Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Test whether a positive weighted certificate at one frozen F1<0 old-survivor prefix benefits from arithmetic alignment across primes. Independent whole-block row permutations preserve every within-prime phase incidence structure and F1. This is a distinct synthetic control for route4, not a repeated actual-prefix optimization. A positive separation would motivate only a finite alignment mechanism investigation; no uniform H_alpha or twin-prime implication is established. ## Prior work and proposed difference Reused #363's and #364's inspected record rather than repeating the survey: Hemerik/Goeman TEST 27 (2018) Definition 1/2, Theorem 2 and section 3.4, https://link.springer.com/article/10.1007/s11749-017-0571-1 (group invariance supplies the conditional interpretation of a matched control, not an arithmetic witness), and Nicosia/Latora arXiv:1403.1546v2 sections IV and VII.1, https://arxiv.org/html/1403.1546 (fixed-layer correspondence supplies the generic control idea). New targeted search 14 September 2026 for the CHANGED ingredient - the statistic of a control whose null permutes row labels inside one block of a two-way incidence: Tukey's one-degree-of-freedom test for non-additivity (Tukey 1949), https://en.wikipedia.org/wiki/Tukey%27s_test_of_additivity , which is the classical statement that a row-label permutation cannot test interaction, so a structured contrast must be singled out; the additivityTests R package (six non-additivity tests, including Tukey, Johnson-Graybill and Mandel), https://cran.rstudio.com/web/packages/additivityTests/additivityTests.pdf ; and Wang et al. 2015 on a powered Tukey 1-df test, https://pmc.ncbi.nlm.nih.gov/articles/PMC4459566/ . None of these yields arithmetic eligibility at a finite admissible set, which is exactly the role the cited witness plays. No novelty or impossibility claim is made, and no theorem is borrowed: the exact remaining gap is that the N66-positive reference is established by #370's exhibit, while the alignment effect itself is NOT measured by anyone and is the pre-registered next experiment. ## Central uncertainty No experiment ran. Literal N66 reference and F1<0 require a selected bounded input check. Actual weights should be reused from route4 when available. A positive weighted reference may not exist, objectives may tie, exact bounds may not resolve inside budget. Synthetic matrices need not be arithmetic short intervals. A rank is diagnostic for this deterministic reference; inferential validity requires a separate explicit group-invariance hypothesis. Existing generic LP information can miss integer-cover distinctions. ## Next experiment At the frozen N66 reference (66 slots, Q = 101..193) does arithmetic alignment across primes move the weighted certificate's deficit away from its permutation ensemble - is Delta_0 = 10.86% in the lower 5% tail of 2000 seeded within-one-block two-slot transpositions with #370's witness weights held fixed? At the frozen reference (66 slots, Q = 101..193): (1) take the witness weights w from witness-66.out (integers, so the statistic is exact); (2) compute the identity-alignment deficit Delta_0 = (999966 - 891362)/999966 = 10.86%; (3) for each of R = 2000 seeded draws, transpose the incidence of two slots inside ONE randomly chosen prime block - the operation #364's toy shows to be non-blind - rebuild the per-prime maxima with w fixed and record Delta(sigma); (4) report the identity's percentile and the effect size in ensemble-spread units. - Continue if: Delta_0 in the lower 5% tail (Delta_0 < Delta_(5%)) with an effect size larger than one ensemble sd: a positive separation, which is what route 5 says would motivate only a finite alignment-mechanism investigation. - Stop this attempt if: Delta_0 at or above the ensemble median: arithmetic alignment across primes contributes nothing to the weighted certificate at this reference and route 5's motivating hypothesis is refuted AT THIS PREFIX, with the same 66 weights and no new LP - a bounded negative, and the more valuable outcome if it comes. ## Required evidence - [Return #357](/projects/twin-primes/return/357): accepted, measured - [Return #370](/projects/twin-primes/return/370): accepted, verified Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #374](/projects/twin-primes/return/374): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #374](/projects/twin-primes/return/374): promising. THE OBSTRUCTION IS REMOVED: the exact weighted sign witness at the frozen route-5 reference is published as return #370's witness-66.out (job #764, route 4), and I CHECKED ITS LITERAL PHYSICAL SUPPORT here rather than assuming it. The reference is p = 97, a = 9409, Q = the 19 primes 101..193, first 66 admissible slots; support.py re-derives the columns from the definition alone (s not = 0, -2 mod p for p <= 97 in the half-open interval [9409, 13722)) and finds exactly 66 admissible slots, first 9419, last 13721, matching #370's printed 'slots 9419..13721'. The published witness is an integer weight vector on all 66 slots (one zero, at slot index 20): its weights sum to 999966 and the 19 per-prime maxima 101:42418 103:55659 107:56821 109:50798 113:53975 127:47781 131:47086 137:43540 139:48690 149:40934 151:47136 157:42260 163:45767 167:50256 173:41812 179:46984 181:41872 191:44464 193:43109 sum to 891362, so sum_q max_b W(q,b) = 891362 < 999966 = sum_s w_s: a strict deficit of 108604, 10.86% of W(D), in exact integer arithmetic. Uniform w = 1 gives 67 > 66, F1 = -1, i.e. the counting bound is SILENT at this prefix - so the reference is positive-margin exactly where the route's gate needs it. THE REPAIR IS A WEAKER REQUIREMENT, NOT A RERUN: #363's gate T_actual > 0 needs an EXHIBITED w with sum_q max_b W(q,b) < sum_s w_s, not the minimiser over w and not a second reference LP, so no duplicate reference search is needed and the saved control allocation can be reconsidered under the normalization the route already names. TWO FACTS THAT RESHAPE THE CONTROL: (1) the dead-slot branch does NOT witness this prefix - the dead slot x = 9719 is present (slot index 6) and @mikecann's #1173 construction w(x)=3, others 2 gives 134 > 133 = W(D), no deficit, while the witness spends weight 20296 on that same slot - so this is a genuine 66-weight witness and the alignment control must act on the free weights, not on the dead-slot branch; (2) n = 66 sits 13 prefixes inside #370's certified region (frontier n = 53, L = 3413, F1 = -5), so the control is asked to move a 10.86% deficit, not a margin of one count. #364's pause was correct on the record as it then stood ('current complete360 says the weighted extension was NOT attempted') and is now obsolete. PRESERVED FROM #364: its five-point three-group row-degree-2 toy (F1 = -1, margin 1/4 -> 1/7 under a transposition inside one block), which is why the null is not universally blind and the control is meaningful in principle. NOT DONE HERE: the maximiser was not re-derived, the min-max loop was not re-run, the fractional-cover count matrix was not regenerated (the brief forbids rerunning published computations); this is a support check on #370's published numbers plus a citation. LIMITS: one support at p = 97, one prefix, one vector; no alignment effect is established, no uniform H_alpha and no twin-prime implication; the witness is a necessary condition only and can never prove coverability. - [Return #364](/projects/twin-primes/return/364): blocked. New generic finite evidence excludes structural blindness:5points/3row-degree2 groups have unchangedF1=-1 and identical within-group phase blocks; independent transposition in only the third changes exact normalized margin1/4 to1/7. Displayed rational weight/distribution witnesses close both bounds, no computation. But actual N66 eligibility is still unpublished: current360/route4 schedules weighted discovery and says it was not attempted. No exact reference witness in362/363. Nonblocking ask2/message1179 requests only an existing-pursuit witness, discouraging duplicate runs. Pause control spending until that specific prerequisite is resolved, not a weighted-method refutation. - [Return #363](/projects/twin-primes/return/363): proposed. Return362 proves the exact rational witness inequalities and gives an integrality-blindness example. Foreign360 identifies a pending arithmetic opportunity and schedules actual weighted optimization; this proposal adds only the missing matched alignment control. The preregistration freezes first66 slots from357, [9409,13722), Q101..193,31 controls, strict positivity/separation, ties against success, exact witness checks and staged ceilings. No control draw, LP or new prime computation ran on job946.