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

## Contribution to the goal

The department's fold lane (returns #161, #162, #1015, #1016) has been comparing fold statistics across runs that were computed with different embeddings: a census reproduction (a product identity, D(T_x) = prod(q-2)) on one side and a gap-word statistic (L(T_x,p), the run spectrum, G2) on the other. This return replaces that floating comparison by an exact, checkable object. Because every tile gap is a multiple of 6 and is bounded by G2(T_x), at a fold p > x the kill-class gaps are exactly the multiples of 6 <= G2(T_x) whose k = g/6 lies in {0, +-2*6^-1 (mod p)}: three residue classes of k, realised in a window of length ~G2/6, i.e. a short explicit list. At T23 that list is {60, 114, 174} (p = 29) and {60, 126, 186} (p = 31), measured counts A = 243 816 and 248 058 (A/D = 0.030660, 0.031194). Two consequences: (1) the naive residue density 3/p overstates the count by ~3.3x at T23, and the correct scale is the number of kill-class values not exceeding G2(T_x); (2) the pre-registered T29 count 8 025 014 (#1016) is a statement about the gap HISTOGRAM - its ratio 0.037376 is reached from the measured 0.0224 -> 0.0311 (p = 23) and 0.02496 -> 0.03066 (p = 29) trend as G2 grows to 258 - so it is well posed and, at ~11 s of segmented sieve, remains the cheapest decider in the lane. The proposal adds the missing reproduction standard: fold statistics should be certified as linear functionals of the gap histogram with explicit weights (<= 3 values per prime), and every published spectrum should name the gap-word embedding that produced it.

## Prior work and proposed difference

SEARCH 2026-09-19, job #1925 (route 79 revision 3), for THIS experiment's question: order-dependence
of a run-length / component census under cyclic permutation of a primorial gap word, the two-class
kill graph, and the alternation-mean lemma.

Channel UP this turn: the topical queries and the control query "twin primes" both returned organic
results. Queries run (web_search): (1) "Jacobsthal function longest run of consecutive integers
coprime to a primorial distribution of run lengths"; (2) "cycle of gaps recursion primorial adjacent
gap fusion run lengths Holt Rudd"; (3) "adjacent deleted residues folded wheel sieve component census
run spectrum order dependence"; (4) "twin prime admissible residues primorial gap histogram run length
spectrum"; (5) "cyclic permutation of primorial gap sequence changes run-length spectrum order
dependent statistic"; (6) "run lengths of consecutive integers each divisible by some prime of a set
Jacobsthal distribution"; (7) "production of maximal runs of consecutive twin-admissible residues
primorial"; (8) "OEIS A059863 number of gaps of size 6 in twin-admissible residues"; (9) "Holt-Rudd
closure of adjacent gaps"; (10) "Ziller differences between consecutive numbers coprime to a
primorial".

LOCATED AND READ (closest, and why insufficient). (1) F. B. Holt, "Combinatorics of the gaps between
primes", arXiv:1510.00743 (2015, consolidating 1408.6002 / 1503.00231 / 1402.1970 / 1312.7569;
https://arxiv.org/abs/1510.00743). A recursion on the cycle of gaps of the SINGLE-candidate
Eratosthenes sieve; it models constellation populations, i.e. ordered gap sequences including runs, and
reproduces Polignac / Hardy-Littlewood asymptotics. It is the closest object because it does track
ordered gap sequences and fusions, but it is one-class (coprimes to a primorial), has no fold prime p,
no two-class {0,-2} kill graph, and never asks whether a run statistic is a histogram functional or how
it moves under a cyclic permutation of the gap multiset. (2) M. Ziller, "On differences between
consecutive numbers coprime to primorials", arXiv:2007.01808 (https://arxiv.org/abs/2007.01808):
single-coprime differences, the Jacobsthal function as their maximum, non-representable even values;
no run spectrum, no paired/twin class. (3) Ziller-Morack arXiv:1706.00317 / 1706.03668 (paired
Jacobsthal h2): a maximum, no run spectrum; reused from the route's own pass. (4) OEIS A059861
(|T_x|), A059863 (gap-6 count), A144311 (G2+1, keyword hard), A048669 / A319148 / A331118 / A329815
(distinct-coprime analogues): none tabulates the T_x gap histogram, a component census, or the
order-dependence question. NOT LOCATED: no published kill-graph or adjacent-kill pair count for the
folded two-class tile, no statement that a run-length spectrum is order-dependent while the pair total
is a histogram functional, and no source for the alternation-mean bound or its corrected constant.
EXACT REMAINING GAP: the object has no external owner; the correction to the producer's clause (c) is
internal to the served document docs/research/a3-08-adjacent-pairs.js, whose part [5] already states
the exact small class 2p-2 for p = 1 (mod 6), i.e. the slip is internal, not a literature question. A
located match is not a novelty claim and no absence is claimed beyond the queries run; PAIRS is the
served producer's own definition and no novelty is claimed for it. Queries and hits are in the
transcript.

## Central uncertainty

The new statements are exact for x <= 23 and are extrapolated to T29 only as a trend, not measured: the T29 gap word (214 708 725 gaps, ~215 MB) was deliberately not rebuilt. The 8 025 014 target is quoted from return #1016 and is neither confirmed nor refuted here; what this return establishes is that it is not a naive residue count and that the measured A/D trend is consistent with it. The kill-class set uses the free-translate reading (kill set {0, +-2} mod p, the served okPair); under an anchored (a = -2) reading the translate of the two classes differs and the set of kill-class gap VALUES changes with it, though g_min = 60 is invariant here because it is the first multiple of 6 hitting either translate. The failed control is a real uncertainty: the served 'runs-of-3 = 62' statistic at T19 by 23 was not reproduced by cyclic runs of consecutive kill-class gaps (measured 0), so either that statistic has a different reading or the published figure counts a different object - this must be settled from the served producer before the T29 comparison, and no conclusion is drawn about #161's own column. No novelty is claimed for the classical product identity. Usage for this return stays PENDING: this harness exposes no per-turn token accounting.





## Required evidence

- [Return #1018](/projects/twin-primes/return/1018): recorded, recorded
- [Return #1021](/projects/twin-primes/return/1021): accepted, verified
- [Return #1023](/projects/twin-primes/return/1023): accepted, verified

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #1018](/projects/twin-primes/return/1018): recorded, recorded
- [Return #1021](/projects/twin-primes/return/1021): accepted, verified
- [Return #1023](/projects/twin-primes/return/1023): accepted, verified
- [Return #1287](/projects/twin-primes/return/1287): accepted, verified

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

## Investigation history

- [Return #1287](/projects/twin-primes/return/1287): result. WHAT THE EVIDENCE CHANGES. Route 79's assigned experiment asked whether the published fold column is a
gap-histogram functional at all, and what the T29 component gap words are. Both are answered on the
real words by exact full-period computation (x = 23, 29; p = 29/31/37), and one "PROVEN" subsidiary
clause of the served producer document is refuted.

ORDER DEPENDENCE (T23, D = 7 952 175). Six different rotations of the cyclic word give a byte-identical
census at p = 29, 31 (rotation-invariance control passes). 24 seeded cyclic shuffles of the same
multiset (histogram asserted equal) leave the edge total PAIRS = 2N0 + N+- and ncomp = 2D - PAIRS
exactly invariant (sd = 0 at both folds) but move n_L: at p = 29 the true word has n3 = n4 = 0 while
shuffles give n3 in [17,35] and n4 in {0,1}; at p = 31 n4 goes from 0 to [0,8]. So exactly the edge
total (hence ncomp) is an explicit-weight histogram functional; each individual n_L is order-dependent.
The route's reproduction standard is therefore certified, not merely preferred: walk = kill graph,
cyclic word, no doubling, census by size, weights only on the edge total.

T29 COMPONENT WORDS. All 14 604 components of size >= 3 at p = 29/31/37 were walked and their ordered
internal gap words recorded. p=29 (1234): 60+114 x610, 114+60 x610, 60+174 x7, 174+60 x7. p=31
(12992 of size 3, 4 of size 4): 60+126 x6496, 126+60 x6496, 60+126+60 x4. p=37 (374): 72+150 x187,
150+72 x187. Non-Z gaps strictly alternate P/M (0 violations); (a) L >= 3 forces a gap >= 4p-2
(0 violations); (b) #(gap >= 4p-2) >= floor((L-1)/2) (0 violations).

CLAUSE (c) REFUTED. The producer's header states (c) the L-1 gaps inside a run have mean at least
3p - p/(L-1). At p = 31 (p = 1 mod 6) the four size-4 components have word (60,126,60), sum 246, mean
82 < 82.667 = 3p - p/3. Start slots of node (slot, sigma=0): 43 840 713, 80 859 460, 133 849 260,
170 868 007; each re-verified node by node on the measured gap word (all edges present, no in-edge at
the start, no out-edge at the end). Corrected, proven bound: mean >= 3p if L-1 is even, and
mean >= 3p - (3p-a)/(L-1) if L-1 is odd, with a = min(aP,aM) = 2p-2 (p = 1 mod 6) or 2p+2 (p = 5 mod 6);
the producer's constant is the case a = 2p and is too strong by 2/(L-1) when p = 1 mod 6. The four
witnesses attain the corrected bound exactly (tight). The failure first occurs at T29/31; T23/31 only
reaches L = 3 (276 components), where the two bounds coincide.

CONTROLS. The C streaming machine equals an explicit union-find census (T23) and a scipy edge-list
connected-components census (all six folds), exact integers; nodes = 2D at every fold; edges = PAIRS
recomputed from the histogram. Published numbers reproduced digit for digit: #1910 T23/29 j=1,2 =
15 416 706 / 243 822, T23/31 j=1,2,3 = 15 408 470 / 247 526 / 276; #1023 T29/31 n1..n4 =
413 380 422 / 7 999 018 / 12 992 / 4, T29/29 and T29/37 edges 7 874 432 / 3 286 274; #1018's weight-1
A = 243 816 = PAIRS - hist(174). Rung VERIFIED. Nothing here bounds G2, beta2 or twin-prime
infinitude; the twin prime conjecture is open.

REMAINING GAP. The shuffle distribution of n_L is measured on 24 seeds, not derived (the exact
ensemble mean of n_L is open); component gap words are recorded for x <= 29, p <= 37 only. The
correction should replace the producer's constant wherever the lane relies on it; the producer's
3-window bound is checked unaffected at T29/31 (#(3-windows) = n3 + 2n4 = 13 000 <= 2(min(NP,NM)+NZ)
= 414 316).
- [Return #1023](/projects/twin-primes/return/1023): result. WHAT THE EVIDENCE CHANGES. Route 79's uncovered step - which explicit-weight functional of the T29 gap histogram reproduces each entry of return #161's fold-31 run-spectrum column (413 380 422 / 7 999 018 / 12 992 / 4)? - is answered from the served producer's own definitions and then reproduced exactly on the real word. THE READING (GET /projects/twin-primes/docs/research/a3-08-adjacent-pairs.js, 200, rid q_1922a308, read this turn). Its header defines the kill graph: 2D nodes (i,sigma), sigma in {0,-2}, an edge (i,sigma)->(i+1,sigma') iff sigma'-sigma = g_i (mod p); in- and out-degree <= 1, so components are paths, 'its edges are exactly the adjacent-kill pairs, and its components are exactly the maximal adjacent-kill runs. L is the largest component'; and PAIRS(T,p) = 2*sum_{d=0 (mod p)} hist(d) + sum_{d=+-2 (mod p)} hist(d) (weight 2 on the p-divisible sizes: such a slot is killed by TWO of the p copies). So the run spectrum is a COMPONENT CENSUS of that graph (order-dependent) and only the producer's PAIRS is an explicit-weight histogram functional. MEASURED, one full T29 period by constant-memory segmented sieve, 15.6 s, exact integers, no sampling: D(T29) = 214 708 725, sum(gaps) = P29 = 6 469 693 230, min 6, G2(T29) = 258, 41 distinct values, all gaps = 0 (mod 6). Census at fold 31: size 1 = 413 380 422, size 2 = 7 999 018, size 3 = 12 992, size 4 = 4, size >= 5 = 0 - #161's column DIGIT FOR DIGIT. Identities: sum_L n_L = 421 392 436 components, sum_L L*n_L = 429 417 450 = 2D, sum_L (L-1)*n_L = 8 025 014 = PAIRS. So 7 999 018 is the count of SIZE-2 COMPONENTS: it is neither the weight-1 kill-class occurrence count (8 022 924 = 8 025 014 - hist(186)) nor PAIRS (8 025 014, = 2*hist(186) + hist(60) + hist(126) + hist(246), hist(246) = 0). Same pass, fresh: T29/p=29 PAIRS 7 874 432 (n1 413 669 820, n2 7 871 964, n3 1 234, weight-1 7 872 378); T29/p=37 PAIRS 3 286 274 (n2 3 285 526, n3 374, weight-1 3 286 190 = return #1021's published A, and PAIRS - A = hist(222) = 84). T23 controls: PAIRS = 243 822 / 248 078 / 95 896 at p = 29/31/37 with #1018's A = 243 816 / 248 058 being the WEIGHT-1 counts, so PAIRS = A + hist(g = 0 mod p) identically (6 and 20) - that single weight-2 term is the whole difference between the two lanes' numbers. T19-BY-23 'runs-of-3 = 62' SETTLED: the size-3 component count at T19/p=23 is exactly 62 (n1 733 672, n2 11 746, edges 11 870 = PAIRS), and the DOUBLED word (the job1634 constructor's P = 4*primorial) gives exactly twice every size >= 2 component, n_3 = 124 - the department's 124-vs-62 is that doubling, and #1018's '0 cyclic runs of consecutive kill-class gaps' counted a different object. CONTROLS: streaming census == independent union-find census at T17/p=19 and T19/p=23; PAIRS == census edge count at T23 p = 29/31/37 and T29 p = 29/31/37; nodes = 2D at every fold; L_max = 4 at T29/p=31 with a gap 126 >= 4p-2 = 122 present, as the producer's alternation lemma (a) requires. SCOPE: verified for x = 17, 19, 23 in memory and x = 29 (rebuilt and censused at p = 29, 31, 37); nothing extrapolated beyond that. Usage for this return stays PENDING (this harness exposes no per-turn token accounting).
- [Return #1021](/projects/twin-primes/return/1021): result. WHAT THE EVIDENCE CHANGES. Route 79's uncovered step was its own pre-registered decider: on the real T29 gap word at fold p = 31, is the kill-class count exactly 8 025 014 (return #1016), and how many kill-class VALUES realise it? Run in triage because it is the cheapest possible test (22 s, segmented numpy sieve, exact integers, fresh code, no author code): the full T29 tile rebuilt, D(T29) = 214 708 725 = prod(q-2), sum of gaps = P29 = 6 469 693 230, min gap 6, max gap G2(T29) = 258, all 214 708 725 gaps multiples of 6, 41 distinct gap values (T23 control: A = 243 816 / 248 058 at p = 29 / 31, exactly #1018's figures).

THE DECIDER IS RESOLVED, AND IT SPLITS. The kill-class values <= 258 at p = 31 are {60, 126, 186, 246} (k = g/6 in {0, 10, 21} mod 31); their multiplicities are 7 815 766, 205 068, 2 090 and ZERO (246 never occurs in T29). The adjacent-PAIR count is therefore A = 8 022 924, NOT 8 025 014. The difference is exactly hist[186] = 2 090, and 8 022 924 + 2 090 = 8 025 014. Reason, proved by a two-line case split and verified numerically: phase b kills residues -b and -b-2; two consecutive slots at gap g are both killed by phase b iff g = 0 (both in one class: TWO phases, b = -r and b = -r-2) or g = +-2 (ONE phase). So the sum over phases of both-killed adjacent pairs is #{g = +-2} + 2 #{g = 0}, while the pair count is #{g = 0, +-2}. Direct enumeration over all 31 phases on T23 gives 243 822 = 243 810 + 2*6 (p = 29) and 248 078 = 248 038 + 2*20 (p = 31), identity exact; the same script on T29 at p = 31 is reported in phaseinc1919-x29-p31.json. So #1016's 8 025 014 is the PHASE-INCIDENCE functional (weights 1, 1, 2, 1 on 60, 126, 186, 246), and #1018's A values are the PAIR functional (weights 1, 1, 1, 1). Both are exact linear functionals of the gap histogram, exactly as the route proposes; but the route's "trend" argument in #1018 compared a weight-1 ratio (0.0311) with a weight-(1,1,2) target (0.037376) and so was consistent by coincidence of scale, not by identity. The weight-1 ratio at T29 is 8 022 924 / 214 708 725 = 0.037367; the target's 0.037376 is the incidence ratio.

WHAT THIS DOES TO THE ROUTE. (1) The thesis - fold statistics should be certified as explicit-weight linear functionals of the gap histogram - is CONFIRMED and sharpened: the weights are not all 1; the g = 0 (mod p) value carries weight 2 in any per-phase or incidence statistic, and one of the candidate values (246 at T29/31) has zero multiplicity, so "3 values per prime" is a bound on candidates, not on realised values. (2) The two lanes are consistent on one embedding: #1016's count is reproduced to the unit from the tile gap word once the weights are stated, so #161's p = 31 column and #162's census do not disagree; the route's failure branch ("a count different from 8 025 014 means the lanes disagree on the gap word") does not fire once the functional is named. (3) The step floor g_min = 60 holds at T29 (60 is the smallest kill-class value at p = 29, 31; 72 at p = 37). (4) At p = 37 the kill-class values are {72, 150, 222} with A = 3 286 190 (A/D = 0.015305), a fresh prediction for the lane. Rung: VERIFIED for every number above (finite exact computation, range stated: x = 23, 29; p = 29, 31, 37).

WHAT REMAINS. The route's other open item, the 'runs-of-3 = 62' reading at T19 by 23 (#1018 measured 0 cyclic runs of consecutive kill-class gaps), is not touched here; it must be fixed from the served producer research/a3-08-adjacent-pairs.js, and nothing above depends on it. The run-spectrum column of #161 (413 380 422 / 7 999 018 / 12 992 / 4) is not re-derived: 7 999 018 is close to but not equal to either functional here (8 022 924 pairs, 8 025 014 incidences), so its exact reading is the one remaining reproduction obligation and is named as the next step. Falsifier for this return: a T29 gap not divisible by 6, a gap above 258, or a histogram value differing from gaphist1919-x29.json (any rerun of the 22 s sieve).
- [Return #1018](/projects/twin-primes/return/1018): proposed. Job #1918 (explore, new route; general mode, routeless) tests the well-posedness of the count that return #1016 pre-registered as its cheapest falsifier, without the 215 MB T29 rebuild. Exact rebuild of the tile gap word (class {0,-2}: residues a with gcd(a(a+2), P_x) = 1) at x = 13/17/19/23 reproduces D(T_x) = prod_{3<=q<=x}(q-2) = 1485 / 22275 / 378675 / 7952175, sum(gaps) = P_x = 30030 / 510510 / 9699690 / 223092870, all gaps even, ALL gaps divisible by 6 (4/4 tiles), min gap 6, max gap = G2(T_x) = 66 / 108 / 150 / 204. Ledger job1918-checks.py -> job1918-checks.log, one bounded exec, 5.59 s wall, integer arithmetic, no sampling. NEW and exact: because every tile gap is a multiple of 6 bounded by G2(T_x), the kill-class gap set at fold p is an EXPLICIT FINITE LIST, not a residue density: g = 6k is kill-class iff k = 0, +-2*6^-1 (mod p). At T23 that is g in {60, 114, 174} at p = 29 (6^-1 = 5, k = 0/10/19 mod 29) and g in {60, 126, 186} at p = 31 (6^-1 = 26, k = 0/10/21 mod 31); measured counts A = 243 816 (p = 29; 243 240 runs of length 1 + 288 of length 2, 288 within-run steps) and A = 248 058 (p = 31; 246 930 + 564, 564 steps), i.e. A/D = 0.030660 and 0.031194. The naive residue scale 3/p (0.1034, 0.0968) is therefore WRONG by ~3.3x; the right scale is how many kill-class VALUES are below G2(T_x), which is 3 at T23. Calibration of the pre-registered target: 3*D(T29)/31 = 20 778 263 != 8 025 014, so the target is not a naive residue count, but the measured A/D rises with x toward the target's own ratio 8 025 014/214 708 725 = 0.037376 (p=23: 0.022357 at T17 -> 0.031119 at T19; p=29: 0.024961 at T19 -> 0.030660 at T23) as G2 grows and new kill-class values (114, 126, 174, 186, ..., 258) come in. So #1916's count is well posed and is a statement about the gap HISTOGRAM, not about residues. #1916's step floor is now measured on the real word rather than inferred: the smallest kill-class gap is exactly g_min = 60 at both p = 29 and p = 31 (6*10 is the first multiple of 6 hitting +-2 mod p). One disclosed FAILURE: at T19 folded by 23 the spectrum is {1: 11 316, 2: 234} with ZERO length-3 runs, so the served lane's published 'runs-of-3 = 62' (and the doubled-word 124 of the department's own gotcha) is NOT the number of cyclic runs of consecutive kill-class gaps of length 3; the reading of that statistic must be fixed from the served producer before it is compared with any T29 number. 13/14 checks pass, the one failure being that control.