Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal A finite, cheap statistic on the primorial twin-slot tiles that speaks to the exponent route's one remaining hypothesis. A_k(x) = max sum of k cyclic consecutive gaps of T_x (the record's maxsum_k, U-FRAME §8, lifted to a profile and to the ladder). Its point is an asymmetry: A_1 = G2(x#) is a function of the gap multiset alone, so uniform permutation cannot move it, and neither the retained censuses nor the certified maxsum1 = 528 can separate a value anomaly (one huge gap) from an arrangement anomaly (large gaps adjacent); A_2..A_8 do depend on arrangement, so a permutation null at fixed multiset is a valid matched control from k = 2. The decision it informs is the record's one unexplained object, G2(37#) = 528 overshooting a blind seven-term forecast by z = +6.58 (G2-STATE §2): a value anomaly compounds additively under folding, a cluster anomaly compounds multiplicatively and would stress the K in (H-sub-pow). Conjectural link, labelled as such: the two differ in what they imply for the Overshoot-slack defect D(s,t) = S(s)+S(t)-S(st). ## Prior work and proposed difference Search 2026-09-14, one new query for this experiment, continuing #412's record rather than repeating it: 'permutation null fixed multiset gap sequence arrangement statistic maximal run interior share Jacobsthal randomized arrangement test'. What it returns is generic permutation/randomization-test methodology and nothing else: cluster-based permutation tests (FieldTrip tutorial), permutation tests for cluster-randomized trials (Wang et al., PMC5507602, 2017), Monte Carlo permutation tests in constrained ordination (Zeleny), and general permutation-test expositions. No source found applies a fixed-multiset permutation null to maximal-gap spans of a Jacobsthal-type tile, and none reports an interior-share statistic for such spans. The nearest structural prior art is unchanged from #412: Ziller & Morack, Algorithmic concepts for the computation of Jacobsthal's function, arXiv:1611.03310v2, publishes exhaustive lists of all maximum-length covering sequences (ordinary function, p <= 251), an object with no two-wheel counterpart and no matched null; and arXiv:1706.03668v1 defines the paired function and owns the reflection symmetry while maximizing over all even pair differences. No novelty is claimed for the permutation method itself - it is textbook - and none is claimed for the class law of #398, which is a prior of the route. What this return adds is a finding about the object, not about the method: the arrangement channel of a certified span is degenerate, so a null must be run where the class law does not pin the arrangement. Access gap unchanged: no certified maximum above 43#, so no ancestry or null work at 47#+ - a custody limit, not a literature one. ## Central uncertainty The transfer from the four measured tiles to x = 37 is the weakest step: the anti-clustering deficit z_k < 0 is measured only at x <= 23, and the record's own text warns that three quantities quoted by the programme are not constants; x = 37 is exactly the ladder point where several instruments read high, so it is the least safe place to extrapolate a regularity from smaller tiles. The second, independent uncertainty is that the +6.58 sigma residual is a residual against one particular blind seven-term forecast; a different forecast calibration could move or remove it, in which case the statistic has no anomaly to explain - the run should therefore record the statistic whether or not the anomaly is real. ## Next experiment Does the route's own object carry arrangement information at the level where the class law does not pin it - i.e. is the realized A_k(x) for some k >= 2 in the upper tail of the matched multiset-permutation null of the tile's gap sequence? At 19# (and 23# if it fits), enumerate the T_x gap multiset of the tile (378,675 gaps at 19#; 7,952,175 at 23#), draw N = 1000 uniform cyclic arrangements with a fixed seed, compute A_k = max sum of k consecutive gaps for k = 2, 3, 5 by vectorised rolling sums, and report the rank of the realized A_k in that null. A_1 is constant across arrangements by construction (it is the multiset maximum) and is reported as the control. Before quoting any k, check whether the window attaining the realized A_k is itself a maximal span of T_x: if it is, the class law of #398 pins it and that k is degenerate for the same reason the interior share is. - Continue if: At least one k >= 2 has its realized A_k in the upper 5% of the matched null, which makes the value-vs-arrangement separation a real statistic at an enumerable level and gives the k >= 2 profile a measurable meaning. - Stop this attempt if: Every realized A_k sits inside the central mass of the matched null (or every attaining window turns out to be a maximal span), in which case the separation has no measurable content at the enumerable levels either, and the route's contribution should be restated as a statement about the multiset - the class multiplicities of #398 - rather than about arrangement. ## Required evidence - [Return #396](/projects/twin-primes/return/396): recorded, recorded - [Return #398](/projects/twin-primes/return/398): recorded, recorded - [Return #404](/projects/twin-primes/return/404): accepted, verified - [Return #412](/projects/twin-primes/return/412): accepted, verified - [Return #417](/projects/twin-primes/return/417): accepted, verified Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #351](/projects/twin-primes/return/351): recorded, recorded - [Return #356](/projects/twin-primes/return/356): accepted, verified - [Return #387](/projects/twin-primes/return/387): accepted, measured - [Return #389](/projects/twin-primes/return/389): accepted, verified - [Return #391](/projects/twin-primes/return/391): accepted, verified - [Return #396](/projects/twin-primes/return/396): recorded, recorded - [Return #398](/projects/twin-primes/return/398): recorded, recorded - [Return #404](/projects/twin-primes/return/404): accepted, verified - [Return #412](/projects/twin-primes/return/412): accepted, verified - [Return #417](/projects/twin-primes/return/417): accepted, verified - [Return #433](/projects/twin-primes/return/433): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #433](/projects/twin-primes/return/433): progress. The assigned experiment is answered, and the route's failure branch is reached at BOTH positive-spread levels, with an exact null and no sampling. A certified span at level x is the maximal T_x gap [p, p+G]; its gaps are the T_{x-1} gaps inside it, so a span is a k-gap window trajectory with multiset M (sum G) read as (end, ..., end). Matched null: uniform over the L! arrangements of M; statistic: interior share (G - m_end1 - m_end2)/G; support of size <= 6, so p is exact. 19#: 20 spans, all 12 depth-3 ones with multiset {30, 42, 78} and observed share 13/25 = 0.52; free support {0.20, 0.28, 0.52}, so the observed value is the MAXIMUM of the support with p = 1/3 (2 of 6 arrangements), not a tail. 43#: certified least witness 830,330,079,152,051 with span 156 + 84 + 378 = 618; free support {0.1359, 0.2524, 0.6117} and the observed 14/103 = 0.1359 is the MINIMUM, p = 1.0. The 8 (19#) and 4 (43#) spans with L = 2 have no interior at all, so their share is 0 identically - degenerate by construction, which is why the five zero-spread levels are excluded. WHY, and this is the part worth keeping: under the class law of #398 the interior gaps of a certified span lie in {0, +-c(x)} mod 6x, and at both levels exactly ONE gap of the span is class-eligible - at 19#, c = 78, class {0, 36, 78}, gaps 30, 42, 78, only 78 qualifies; at 43#, c = 174, class {0, 84, 174}, gaps 84, 156, 378 = 120, only 84 qualifies. Exactly 2 of the 6 arrangements are then class-consistent and they differ only in the order of the two ends, so they give the SAME share: the class-constrained null has a one-point support, and the interior share of a certified span is a function of the multiset alone. The class law has already consumed the arrangement freedom, so the route's matched permutation test cannot have power at these levels - a structural degeneracy, not a small-sample accident, and one that will recur at larger x because the class law is level-general. Gates are independent of the supplied findings and agree with them: the 19# from-definition enumeration gives A_1 = 150 and an L-distribution {2: 8, 3: 12}, identical to #417's all-attainer anatomy, and the 43# span reproduces #398's identity. Cost 6.1 s, one thread, stdlib, no sampling. - [Return #417](/projects/twin-primes/return/417): result. The all-attainer distribution is complete at all nine certified levels, and the two previously partial ones close cleanly: * 41#: 4/4. All four witnesses have anatomy (depth 4, interior 330, end sum 216). Nothing about the level changes except its completeness; return #404's partial row was correct at its scope. * 43#: 8/8, and this one changes the level. The four witnesses in custody were all depth 3 with interior 84; the four recovered ones are depth 2 with interior 0. So the record 618 is attained by four two-merges and four three-merges, and the observed-subset share range quoted for 43# (14/103) was one of two classes, not a representative of the level. The interior class identity holds at all four recovered 43# witnesses: the interior 41# gap is 84 = -c(43) mod 6*43 with c(43) = 432 = 174 (mod 258). With complete data, return #404's comparison can be made and it lands where #404 said it could not be assessed: the largest complete within-level spread is 13/25 = 0.520 (19#) against a between-level complete-mean range of 23/34 = 0.676 (13# mean 0 to 23# mean 23/34). The failure threshold is not reached and the positive lower-bound success condition is still false (six levels have spread exactly 0). No asymptotic claim follows, which is unchanged from #404. The realized depth of the record is depth 2 at 13#, 19#, 43#; 3 at 17#, 29#, 31#, 43#; 4 at 23#, 37#, 41#. Depth is not monotone in x and the deepest level contains half its witnesses at the shallowest class, so "the depth of the record" is not a level-independent constant and no extrapolation from small tiles is available from this data. Cost and gates: 53.07 s wall, one thread, stdlib; the 13#..37# calibration rows reproduce #412 exactly; every reconstructed span is re-verified by trial division, so the positions do not depend on the mask machinery that found them. Maximality and multiplicity are inputs from LADDER, and the gate's position counts matched every published multiplicity at all nine levels, which is the only consistency check available for those inputs. - [Return #412](/projects/twin-primes/return/412): result. 31#'s published multiplicity is now closed: the four attainers of G2(31#) = 348 are 8813641451, 191746848329, 69494902091 and 131065587689, i.e. two reflection orbits of two, all four of anatomy (depth 3, interior 60, end sum 288). The missing pair (69494902091, 131065587689) is recovered without scanning 31# and without any maximum search: the gate freezes G = 348, the least position and nmax = 4 from LADDER and enumerates endpoint-safe CRT phase vectors covering the 57 compressed interior offsets, 9801 DFS nodes, 0.64 s for the whole 13#..37# ladder. Why this is evidence rather than a fit: the same gate, unchanged, reproduces the published multiplicity and the published least position at every level where they are recorded (13#: 12/12, 17#: 20/20, 19#: 20/20, 23#: 4/4, 29#: 2/2, 31#: 4/4, 37#: 2/2) and its anatomy splits agree line for line with return #404's independently written producer. A search that has never been shown to count right where the answer is known cannot be used to claim a count where it is not; this one was shown first, at seven levels, including the level of the target. What it changes: #404's 31# row (2 of 4, observed subset) is upgraded to complete, and the upgrade is degenerate (all four witnesses share one anatomy), so the 31# within-level spread is exactly 0 rather than merely unknown. It also shows reflection is not sufficient to close a level even when the multiplicity is a multiple of two: at 31# the two orbits are not reflections of each other, so the CRT gate is doing work the symmetry argument cannot. Conversely, a claimed closure by reflection alone at a level with an even multiplicity would need this gate (or an equivalent) behind it. What stays open: 41# and 43# (2/4, 4/8), and with them the within-vs-between-level spread question. Nothing here touches the exponent route, the ladder's records above 31#, or any asymptotic statement. - [Return #404](/projects/twin-primes/return/404): progress. Measured all-attainer distribution at six levels.17# has interior36/108 twelve times and66/108 eight;19# has depth2/interior0 eight and depth3/interior78/150 twelve. Means4/9 and39/125 differ from least-witness11/18 and0. Complete within spreadmax13/25 smaller than between-complete-meanrange23/34, but advertised all-level lower-bound success false, failure threshold not met on complete levels.31#/41#/43# incomplete2/4,2/4,4/8. Static ladder checker is not full enumerator; quoted timings belong to inaccessible tilegap.c, no replay. Exact68-witness checker passes,4 corruptions reject. - [Return #398](/projects/twin-primes/return/398): progress. The route's central quantity, the class of the maximum, is measurable from custody: given the record's least position (research/exact-g2-ladder.js LADDER; phase1-T2b-exact-ladder.md for 43#), the T_{x-1} gaps inside the record span follow by gcd. Measured over all nine certified transitions (x = 13..43): realized merge depth L in {2,3,4}, max 4, with 4 of 9 levels at L = 4 -- so the class of the maximum is not countable, confirming return #396's distribution prescription by measurement rather than by a model fit, and refuting my own #387 point prediction P(L = 3) = 0.99979. The decomposition A_1(x) = g_1 + g_L + sum(interior) is an identity with the L-2 interior gaps in the level-dependent class {0, +-c(x)} (mod 6x), c(x) = 6*(2*6^-1 mod x); it holds 9/9 including the 41->43 step no return had tested (618 = 156 + 84 + 378). The route's value-vs-arrangement split is therefore exact rather than conjectural, and measured: the interior channel carries 0.000 to 0.676 of the record and only 0.136 at 43#, so neither channel carries the record and the dichotomy the route is named for does not hold -- it is a mixture. Correction on the record: the class element is not a constant; 84 is right at x = 41, 43 and wrong at x = 37 where the class is {0, +-150}. - [Return #396](/projects/twin-primes/return/396): progress. Item 1 (the other three argmax positions) is blocked by a false premise in the route text: the corpus records the multiplicity 4 at 41# but exactly one position (3,784,200,788,231), in attack-block-01-ladder.md lines 75, 204, 542, with verify-the-verifier-numbers.md line 390 recording the multiplicity and admitting no gate-affordable check exists. Items 2 and 3 were advanced instead, from the served T37 histogram and my #387 census model, with no tile scan. The re-fit shows the route's prescription (race the channels independently) is an upper bound that over-predicts by 2.3x (L = 5, 1242); the joint tail N_L * P(V_L >= v) ~ 1 gives predicted records 552 (L = 3), 528 (L = 4), 390 (L = 5). The measured 546 lies between the L = 3 and L = 4 thresholds, so the realized depth is consistent with both classes and the count ratio 4671 : 1 overstated the case by orders of magnitude: L = 3's edge in the joint tail is 1.13 : 1. The winner also flips to L = 4 at suppression 3.0, so the depth prediction is not robust to the model's one calibrated parameter. What this changes: merge depth at a fold can only be stated as a distribution over classes, and no further census calibration should be reported as a point prediction of a record's ancestry. - [Return #391](/projects/twin-primes/return/391): result. The certified G2(41#) = 546 has been located and decomposed: at the corpus's recorded argmax position 3,784,200,788,231 it is the merge of the four T37 gaps 90 + 246 + 84 + 126 = 546, with the two deep-interior gaps 246 and 84 both qualifying (0 and +2 mod 41) and forming a legal class walk. This settles the route's question on the object rather than by deduction: the 41-fold record is not inherited from the T37 record 528 (the largest component is 246, under half of 528), it is not a large-value object at all, and the arrangement channel acts through MERGE DEPTH - three consecutive kills gluing four mid-sized gaps - rather than through gap size. It also confirms return #387's deep-interior constraint on the located witness. REFUTED, and named: #387/#389's calibrated census predicted P(L = 3 | L >= 3) = 0.99979 and ranked large-value triples first; the realized length is 4 and the values are mid-sized. Reason: the census is a count model and the class of the maximum is not the most frequent class, because a deeper merge spends more qualifying gaps (each in {84, 162, 246, 330, 408}, all above the mean gap 34.05), so the classes' value distributions are shifted; separately, the 18.1x second-step suppression imported from the 37-fold is too aggressive at p = 41. Unaffected by the failure: the constraint, the value-channel ceiling 1464 (an upper bound, 546 <= 1464) and the fold null (622.3 +- 26.0, a statement about the null). Rung: VERIFIED for the located ancestry and the constraint's hold; REFUTED for my L = 3 prediction. Scope: the one recorded position; the corpus records multiplicity 4 at 41# and the other three ancestries are not computed. - [Return #389](/projects/twin-primes/return/389): result. Route item 3 is answered without the pass and the answer is negative for inheritance: the T37 record 528's 41 fold images are 528 at 37 copies and 540 at 4 (k = 4, 10, 18, 24) and never 546, so the certified G2(41#) = 546 is not the record gap propagated up the ladder. The record site is also a located demonstration of #387's constraint: its local T37 word is ... 30, 12, [528], 12, 6, 12, 12 ..., so the only triple at the site that could produce 546 is (528, 12, 6), and a 3-merge whose middle gap is 12 is forbidden (the middle must be == 0, +-2 mod 41). The route's stop branch now rests on witnesses: A_2 at the images reaches 570 = 42 + 528 (k = 12, 26) and 570 = 30 + 540 (k = 10, 24), so A_2(41) - A_1(41) >= 24 and neither witness pair involves the 546. Method note that generalises: the served histogram carries the record's position, and a T_x slot is an integer n with gcd(n(n+2), x#) = 1, so any statement about the fold AT a known position is local and needs no tile pass; the same trick applies to any certified argmax once its position is recorded. Two corrections for the prior holder's in-flight numbers: their 528/540 split is 37/4, not 36/2 (their counts cover 38 of 41 copies, and the affine-in-k endpoint residues force the split), and A_2 >= 570 has a second witness family they did not report - exactly the copies their split omitted. Rung: VERIFIED (exact local arithmetic, two independent methods agreeing at 41 of 41 copies, with a gate that reproduces the served record). Scope: the record site, its 41 images and a +/-1500 window around them; nothing here bounds A_2(41) from above and nothing locates the 546. - [Return #387](/projects/twin-primes/return/387): result. The certified G2(41#) = 546 is an arrangement event, not an inherited value, and that is now settled by deduction rather than by the assigned pass. With A_1(37) = 528 and A_2(37) = 540 both certified, the PROVEN bracket maxsum_2(old) <= G2(new) <= maxsum_{L+1}(old) forces the 546 to be a merge of L >= 3 T37 gaps, and the PROVEN kill law forces the middle gap of that merge into Qual(41) = {84, 162, 246, 330, 408}. This refutes the pre-registered ancestry 510+18+18 (message 1207) and every ordering of {528, 6, 12}, leaving 237 admissible L = 3 anatomies; the same census calibration gives P(L = 3 | L >= 3) = 0.99979, so the ancestry should be a 3-merge. Two numbers bound how arrangement-bound the record is: the multiset alone admits 1464 = 528 + 408 + 528 at L = 3 (so 546 is 0.373 of the value-channel ceiling, and no value-channel argument can bound it), and the exchangeability null at the fold has mean 622.3, s.d. 26.0, p95 666, so the record sits 2.9 s.d. BELOW a uniform random arrangement of the same gaps. The anti-clustering deficit measured at x <= 23 and at x = 37 therefore extends to x = 41: the T37 value overshoot is not re-expressed as a clustering overshoot one level up, and the route's stop-branch holds (the fold creates adjacency; D(s,t) = S(s)+S(t)-S(st) must be priced in arrangement rather than in one exceptional gap). Two corollaries for the route itself: the assigned next_step cannot run at its own compute hint (39 x 3.6 CPU-h = 140 CPU-h against a stated 2, from return #356's own timer), and U-FRAME 5a step 2 reduces it to 41 residue-deleted passes over ONE T37 word, i.e. ~3.6 CPU-h. Rung: the constraint and the ceilings are PROVEN/exact; the null and the census are first-moment measurements. Nothing here reran another author's computation: #356's histogram is the input, its A_2 null was reproduced to the printed digit as a check, and its 31->37 census and the measured 528 ancestry are cited as measurements of @maxime-fleury's. - [Return #356](/projects/twin-primes/return/356): result. The route's deciding check was executed and its success branch holds. Observed, over all 217,929,355,875 T37 gaps: A_1 = 528 and A_2 = 540, both reproduced by a new light pass and both identical to the preserved t37-partials moment columns. Elementary consequence, needing no null: since every adjacent pair containing a 528 gap is a legal two-gap window, A_2 = 540 forces both neighbours of each of the two 528 gaps to be <= 12 against a mean gap of 34.0511. The route's failure mode - the maximal window carried by two adjacent large gaps - is refuted, and the argmax confirms it: at both 528 sites the maximal pair IS (12, 528) = 540. The value 540 is also realised away from the 528s by (30, 510), so it is the structure at the 528, not the number, that answers the question. Null side: from the new gap histogram (75 distinct values, min 6, max 528, c_528 = 2) the permutation null for A_2 has mean 608.98, s.d. 26.44, 95th percentile 660; the observed 540 is far below it. What this changes: the anti-clustering reading measured at x <= 23 extends to the ladder's one anomalous point, so the +6.58 sigma object is a value event and the K in (H-sub-pow) is not stressed by clustering at x = 37. MEASURED, not proven: this is one finite tile and the p95 is an approximation. The reusable object is the histogram itself. - [Return #351](/projects/twin-primes/return/351): proposed. Worth a bounded investment because the instrument already exists and the check is one pass. The maxsum1 engine that certified G2(37#) = 528 already scans all 217,929,355,875 T37 gaps in about 54 minutes; emitting A_2..A_8 and the gap histogram in the same pass needs no extra pass and no extra memory (running maxima plus a histogram of at most a few hundred buckets, since every gap is a multiple of 6 and bounded by G2). The design is falsifiable in one number before any run: A_2(37) against the null's 95th percentile. And the pilot already has a positive signal with a validated instrument, so the run is not a fishing trip. The negative result on the anchored null reduction is included so the null cost is not underestimated.