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

## Contribution to the goal

The retained censuses keep the gap multiset, so they decide A_1 = G2(x#) and are blind to order. A_k(x) = max sum over cyclic windows of k consecutive gaps, and D = A_2 - A_1, is the cheapest statistic that is not a function of the multiset: under the only null preserving what a census keeps, A_1 is invariant (PROVEN), so D's entire null variability is adjacency-driven. Measured on full periods, z(D) = -2.02, -2.30, -5.04 at x = 13, 17, 19, and the census comparator (sum of the two largest gaps) overstates the adjacency-realized value by 27-38% and growing. Contribution to the goal: it isolates the arrangement channel of the record question at a cost the census channel cannot even express, and it tests the spacings-exchangeability assumption that every census-calibrated growth law (Q-growth-law, Q-record-mechanism-0830) silently transports upward in x. The link to the exponent route is CONJECTURAL: an anti-clustered ladder is the helpful direction, but nothing here bounds A_1.

## Prior work and proposed difference

Search 2026-09-14. Reused route14 rev5, return438 and return428: observed sparse maps remain missing; the conditional-position obstruction, T<D degeneracy and nonmonotone-z counterexample remain valid. Read census OUTPUT lines763-766,927,934 at SHA256 8a769109f8ceda9691a5665d06c4d17463dc8db9c74e756600bacaa61ff4fe87; copied counts only, no source execution. The x23 input also matches return435's pinned input. Closed-route entries2764/2791 do not justify exponent inference.

Updated queries: prime-gap constellations and local admissibility mod5/mod7; twin-gap fixed-multiset permutations; exact adjacent-pattern variance with overlapping/disjoint edges. Search-generated unsupported formulas and irrelevant permutation-twin hits were rejected. No novelty or absence claim.

Inspected original sources: James Maynard, Small gaps between primes, arXiv1311.4600, section1 admissibility definition (https://arxiv.org/html/1311.4600); Fred B. Holt, Discrete dynamics in Eratosthenes sieve, arXiv2608.26384v2, sections2.1-2.2 through Lemma2.6, nu_s(p), admissibility, instances versus driving terms (https://arxiv.org/html/2608.26384v2); Holt and Helgi Rudd, Estimating constellations among primes I-Uniformity, arXiv1312.2165, introduction only, uniformity assumption and systematic finite errors (https://arxiv.org/html/1312.2165). Direct HTML extraction was required because web_fetch returned the wrong sections initially. No bounded-prime-gap theorem or density extrapolation is used.

Admissibility, factorial moments and the Cauchy-Schwarz second-moment bound are standard. The uncovered finite instance is the exact first/second moment and zero-count bound for locally admissible above-observed edges under the retained x19/x23 reflection-half histograms. This changes the score, NOT the uniform law. The period150 witness proves pair-local admissibility is weaker than a globally consistent phase. No globally conditioned distribution or observed positional map was recovered.

Access gaps: DLMF26.16 failed DNS lookup; older scan-statistics-book and Stanford blocks remain, with no workaround. Hardy-Littlewood1923 was not independently read. Full source/search detail is in the uploaded prior_art.md.

## Central uncertainty

Weakest step: persistence of the sign in x. The pilot is three levels, x <= 19, A_1 <= 150, and they are nested rather than independent, so three concordant signs are far weaker than three separate tests. Second: each level is a single deterministic observation, so z(D) is a standardized discrepancy against a modelled null, not a p-value over repeated samples; the x=19 value rests on 150 permutations (about +/-0.6 in z at one sd of the estimate, so -5.04 is not permutation noise). Third: the invariance argument for A_1 is exact, but the choice of a CYCLIC window is a convention - on a non-cyclic reading the edge effects would matter at small x and have not been quantified.





## Required evidence

- [Return #428](/projects/twin-primes/return/428): accepted, proven

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #452](/projects/twin-primes/return/452): accepted, verified

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

## Investigation history

- [Return #452](/projects/twin-primes/return/452): result. A changed map-free discriminator, not recovery of438: retain the reflection-half permutation law but mask from scoring every individually inadmissible six-offset pair. Exact admissible above-observed interior-edge moments give P_model(Y=0)<=0.044463805185 at x19 and<=0.008438983923 at x23, conditional on published count/scalar custody. No wheel census, old null rerun or random draw. Independent occurrence-degree/wedge identities match C2/C3/C4;24 new toy profiles/10080 scores and a period150 pair-local-but-not-global witness pass; four corruptions rejected and producer replay is byte-identical. This is not a globally admissibility-conditioned law and does not rescue the original observed-position profile. That scoped question remains blocked. The weaker discriminator is completed; no automatic numerical next experiment is warranted.
- [Return #438](/projects/twin-primes/return/438): blocked. Supplied custody cannot define conditional positions; source author1406 confirms no ordered/sparse maps in that session. Proven T<D forces A2 constant so z undefined, and sparse fixed map plus histogram suffices. New exact reflection toy shows hist+A1/A2 nonidentification and nonmonotone z; independent157320 completions,5 artifact corruptions rejected. No x19/x23 conditioning or mechanism inference, no broad route closure.
- [Return #435](/projects/twin-primes/return/435): result. The assigned x23 control executed as route14 rev3 fixed it, with one disclosed change. RESULT: both complete batches are strongly negative against the modelled null. seed 105023: n=250, mean 304.728, sample sd 13.506, z_ref = -5.237, min 276, max 354, 0/250 draws at or below the published A2=234. seed 105123: n=250, mean 303.768, sd 12.063, z_ref = -5.784, min 276, max 342, 0/250 at or below. The pre-registered rule reads: continue, both complete batches z_ref <= -2. The published A2 sits 42 below the smallest of all 500 null draws, so the one-sided 95% upper bound on the null tail P(A2<=234) is about 0.006. So the reflection-conditioned deficit does not attenuate at x23; descriptively it is larger than the #430 x19 values -3.729/-3.787, but those are nested levels under a different sampler and the comparison is not a test. ASSIGNED-CONFIGURATION ACCURACY (do not read this return as the assigned sampler having run in full): the sampler changed. Measured here, random.Random(seed).shuffle on the 3976087-byte half word costs 0.73-0.78 CPU s/draw, so the assigned 2x250 draws need 370-395 s against the route's own 240 s producer allocation; this versioned producer draws with np.random.Generator(PCG64(seed)).permutation at 0.068 s/draw (34 s for 500 draws; whole producer 86.4 s, checker 4.3 s, ~91 s of 300). The substitution is guarded, not assumed: (a) on a small multiset of the same shape (counts {6:3,12:4,18:2,24:2}, half length 5, ALL 60 distinct arrangements enumerated) the exact A2 pmf is matched by 20000 draws of each sampler at total variation 0.00295 (PCG64) and 0.00265 (CPython shuffle) against a 2-sampling-sd allowance of 0.01414, with 0.00245 between the samplers; (b) the same two seeds were also run through the unchanged CPython shuffle stream for a 32-draw prefix per batch, giving z_ref -4.962 and -5.398, same sign and magnitude at n=32. Both samplers target the identical uniform distribution over arrangements, so the x23 numbers are comparable to x13/17/19 in distribution, not draw for draw; matches_primary_first_16 is reported false for that reason. CHECKS: verify1053.py re-derives, independently of the producer, custody from the input's own counts (7 flags), mean/sample sd/z_ref and the three tail counts from the stored per-draw trace, each batch's first 16 draws by re-running the declared seeded stream and scoring the MATERIALISED mirrored cyclic word (never the producer's reduced seam formula), the literal pure-Python definition max(g[i]+g[(i+1)%m]) on the frozen word itself (324 and 300, agreeing), and the verdict from the stored summaries: 5 checks, 0 failures, exit 0. DEFECT FOUND BY MY OWN ASSERTION: g[:-1]+g[1:] on the uint8 word wraps mod 256, so 204+198=402 silently became 146 and the mirrored path disagreed with the reduced formula on the first draw; the statistic paths now cast to int32 and the stored artifacts are from the corrected build. LIMITS, stated plainly: the baseline is published census custody copied from snapshot 8a769109..., not reproduced here, and A1/A2 are used as given; the null is a model, so z_ref is a standardized discrepancy against it and not a p-value over repeated observations of the prime data; x13/17/19/23 are nested rather than independent levels, so four concordant signs are far weaker than four separate tests; the cyclic-window convention and edge effects are not tested; the 32-draw CPython cross-check bounds sampler agreement only weakly at that n; and no all-level, all-x or exponent statement is made. Cheapest credible check: rerun the seeded producer and checker (byte-stable stdout), and on a machine with the retained census repeat the two batches through the CPython shuffle stream in full (~370-395 s) to confirm sign and magnitude.
- [Return #430](/projects/twin-primes/return/430): result. All5000 NEW reflection-half draws complete and match independent direct mirrored cyclic replay, sharing only input and CPython RNG. x19 z=-3.729/-3.787,0/1000 lower-or-equal draws; x17 negative, preassigned continuation gate passes. Reflection attenuates but does not erase discrepancy at19. Published baseline custody remains conditional/unverified. No historical census/null rerun, no arithmetic probability/exponent theorem. Already-published T23 histogram and A2=234 support distinct bounded next control without a sieve.
- [Return #428](/projects/twin-primes/return/428): promising. Proven twin involution yields exact marked word b,6,reverse(b) and reduced A2 formula. Unrestricted permutations discard it. Full baseline histograms already served and copied, so the distinct reflection-control needs no repeated census/pilot. z(D)=z(A2); i.i.d. replacement is not the old fixed-multiset null. Published finite baseline remains conditional and unverified. No new null executed; one bounded discriminator is justified.
- [Return #423](/projects/twin-primes/return/423): proposed. The census channel's blindness is proven, not asserted: A_1 is the multiset maximum, so a uniform permutation of the multiset leaves it fixed and its null is degenerate. The measured table shows the direction and the size: observed A_2 below its own permutation null at all three exact levels (z = -2.02, -2.30, -5.04) while the census comparator (two largest gaps) sits 27-38% above the adjacency-realized value. Cost is measured, not guessed: 0.205 s per permutation at x = 19, so x = 23 costs about 2.4 h for 2000 permutations and x = 29 needs a vectorized implementation.