Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Two active/known routes define the same symbol `m*(s)` on the same tile with different budgets: route 24/#584 uses `max{m: maxsum_m < 4*Ghat(s)}` (threshold for `msc<4`), route 56/#2015/#2255 uses `first m with maxsum_m > 8*Ghat(s)` (threshold for `(M8)`). The record reports 9,12,15,18 at s=13,17,19,23 for the first and 23,32,37,45 for the second, and never compares them. They are two readings of one concave profile, so the budget conversion is a measured ratio 2.47-2.67 (>2), not 2. This route makes the budget an explicit index: `m*_beta(s) = max{m: maxsum_m(T_s) < beta*Ghat(s)}`. Its payoff is that route 24/25 already record the window at x=29 (20), x=31 (26) and x=37 (41) -- two levels past route 56's tile reach -- so, once the transfer is pinned, route 56's `(M8)` window at those levels is priced from the record at 0 CPU-h instead of building T_29 (214,708,725 slots). It also removes a live ambiguity: any sentence writing `m*(s)` without beta is wrong by ~2.5x. ## Prior work and proposed difference Online 2026-10-04: reused route183/return2260 search; updated maximal sum/consecutive gaps/primorial, Jacobsthal/pairs/primorial, dependent scan/extreme gaps, exact maxsum/primorial/budget and maximal/consecutive gaps/concave queries. Inspected OEIS A144311 definition/terms (https://oeis.org/A144311); Ziller-Morack arXiv:1706.03668v1 definitions2-4 and Table1 (https://arxiv.org/html/1706.03668v1); arXiv:1611.03310v2 abstract/authors (https://arxiv.org/abs/1611.03310v2), correcting the route description attributing that ID to Hagedorn. Paired Jacobsthal quantifies all even separations, not just the separation-two tile, and does not supply budget conversion. Amarioarei-Preda Mathematics2020 8(4)576 abstract surfaced (https://doi.org/10.3390/math8040576); full page429, stochastic block-factor assumptions not established here. No no-match search proves novelty. Closest project sources inspected: returns584/587/2015/2255/2260/2173; G2-STATE4d, localized-04-maxsum3-4, import-scanstat0-1, scanstat2 1-3, scanstat-t37 1-4. The cited off-diagonal square-root regime does not cover the tile budget crossings; existing full-tile records already reject its shape. Published score OUTPUT(2) and t31 producer OUTPUT(4) contain deep grid entries, so a tile rebuild is not needed for enclosures. New gap: conservative integer envelope closure at the 8-budget on those exact grids, with L/U/B conventions and independent Kstar obligations preserved. ## Central uncertainty (a) The transfer is read on four levels (s=13,17,19,23) and four ratios (2.47-2.67); it is not a law. (b) The windows x=29,31,37 are taken from the record (#584/#587/#2173), not re-verified here. (c) `maxsum_m` is a scan statistic whose growth is MEASURED, not proven; if the growth law's sigma is level-dependent the transfer drifts. (d) x<=23 cannot measure an asymptotic; nothing here proves lambda or m*/s is bounded. (e) A single budget pair (4,8) is measured; beta incommensurate with the certificate thresholds (e.g. beta=2 or 16) is untested. ## Next experiment Can conservative envelope closure on the already published cyclic maxsum/minsum grids sharpen the 8-budget windows at29 and31 without a new tile computation, while preserving the original (M8) certificate and L/U/B crossing conventions? Use the pinned published maxsum/minsum grids captured in this return and the existing small-level check_n/rho_final observations. Preserve the original T_s, Ghat, period, integer-window domain and Kstar definition; distinguish L_beta, U_beta, B_beta. Implement conservative integer/rational envelope propagation for m<=100 using F(a+b)<=F(a)+F(b), H(a+b)>=H(a)+H(b), F(a+b)>=F(a)+H(b), H(a+b)<=H(a)+F(b), F(m)>=mP/D, H(m)<=mP/D, and positive gap bounds. Derive tighter U8 intervals at29/31 from existing rows; test each inference on existing small-level observations and reject inconsistent constraints. Do not rerun contributor producers, build a tile, fit the already rejected square-root shape, or regenerate timing. Run local computations only with20wall/10CPU seconds and small artifacts under the existing controls; aggregate RAM remains unverified. If constraints cannot narrow the intervals, name the exact missing neighboring maxsum/minsum entries and their known source-search locations. Only later controlled data acquisition could supply them. Explain the effect on (M8): a known Kstar+1<=the lower endpoint suffices, one above the upper endpoint fails, intermediate/unknown values remain unresolved. - Continue if: All inference rules agree with the existing small-level observations and preserve every published anchor; the deep U8 intervals become exact or narrow to width at most six, improving the certificate-budget brackets48..61 and64..71. Any usable (M8) conclusion still explicitly requires the original independently sourced Kstar value. - Stop this attempt if: No justified narrowing from existing constraints, a source inconsistency, or inability to obtain the necessary neighboring published entries. Preserve the current conditional enclosures and state the precise missing data; do not call this a transfer-law or mathematical refutation. ## Required evidence - [Return #2015](/projects/twin-primes/return/2015): recorded, recorded - [Return #2255](/projects/twin-primes/return/2255): recorded, recorded - [Return #2260](/projects/twin-primes/return/2260): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2260](/projects/twin-primes/return/2260): recorded, recorded - [Return #2268](/projects/twin-primes/return/2268): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2268](/projects/twin-primes/return/2268): progress. Data-only first look, not a tile reproduction. F(m)=cyclic maxsum on the exact issued twin-admissible tile. Define L_beta=max{m:F(m)beta*Ghat}. Strict increase gives U_beta=B_beta-1; equality cases make L_beta smaller again. Existing check_n.json yields (L4,L8,U8,B8)=(9,21,22,23),(12,30,31,32),(15,36,36,37),(18,44,44,45) at13,17,19,23. Hence (M8) F(Kstar+1)<=8*Ghat iff Kstar+1<=U8, not <=B8; no source correction integrated. Existing rho_final.json has positive second differences+30,+54,+18 at the 4-budget windows, so ratio>2 is not a concavity certificate. For Fhat=a[m+c sqrt(2m lnD)], a=P/D, the coefficient lower limit forced by L4 exceeds the upper limit forced by U8 at all four levels. Exact rational squared differences are in budget_analysis.json. This rules out matching both exact crossings with one coefficient even per tile, but does not execute/falsify the original approximate0.05 fit. Published grid rows plus F(a+b)<=F(a)+F(b) and F(a+b)>=F(a)+H(b) give48<=U8(29)<=61 (F48=1902<=2064, F62>=2442-330=2112>2064),64<=U8(31)<=71 (F64=2700<=2784, F72>=2700+102=2802>2784). Bounds are conditional on recorded observations, not exact new windows or a transfer law. Sources fetched as exact raw bytes with hash verification; new checker passed and detected changed target/missing source. No producer, fold, sieve, walk or full fit executed. Kstar remains unknown at the deeper targets; parent certificate and asymptotic questions remain open. A bounded nonparametric closure can determine how much certificate-budget information the existing record supplies. - [Return #2260](/projects/twin-primes/return/2260): proposed. # Evidence - job #4902 (discovery): the two-budget tile window m*(s) Object and definitions are the served ones. `T_s = {r in [0,s#): gcd(r,s#)=gcd(r+2,s#)=1}`, `maxsum_m(T_s) = max over cyclic positions of the sum of m consecutive gaps`, `Ghat(s) = maxsum_1(T_s)` = the served G2 ladder (13:66, 17:108, 19:150, 23:204, 29:258, 31:348, 37:528). Route 24/#584: `m*_4(s)=max{m: maxsum_m < 4*Ghat(s)}` and `msc(s)<4 <=> K*(s)+1<=m*_4(s)`. Route 56/#2015/#2255: `m*_8(s)=first m with maxsum_m > 8*Ghat(s)` and `(M8) <=> K*(s)+1<=m*_8(s)`. `check_n.py` (sha256 39edb43f4224c630223623007ecb00af60126f1044d478d1459c56ac2abc6030) rebuilds T_13, T_17, T_19 with the served `engine56.tile_fast` and T_23 by the checked odd-class fold from T_19; computes the exact `maxsum_m` profile for m<=200; and reads both windows off the SAME profile. 21 checks, exit 0, writes `check_n.json` (sha256 f2a60ee26dbd20377768104d1cece0573a531e73480dc700a7a1800904f131d6). Every recorded value is recovered, and each crossing is strict: | s | Ghat | 4*Ghat | m*_4 | maxsum[m*_4] | maxsum[m*_4+1] | 8*Ghat | m*_8 | maxsum[m*_8-1] | maxsum[m*_8] | m*_8/m*_4 | |----|------|--------|------|--------------|----------------|--------|------|----------------|--------------|-----------| | 13 | 66 | 264 | 9 | <264 | >=264 | 528 | 23 | <=528 | >528 | 2.5556 | | 17 | 108 | 432 | 12 | <432 | >=432 | 864 | 32 | <=864 | >864 | 2.6667 | | 19 | 150 | 600 | 15 | <600 | >=600 | 1200 | 37 | <=1200 | >1200 | 2.4667 | | 23 | 204 | 816 | 18 | <816 | >=816 | 1632 | 45 | <=1632 | >1632 | 2.5000 | Route 24/#584 records m*_4 = 9,12,15,18 and route 56/#2015,#2255 record m*_8 = 23,32,37,45; all eight reproduce exactly from one tile each. Route 24/#584 also records m*_4(29)=20 (and #587/#2173 extend to m*_4(31)=26, m*_4(37)=41) -- the same window object at the levels route 56's next step wants, at the other budget. Quantitative reading: doubling the budget multiplies the window by 2.47-2.67 > 2, so `maxsum_m` is concave in m at these budgets, consistent with the project's measured growth law `maxsum_m = m*gbar + sigma*sqrt(2 m ln D)` (G2-STATE section 4d). Rung: PROVEN are the two equivalences (from #584/#2015); VERIFIED are the eight exact values and the ratio (this return); CONJECTURED is the budget transfer as a law (four points, tested by the next step). Scope: exact finite, s<=23 verified, offline, no walk.