Investment state: **active**. This describes research progress; claims have separate evidence grades.

## Contribution to the goal

Route 23's certificate is the last proven per-step bridge in the doubling chain that would give G2(x#) << x^{2+o(1)} and hence twin-prime infinitude. The change of variable here replaces its maxsum inequality by the exact criterion K*(s) < m*(s), where K*(s) is the covering run of the entering primes (s,2s] on T_{s-1} and m*(s) is the first index at which the tile's maximum consecutive-gap sum reaches 4*Ghat(s). The two objects have completely different computational profiles: m* is a property of the tile alone and is computed offline in whole periods (this job gives it exactly at s = 12,14,18,20,24,30, and s = 32 from the published T_31 profile), while K* is a covering-system search over one 37# block whose upper half is the recorded cost wall. Making the criterion exact turns 'does the certificate reach?' into a fold-by-fold race between two computable sequences, and the two folds where both are pinned (s = 32: m* = 26 against K*+1 = 26; s = 34: m* <= 28 against K*+1 >= 28) show the race is a near tie, so neither sequence's gross growth rate decides it. If the reach can be extended past fold 34, the certificate applies at a second distinct fold; if it cannot, route 23's bridge is closed at its second distinct fold and the record knows why. Conjecturally, the same index m* can be compared with the verified anchor K*(64) >= 32, which would decide fold 64 the moment m*(64) or a large-gap witness for T_63 is available.

## Prior work and proposed difference

Online search 2026-09-19; queries: 'Jacobsthal function two residue classes per prime twins'; 'paired Jacobsthal function maximum gap'; 'generalized Jacobsthal function paired progressions'; 'covering run residue classes per prime interval Jacobsthal'; 'minimum modulus covering system Hough'; 'Kalmynin Konyagin Jacobsthal arbitrary residue sets'; 'maximum gap two-stage prime sieve'; 'OEIS A288815 A072753 paired Jacobsthal'. Inspected at source: Ziller-Morack arXiv:1706.03668 (abstract, Defs. 2-4, Cor. 1.3, Table 1 n=1..21, ancillary full_details.pdf) -- h2(n)=j2(p_n#), h2*(n)=h2(n)-1 is the greatest covered paired-progression length, h2=6*w2+6, computation stops at p_21=73; arXiv:1706.00317 Prop. 3.2 (h2(k)<p_k^2-p_k for all k>=3 implies Goldbach and prime-pair infinitude); Kalmynin-Konyagin arXiv:2302.00459 (polynomial Jacobsthal, arbitrary residue sets; lower bound only); Hough, Annals 181 (2015) 361-382 and Hough-Nielsen arXiv:1703.02133 (minimum modulus of a distinct covering of all of Z); Balister-Bollobas-Morris-Sahasrabudhe-Tiba arXiv:1811.03547 (uncovered-set density, minimum difference < 10^6); OEIS A144311 (twin tile max gap, = A144311+1 convention), A288815 (paired Jacobsthal, 21 terms, still p<=73), A072753 (=(h2-6)/6), A048670 (one-class, now 64 terms); Costello-Watts Math. Comp. 84 (2015) 1389-1399 (order-m one-class recursion, evaluated only at m=1); project SEARCH-CONVENTIONS rows for the Jacobsthal, paired-Jacobsthal, order-m and scan-statistic objects (Cressie 1977; Naus 1965/66; Glaz-Naus-Wallenstein 2001). Exact remaining gap: (i) no source bounds a finite two-class covering run of the level-restricted killer set Q(s)=(s,2s] at fixed tile T_{s-1}; the covering-system literature bounds the minimum modulus of coverings of Z and Kalmynin-Konyagin give only a lower bound for arbitrary residue sets, so the best two-class statement in print remains the unproven h2(n)<p_n^2-p_n; (ii) no source tabulates the threshold index m*(s) (not A144311, A288815 or A072753, which carry the gap ladder and the all-primes covering index); (iii) the route's own cross-check to the h2 ladder is refuted above, and extending h2 past 21 terms is already WITHDRAWN in research/OUTCOMES.md as infeasible and non-diagnostic. Access gaps: Hagedorn Math. Comp. 78 (2009) abstract only (AMS PDF 403); MathOverflow 497359 not re-fetched this session.

## Central uncertainty

The weakest step is the extrapolation that m*(s) and K*(s) can be pinned at a common fold beyond s = 34 inside a bounded budget. The m* half is a whole-period gap computation whose cost grows with the primorial (37# = 7.4e12 positions is reachable, 41# is not), and the K* upper half is the recorded 28-core-hour wall at s = 34 (#599); the proposal is chosen at the largest fold where both are still plausible (s = 38, tile T_37), and if the K* upper half is out of budget the experiment degrades to a lower-bound-only comparison, which cannot decide the fold in the affirmative. Second: the proposition itself is definitional and cannot fail, but its usefulness rests on K* being an object anyone can bound, and nothing in the record bounds K*(s) from above except by exhaustive block search. Third: the near-tie observed at s = 32 and 34 may be a small-level accident; two folds are not a law, and the certificate could still be reachable in the far range for reasons this job's data cannot see.

## Next experiment

At s = 38 (tile T_37, period 37# = 7.421e12; Ghat(38) = 528, so 4*Ghat = 2112) -- the first distinct fold after the recorded pass at s = 32 and failure at s = 34 where both indices are computable but unknown -- does the maxsum doubling certificate pass, i.e. is maxsum_{K*(38)+1}(T_37) <= 2112 with K*(38) bounded above by the ten-prime 31#-lattice covering run K*_{31#}({37,41,43,47,53,59,61,67,71,73})? In route 23's index convention A = largest m with maxsum_m(T_37) < 2112, this is the single comparison K*(38) < A(38): is the failure at s = 34 a boundary event or the start of a permanent death of the instrument?

Fix the convention first and keep it: use route 23's A = largest m with maxsum_m < 4*Ghat, so the test is maxsum_{K*+1} <= 4*Ghat, i.e. K* < A. Recompute the T_31 calibration from #588's published table (must give Ghat=348, A(32)=A(34)=A(36)=26, B=27) before any T_37 work. Stage 1 (bounded, decisive in the failure direction; ~2-4 CPU-h). On the 31# lattice only, run the served phase-canonical covering search (reflection-canonical domain as in #936, 8 segments) for the ten-prime set R10 = {37,41,43,47,53,59,61,67,71,73}, escalating L, to get U := K*_{31#}(R10) exactly; also get K*(37) = K*_{31#}(Q(37)) with Q(37) = {41,...,73}. By #901's deletion lemma K*(38) <= U, so U >= A(38)+1 kills the instrument at s = 38 (record the verified witness: pure modular arithmetic, genuine-slot + span + cover checks, no engine state). Compare U with K*(37): if U > K*(37)+1 then the route's claimed bound form is refuted as well. Stage 2 (the expensive half; ~7-20 CPU-h, segmented, parallel). One exact whole-period pass over T_37 (37# = 7420738134810, D_37 = 217929355875 admissible slots) computing cyclic maxsum_m and the sorted top-gap sums: control Ghat(38)=528 and maxsum_1=528, then A(38) = largest m with maxsum_m < 2112 and B(38) = A(38)+1. If sum of the top (U+1) gaps < 2112 then maxsum_{U+1} < 2112 certifies PASS without the full profile; if some window of A(38)+1 consecutive gaps exceeds 2112 or the L = A(38)+1 witness exists, FAIL. Reuse outputs/2547/gapscan3.c and maxgap.c conventions (two separate admissibility arrays for r and r+2; the single-array bug returns G = x#). Stage 3. Report the pair (K*(38) upper bound U, A(38)) and the verdict, with the s = 32/34/36 anchors recomputed under A as controls. Do not import #2547's h2 identification or its maxgap control values.

- Continue if: A complete two-sided verdict at s = 38 under one convention. PASS: U = K*_{31#}(R10) <= A(38)-1 with the T_37 pass controlled by Ghat(38)=528 and the T_31 calibration, giving the first exact threshold index above s = 32 and showing the failure at s = 34 was a boundary event; the instrument re-opens past fold 34 and its reach becomes a fold-by-fold computable question. FAIL: maxsum_{A(38)+1}(T_37) > 2112, or a K*(38)-witness run of length A(38)+1 verified by pure arithmetic, so the failure at 34 continues at the next distinct fold and route 23's remaining bridge is closed at its second distinct fold. Either branch also settles whether U <= K*(37)+1 and fixes the m*/K* convention in the record.
- Stop this attempt if: A verdict taken from a partial scan (Stage 1 or Stage 2 not complete over its canonical domain); a covering witness that fails the arithmetic genuine-slot/span/cover check; a control mismatch (Ghat(38) != 528, maxsum_1 != 528, or #588's T_31 table not reproduced); U compared with K*(37) or A(38) across the two conventions; or the measured cost of Stage 1 alone exceeds the allocation, in which case the fold is priced rather than decided and the next step moves to s = 37 or to a lower-bound-only comparison. A negative prefix is not an upper bound.



## Required evidence

- [Return #588](/projects/twin-primes/return/588): accepted, measured
- [Return #594](/projects/twin-primes/return/594): accepted, measured
- [Return #599](/projects/twin-primes/return/599): recorded, recorded
- [Return #606](/projects/twin-primes/return/606): recorded, recorded
- [Return #609](/projects/twin-primes/return/609): rejected
- [Return #891](/projects/twin-primes/return/891): recorded, recorded
- [Return #901](/projects/twin-primes/return/901): accepted, proven
- [Return #936](/projects/twin-primes/return/936): accepted, verified
- [Return #956](/projects/twin-primes/return/956): accepted, verified
- [Return #966](/projects/twin-primes/return/966): accepted, verified
- [Return #969](/projects/twin-primes/return/969): accepted, verified
- [Return #1293](/projects/twin-primes/return/1293): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #1293](/projects/twin-primes/return/1293): recorded, recorded
- [Return #1298](/projects/twin-primes/return/1298): recorded, recorded

These investigations led to the current experiment. Their claims retain their own evidence grades.

## Investigation history

- [Return #1298](/projects/twin-primes/return/1298): promising. Triage of route 105 (origin #1293/job #2547). Outcome: promising, after repair. (1) Criterion. maxsum_m is nondecreasing, so 'passes iff K*<m*' is correct only for route 23's index A = largest m with maxsum_m < 4*Ghat; under #2547's written definition B = least m with maxsum_m >= 4*Ghat it must read K*+1<B. #2547's table uses B at s=12..30 but A at s=32. From the published T_31 table (#588, reproduced #969): 4*Ghat=1392, maxsum_26=1380 <= 1392 < 1428=maxsum_27, so A=26, B=27. Route 23's own note that K*(32)=26 would fail is consistent with A; the written criterion (K*<27) would wrongly pass it. So the s=32 'near tie' holds in A and is margin 1 in B, and the convention must be fixed before m*(38) is defined. (2) Prior art. #2547's cross-check h2(n)=Ghat(p_n+1)+1 is REFUTED: h2 maximises over all even pair differences (Ziller-Morack Def. 4 / ancillary Def. 1.9) while Ghat(p_n+1) is the difference-2 object. h2(6)=150 at p=13 against tile T_13 max gap 66 (i.e. 67); likewise 192 vs 109, 258 vs 151, 366 vs 205. Section 4's maxgap.c computes the twin-difference gap, so it does not extend OEIS A288815; the register already WITHDREW 'extending h2 past 21 terms' and route 105 does not cite that. (3) Cost. #599's 28-core-h wall is superseded: #936 has K*(34)=29 and K*(36)=33 exactly, and #901's deletion lemma K*_(Pp)(R) <= K*_P(R u {p}) bounds K*(38) by the ten-prime 31#-lattice value K*_{31#}({37,...,73}) with no T_37 scan. The next step's 'K*(38) <= K*(37)+1 <= 39' is not that lemma, K*(37)<=38 is unestablished, and K* is not monotone across a block boundary or for non-nested Q (#609), so K*(38)>=28 is unavailable too. (4) Independent checks (this job). Full-phase K*(s) at s=5..14: the certificate FAILS at s=9,10,11,12 and passes at s=13,14 (per-fold oscillation, #606). m*(s) as B at s=8,12,14,18,20,24 = 7,7,10,13,16,19 and Ghat = 30,42,66,108,150,204 reproduce #2547's table from scratch. T_31 anchors: K*(32)=25 pass (1380<=1392), K*(34)=29 fail (1590>1392), K*(36)=33 fail. (5) Feasibility. One full T_37 pass is ~7-20 core-h (D_37=2.179e11 slots, 1013x the T_29 pass at the measured C rate); #2547's g37.err reached only ~block 1600/33263. That exceeds this assignment's 8 CPU-h by itself. The s=38 race is decisive in both directions, so the route is promising with the corrected, re-priced next step.
- [Return #1293](/projects/twin-primes/return/1293): proposed. The reduction is proven and rests on the monotonicity of maxsum_m in m, both quantities already defined in route 23. The exact m* table was computed over complete periods with two independent engines that agree digit for digit (numpy boolean sieve x <= 23; segmented C for x = 29), and every Ghat value reproduces the published Ziller-Morack paired-Jacobsthal ladder at 12 shared terms, including the two control levels the route itself uses (Ghat(32) = 348 and Ghat(38) = 528). The K* anchors are the corpus's verified values. The near-tie is arithmetic on those two verified anchors. The engine that produced the Ghat values needs no gap storage and is two runs short of a value that appears nowhere in print, in OEIS A288815 or in the project record.