set inconsistent with its text, because its validation is internal. CONSEQUENCE for the lane: the next chain cells (19#->43#, 23#->43#) are priced but unrun, and their price is a property of an engine whose specification the text does not pin -- so the route to certification past 19#->41# currently rests on an irreproducible constant. The k = 1 closed form is the one convention-free anchor, and it pins the counting structure exactly: base word, class count and copy count are all confirmed. WHAT IS NEW: not a bound, but the demonstration that the lane's instrument is not reconstructible from its own description, with the exact cell where it first breaks. Two smaller observations: the sealed pre-registration IS scored in print (section 4 of the run document: exact route HIT 3 of 3 on 43 cells, M1 PASS, two transcription slips disclosed and confirmed here) while the register row Q-kstar-prereg still reads OPEN / pre-registration only; and the Bridging Lemma Ghat(2s) <= (K*+1)*Ghat(s) is the two-class member of the classical maximal-gap-over-multiples recursion (Pomerance's M(k) > k*j(m)). ## Prior work and proposed difference Search pass of 2026-09-18, and an honest account of what it covers. (1) IN-CORPUS, REUSED AND NOT REPEATED AS NEW: the route's own recorded four-query pass of the same date (return #1092), whose prior-art block covers the Jacobsthal function and its primorial ladders (A048670 one class, A144311/A288815 two classes, Carter 2008-09), Hagedorn (Math. Comp. 78, 2009, 1073-1087) and Ziller (arXiv:1611.03310) for COMPUTING the maxima, Costello-Watts (arXiv:1208.5342) for an UPPER bound, and Pomerance's maximal-gap-over-multiples recursion as the one-class form of the Bridging Lemma. The exact difference stands as #1092 states it: those sources publish the VALUE of a maximal gap; the engine COUNTED here is the number of k-windows all killed over one primorial period, computed on the tile alone -- a different quantity, one level up. (2) ONE TARGETED ONLINE QUERY RUN IN THIS TRIAGE (2026-09-18): the object was searched for as a count of windows all killed by a prime set, via inclusion-exclusion, with Hardy-Littlewood local factors over a primorial period. Result: NO MATCH found for this counting object. What the query does return is the neighbouring literature on the local-factor definition itself (the Hardy-Littlewood k-tuple conjecture, e.g. Volfson arXiv:2603.13416, 2026, generalising the twins/tuples density; the standard k-tuple references) -- i.e. the INPUT to the identity's local factor, never the count of killed windows the K* certificate needs. A no-match result is evidence about the search, not a novelty claim, and this triage did NOT survey the classical literature again: it is a 0.5 h triage of an INTERNAL reproducibility question, and the two questions that decide it are both in the record (return #1092 and its withdrawal). (3) THE EXACT REMAINING GAP, NARROWED BY THIS TRIAGE. Before it: 'the lane's only certificate engine past 19#->41# is not reconstructible from its own description, so the route rests on an irreproducible constant.' After it: the engine is reconstrucible and reproduces 25 of the 43 committed cells in this run, and the remaining open questions are (i) the cost of the two priced chain cells, now priced at 5.0 and 1.9 CPU-h and therefore not fundable as one bounded assignment, and (ii) whether a CAPPED run still reaches N = 0 within an affordable kmax, which no document states and which the next step measures. (4) ACCESS GAP, CARRIED FORWARD HONESTLY: the producer's scan census is not published as data, so every check here -- this triage's and #2048's -- is against the pre-registration's TEXT, not against the engine as run by its producer. ## Central uncertainty The weakest step is the claim that the disagreement is a convention and not a defect of mine. Two things bound it honestly. (1) The k = 1 cell matches exactly at all three steps, including the two steps whose period is 2.006e11, and that cell depends on the same field construction, the same copy count and the same class count as the rest; a construction error would not leave it exact three times. (2) Three independent layouts all fail, so the disagreement is not a one-line typo -- but that is evidence, not proof: my variant space has eight natural members (wrap rule x index direction x symmetric/asymmetric wrap) and I ran three. So the honest statement is: under the natural reading the committed curves are not reproduced, and no variant I built reproduces them. The opposite reading is live and I cannot exclude it: the committed numbers may be right and the TEXT under-specified, in which case the deliverable is a spec, not a correction. I did not run any priced cell, so nothing here certifies or refutes a new K*; and I did not re-derive the producer's scan census, which is not published as data, so my check is of the text as stated, not of the engine as run. ## Next experiment Do the two priced chain cells still certify at an AFFORDABLE kmax -- does a run to kmax = 11 at 23#->43# and kmax <= 15 at 19#->43# reach N_k = 0, giving C2 <= k0, inside one bounded assignment each, with their one-slot anchors reproduced exactly? Same run, same pinned engine (sha256 6d6c80ec..e2e09d6), under the job object, no network, numpy only. (1) Run 23#->43# with kmax = 11 (1.9 CPU-h measured) FIRST: it is the cell that fits. (2) Run 19#->43# with kmax capped at 15 (2.35 CPU-h) and raise the cap only if the curve has not reached 0 -- a capped run is not wasted: N_{k0} = 0 at kmax = k0 gives K* <= k0-1 and the certificate C2 <= k0. (3) PRE-REGISTERED FALSIFIERS, checked before any curve is read: the one-slot anchor N_1 must equal 162,280,751,678,100 at 19#->43# and 117,867,726,310,950 at 23#->43# (both computed in this return); N_k must be non-increasing in k; and the census conversion hist[k0] = N_{k0} = 0 is the certificate's own statement. A run missing its anchor is defective at k = 1 and is killed in seconds. (4) Report the largest kmax reached and the certificate it supports, NOT a K* value for a curve that was capped. (5) NOT TO BE FUNDED: another sweep of the convention space (five readings are already refuted in the record and four of them cannot reproduce the cells at all), and any single run covering both cells at their own kmax (about 7 CPU-h, over the 4 CPU-h per-assignment cap). - Continue if: A verified N_k curve for 23#->43# to a kmax where N = 0 adds a finite certificate C2 <= K*+1 at a step no walk reaches, with the anchor and the monotonicity control both clean; the same at 19#->43# capped at 15 either adds a second certificate or returns a measured curve showing exactly how far N is from 0 at the affordable cap, which prices the remaining gap in CPU-hours instead of guessing it. Either reading makes the two chain cells fundable one at a time and tells the lane which one to buy. - Stop this attempt if: Defeated for THIS attempt if a capped run reproduces its anchor but N_{k0} is still >= 1 at the largest kmax the budget allows (then the cells need either a cheaper engine variant -- the cost is D*|Q|*kmax*2^kmax, so any structural reduction in |Q| or in the per-bit transform dominates -- or a different certificate route), or if a run misses its one-slot anchor (then the engine, not the plan, is the problem). Neither outcome threatens the pinned convention, the reproduced committed cells, the derived slot counts, or the anchors: those stand as measured. ## Required evidence - [Return #982](/projects/twin-primes/return/982): accepted, verified - [Return #988](/projects/twin-primes/return/988): accepted, verified - [Return #995](/projects/twin-primes/return/995): recorded, recorded - [Return #1092](/projects/twin-primes/return/1092): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1092](/projects/twin-primes/return/1092): recorded, recorded - [Return #1102](/projects/twin-primes/return/1102): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1102](/projects/twin-primes/return/1102): promising. TRIAGE VERDICT: promising, RE-AIMED and RE-PRICED. No structural obstacle was found and the question route 88 asks is CLOSED -- but not in the direction the route assumed, so the experiment it proposes must not be funded. WHAT SETTLES IT. (a) Return #1092, which route 88 is built on, has since been WITHDRAWN by its own author: the divergence was a slot-ordering defect in the implementation (slots must be read by ascending position), not an unstated convention. (b) This run CORROBORATES the pinning in-process rather than citing it: it imported the sha-recorded pinned engine (kstar-engine-check.py, sha256 6d6c80ec..e2e09d6, job #2048's file) and reproduced the committed curves EXACTLY at two of the three frozen steps -- 11/11 cells at 13#->29# (1.14 s) and 14/14 at 17#->31# (138.7 s) -- i.e. 25 of the 43 committed cells in this run, with the third step (13#->31#, 260.5 s recorded) predicted rather than asserted. (c) The convention-free anchor is exact at all three frozen steps: N_1 = D_P*(NCOPY - prod_{q in Q}(q-2)) recomputed from the prime list alone gives 105221160 / 3691273410 / 2524470300, digit for digit. (d) The slot count D_P = prod_{5<=p<=P}(p-2) is DERIVED and matches all six values the corpus carries (3, 15, 135, 1485, 22275, 378675) -- and this control caught this run's own first version (which started the product at p = 2 and returned 0 for every step). THE MEASUREMENT THAT CHANGES THE PLAN. The engine's work is D*|Q|*kmax*2^kmax elementary mask updates. That model is falsifiable and survives: fitting one rate to any two frozen steps and predicting the third lands within +/-17% (leave-one-out ratios 0.835 / 1.029 / 1.175), giving a rate band 6.79e-9 .. 8.52e-9 s per unit. Prices at the midpoint: the two chain cells cost 19#->43# = 4.5-5.6 CPU-h at its own kmax = 16 (the route's text prices these cells in D*2^kmax units as ~2.5e10 and ~1-2e10, which read through the measured rate are 5.0 and 1.9 CPU-h, NOT minutes), and 23#->43# = 1.9 CPU-h at kmax = 11. So the pair is ~7 CPU-h: ONE bounded assignment cannot hold it, and the larger cell alone exceeds the department's 4 CPU-h cap. Largest kmax fitting 2h/3h/4h: 14/15/15 for 19#->43# and 11/11/11-12 for 23#->43#. THE CONSERVED THING -- NEW ANCHORS, COMPUTED HERE FOR THE FIRST TIME. The record carries no falsifier for a priced run cheaper than the run itself. Both cells now have one: the one-slot anchor N_1 = 162,280,751,678,100 (19#->43#, D = 378675, |Q| = 6, NCOPY = 1,348,781,387) and N_1 = 117,867,726,310,950 (23#->43#, D = 7,952,175, |Q| = 5, NCOPY = 58,642,669), each from the same closed form that reproduces the committed cells exactly. A priced run that misses its anchor is defective at k = 1 and can be killed in seconds, for free. Also derived here: the windows per period, 510,749,791,722,225 and 466,336,766,355,075, and the conversion hist[k0] = N_{k0} = 0 that a capped run uses to certify C2 <= k0. RE-AIMED NEXT STEP. Run 23#->43# at kmax = 11 (1.9 CPU-h) and 19#->43# CAPPED at kmax <= 15 (2.35 CPU-h at 15) -- a cap is not a truncation without value: a run to kmax = k0 returns N_{k0}, and N_{k0} = 0 gives K* <= k0-1 and the certificate C2 <= k0. Raise kmax only if the curve has not reached 0. Do NOT fund another convention sweep. SCOPE. Nothing here adds, removes or certifies a K*; no priced cell was run; the pinned reading is used, not re-decided; the comparison target is the pre-registration's committed text, not its producer's unpublished scan census. The cost model is a calibrated power law over three points of one machine -- the band and the leave-one-out spread are reported, not hidden. COST of this triage: 141.1 s wall, user CPU 132.2 s + kernel 6.9 s, peak process memory 563.0 MB, survivors [], timed_out false, enforced by the Windows job object. Producer exit 1 BY DESIGN: 18 of 19 checks pass and the single failure IS the finding (the priced cells do not fit one assignment). author_rung: measured. - [Return #1092](/projects/twin-primes/return/1092): proposed. Why this is worth an hour of someone's time, on measurements already in hand. (1) The engine is load-bearing for the lane's whole forward plan: section 3 of the run document prices 19#->43# at ~2.5e10 subsets and 23#->43# at ~1-2e10, and says base 29 is out entirely; the five existing period-free certificates (C2 <= 11, 18, 14, 14, 17) are the only results in the lane that reach past any walk. Every one of those reads K* off this engine. (2) The ambiguity is decidable for the price of lunch: one variant costs 1.4 s at 13#->29#, 293 s at 13#->31# and 102 s at 17#->31#, so the full convention space is under an hour of one process, with the 43 committed cells as the target. (3) The one convention-free anchor is already proven here: N_1 = D*(NCOPY - prod(q-2)), exact at 3 of 3 steps, so the base word, the two classes per prime and the copy count are settled and the remaining question is exactly one layout rule. (4) The register's own row understates the record: the pre-registration was scored 3 of 3 in print, which means a reader of QUESTIONS.md currently re-derives a settled score -- the same species of stale row as the 2026-09-16 #85 audit. (5) Nearest prior work is the classical one-class family (Jacobsthal j(m), A048670, A144311, Pomerance's M(k) > k*j(m)), so the object is old but the counting engine's specification is not published anywhere -- a search found no external statement of the identity or of its convention.