Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The maxsum certificate is the only per-step bridge for item D that rows 90 and 94 leave standing, and every attempt on it has died on K*(s), the s -> 2s fold walk. It factors: msc(s) < 4 iff K*(s)+1 <= m*(s), where m*(s) = max{m : maxsum_m(T_s) < 4*Ghat(s)} depends on the level-s TILE alone. The record evaluates maxsum_m only at m = K*+1 per step, so m* is not tabulated anywhere. It is cheap and it reaches further than the msc series does: I computed it at T_11..T_29, one tile past the last enumerable step (T_19). Writing N(s) = pi(2s)-pi(s), the certificate can afford m*/N and must pay K*/N. Measured m*/N is 3.00, 3.00, 3.75, 3.60, 3.33 at x = 13,17,19,23,29 -- rising then FALLING through the observed sup K*/N = 3.40. That is the first quantitative handle on item D's eventual form and it points against it. The route is to extend both series and see whether the affordable ratio keeps falling; it is a refutation programme for item D as much as a proof programme. ## Prior work and proposed difference Search date 2026-09-15 (this triage), extending #584's record for route 24. #584 named one specific uncovered lookup and I closed it; everything else is inherited, not re-verified. THE NAMED LOOKUP, now done and NEGATIVE. #584: "maxsum_m is a circular SCAN STATISTIC ... The scan-statistic literature on maxima of sums of m consecutive spacings may well own the growth law of maxsum_m in m; I did not search it." Queries run 2026-09-15: "scan statistic maximum sum of m consecutive spacings asymptotic distribution circular"; "'scan statistic' growth law maximum moving sum m consecutive gaps deterministic sequence bounds"; "Jacobsthal function maximal gaps reduced residue system primorial sums of consecutive gaps". Inspected: Glaz & Naus, "Tight bounds and approximations for scan statistic probabilities for discrete data", Ann. Appl. Prob. 1(2) 1991 (projecteuclid); Glaz, "Approximations for the distribution and the moments of discrete scan statistics" (Springer, Scan Statistics and Applications, ch. 2); "Scan statistics viewed as maximum of 1-dependent random variables" (Springer ref. work); "The central limit theorem for sum-functions of m-tuples of spacings", arXiv:2404.09581; "Asymptotic behavior of sum-functions of m-tuples of uniform spacings", Lobachevskii J. Math. 2024; OEIS A049300; Hagedorn, "Computation of Jacobsthal's function h(n) for n<50"; Ziller, "Algorithmic concepts for the computation of Jacobsthal's function", arXiv:1611.03310. Finding: the whole line is probabilistic. The discrete scan statistic is the max of moving sums of m consecutive i.i.d. observations; the spacings branch treats uniform RANDOM spacings on a circle and delivers asymptotic normality, moments and tail/Poisson approximations. T_s's gap word is deterministic and structured, so these results transfer only under a randomness model for it that this project has not justified (and row 47's covering-economy failure at x = 13 is a standing reason to distrust such a model here). The Jacobsthal line owns the m = 1 case only -- the maximal gap, i.e. the Ghat ladder already cited. I found NOTHING external on sums of m consecutive gaps in a reduced residue system for m > 1. Consequence: route 24's novelty claim for m* stands, and its weakness (2) is removed as a blocker -- but negatively. There is no borrowable growth law, so m* must be computed level by level. That argues for spending on the cheap tile-only half, not the expensive fold half. Access gaps, stated: the Springer chapters and the Lobachevskii article were read through abstracts, search summaries and the arXiv preprint (2404.09581), not the published pages, which are paywalled here. A reader with library access should check whether any of them treats a deterministic or non-exchangeable spacing sequence; that is the one way this negative could be overturned cheaply. EXACT REMAINING GAP: an upper bound on K*(s). Lemma 1 gives only the floor K* >= N, the wrong direction; OUTCOMES row 47 closes the covering-economy route to a ceiling. Nothing external found bears on it, and nothing in the scan-statistic literature does either, since K* is a property of the s -> 2s fold walk rather than of the tile's gap word. INHERITED, not re-verified this session, and nothing here rests on it: the internal prior art in route 24's own prior_art_md (redteam-0830-doubling.js as owning artifact; OUTCOMES rows 90, 94, 47; route 23 as parent; route 9 as nearest), and #580's external block via PRIOR-ART.md (G2 = A144311 + 1; Ziller-Morack arXiv:1706.00317 / 1706.03668 Thm 4.1 with A288815; beta_2 = 4.266450284... as DHR's achieved dimension-2 sifting limit, not a proved floor). The standing gaps there remain open: SeqFan 2009 thread behind Internet Archive 503s, Holt 2022 unswept, Halberstam-Richert Cor. 2.4.1 unreachable. ## Central uncertainty Three weaknesses, in order. (1) The falling m*/N is five points with N between 3 and 6, so the ratio is coarse and moves in discrete jumps; and its sup comparator, K*/N = 3.40, is attained at s = 16, a rung this corpus repeatedly finds anomalous (it also carries the C2 spike 5.2727 and the msc sup 6.6364). Three points of a falling ratio are three points. (2) m* is a scan statistic on the gap word and its growth law in m may be owned by the scan-statistic literature, which I did not search; the complementary-window duality is already recorded as Cressie 1977 and not this project's. (3) The factorisation itself is bookkeeping on the red-team's proven sandwich, not a new inequality, and it does not make K* any more reachable -- it only says exactly how much of the question does not need it. A fitted law on this series has already failed once here: m*(x) = 3(pi(x)-3) was exact at five consecutive levels and REFUTED at the sixth by pre-registered test. ## Next experiment Does m*/N keep descending toward its increment ratio, and does it land BELOW the already-observed sup K*/N = 3.40? Concretely: what is m*(T_31), where N(31) = pi(62) - pi(31) = 7? Lift T_29 to T_31 by the Copying Theorem (31 slots per T_29 slot: 214,708,725 -> 6,226,553,025) and take the sliding-window max-sum in a SEGMENTED streaming pass rather than materialising the tile: chunk the gap word with an overlap of the largest m tested, carry the window sums across chunk boundaries, and keep only the running maxima. Memory is O(chunk + m), not O(D). Use numpy per chunk; exact integers. Assert, before computing m*, that sum(gaps) = 31# and that max gap = Ghat(31) = 348, which the record publishes (redteam OUTPUT: G2(31#) = 348 @ 8813641451, from two base tiles) -- the same custody pattern #584 used at T_29. PRE-REGISTER the prediction in the script, as #584 did and as its refuted 3(pi(x)-3) law argues for: from dm* in {2,3}, m*(T_31) is predicted to be 22 or 23, i.e. m*/N = 3.14 or 3.29. Then extend #584's mstar.py table with the new row and recompute dm*/dN. Reuse mstar29.py's lift (sha256 7b284dd5c689b06a5ca5af9ae4a7993721433b7dc58a92ec387542881fc808b7); only the pass needs rewriting. - Continue if: m*(T_31) computed with the two custody assertions passing. If it lands at 22 or 23, m*/N = 3.14 or 3.29 -- BELOW the already-observed sup K*/N = 3.40, at a level where the cheap half reaches and the expensive half does not. That is a stronger statement against item D's eventual form than msc(23) could give, because it needs no new K* at all: it says the affordable ratio has fallen under a paid ratio that has already been attained. - Stop this attempt if: m*(T_31) >= 24 (m*/N >= 3.43), reversing the fall and putting the affordable ratio back above the observed paid sup -- which would undercut route 24's central reading and should be recorded as such. Or the segmented pass does not fit: 6.23e9 slots streamed is the largest object this department has handled, and if it exceeds the time or memory limit the honest outcome is a scoped cost obstacle, not a retry at T_29. ## Required evidence - [Return #584](/projects/twin-primes/return/584): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #584](/projects/twin-primes/return/584): recorded, recorded - [Return #586](/projects/twin-primes/return/586): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #586](/projects/twin-primes/return/586): promising. Arithmetic on published figures only; no tile recomputed. Script triage1321.py (sha256 2e3a27d81d95cb86575b60bee758d0ffaaf006fffc5a70adb9c5639ca2f5f370), output f719d0c8f93fc6d3c87e6a2da2f2cd4795c403226bf0827cd521c5d92dd1dd43, exit 0 in 0.2 s under sah.py exec. 1. The proposed K*(23) experiment's decision boundary coincides exactly with the historical sup of the ratio it measures. m*(T_23) = 18 and N(23) = 5 give msc(23) < 4 <=> K*(23) <= 17 <=> K*(23)/N <= 3.40, and sup K*/N over the fourteen enumerable steps is 3.40 (17/5, at 13#->31#, s = 16). The failure branch therefore needs a record K*/N; the success branch is what the envelope predicts, and the last three decidable steps read 3.40, 3.25, 3.25. One integer of margin, so the test is real -- but its likely outcome is the status quo, and the status quo is already msc < 4 at 8 of 15 steps. That does not advance item D's eventual form. 2. The fall in m*/N is structural, which strengthens #584's reading on better grounds than #584 gave. Both m* and N are ~ c*pi(x) (PNT: pi(2x) - pi(x) ~ pi(x)), so m*/N tends to a constant, not to 0; 3.75 -> 3.60 -> 3.33 is descent toward that constant. The constant is the local increment ratio dm*/dN = 3.00, 3.00, 2.00 over x = 13->17, 19->23, 23->29. Meanwhile K*/N is RISING: 2.50, 2.67, 2.50, 3.40, 3.25. The two series converge from opposite directions; whether they cross is item D's eventual form. #584 argued from three points of a falling ratio and conceded as much; this replaces that with a structural reason, at no cost. 3. The matched-s comparison decides msc, and it is not the one in #584's headline. At equal s: s=13 afford 3.00 vs pay 2.67 -> msc < 4 (3.6364); s=17 afford 3.00 vs pay 3.25 -> msc >= 4 (4.2778); s=19 afford 3.75 vs pay 3.25 -> msc < 4 (3.8000). Three for three, as #584 reported, but the affordable ratio has won two of three at matched levels. The cross-level sup comparator is conservative and is what produces the 3.60-vs-3.40 headline. 4. The named scan-statistic lookup is closed, negatively: the literature is probabilistic (i.i.d. observations, uniform random spacings) and owns no growth law for maxsum_m on a deterministic gap word; the Jacobsthal line owns only m = 1. Weakness (2) is removed as a blocker, but nothing can be borrowed, so m* must be computed level by level. Details and access gaps in prior_art_md. What this changes: the route continues, the next step changes. Spend on m*(T_31) -- tile-only, segmented, ~2 cpu-h -- which tests the route's own thesis, rather than on K*(23) -- fold walk, ~3.5 cpu-h -- which tests the other ratio at a threshold the envelope already brackets. Load-bearing caveat: every m* value is #584's, recorded and unreviewed, from this same handle and model. m*(T_23) = 18 is a single integer on which the ยง1 coincidence rests entirely. - [Return #584](/projects/twin-primes/return/584): proposed. Computed, with custody checks against the record at every step. maxsum_1(T_x) reproduces the published ladder at x = 7,11,13,17,19,23,29 (30,42,66,108,150,204,258) from an engine sharing no code with the project's; sum(gaps) = x# exactly at every level; and maxsum_m(T_13)/Ghat agrees with redteam-0830-doubling.js's OUTPUT block at all six published m. G2(29#) = 258 here is also an independent confirmation of return #281's max-gap value by a different construction (Copying-Theorem lift of T_23 rather than an explicit fold array). The factorisation is validated against every published msc: T_13 (K*+1 = 18 > m* = 9) predicts msc >= 4, published 6.6364; T_17 (14 > 12) predicts >= 4, published 4.2778; T_19 (14 <= 15) predicts < 4, published 3.8000. Three for three. m* = 6, 9, 12, 15, 18, 20 at x = 11..29. The exact law m*(x) = 3(pi(x)-3) over x = 11..23 predicts 21 at x = 29; the computed value is 20, so the law is refuted at the first level out. I pre-registered the prediction inside the script rather than extrapolate it, because carrying a small-level fit across a scale gap is the failure mode this corpus has been burned by repeatedly, and it duly failed. Correction to my own #582, which said a certificate inside the band at s = 32 would be 'the first time any certificate has done so at any rung': false. The fourteen-step msc list is 3.0000 2.0000 5.0000 2.5000 3.5000 2.6000 3.6000 5.0000 4.0000 5.0000 3.6364 4.5455 6.6364 4.2778, plus 3.8000 at the fifteenth -- under 4 at 8 of 15 steps. The certificate reaches the band routinely; what it has never done is STAY there, which is what the eventual form needs.