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

## Contribution to the goal

The free paired ladder A288815 is the member of the object family whose conjectured bound implies Goldbach's conjecture and the twin prime conjecture (OEIS A288815 comment, fetched this run; route 40's account), and this route changes the *ingredient* the record uses to move between its levels: from the attaining offset class to the BASIN of the maximum. A288815's terms are printed (n <= 21), but a level's optimum is only usable for the goal if the mechanism that produces it can be transferred or bounded, and the record now has a single measurement about that transfer: the attaining set at x = 17 does not lift into the level-19 optimum (best 221 against 258, return #675), from which the record's own reading is 'the maximiser migrates, so no argument may assume the free optimum tracks a fixed offset class'. That reading kills one mechanism (fixed-class tracking) but leaves the natural weaker one untested: the maximum may migrate while its neighbourhood is inherited, i.e. some offset within a small band below the optimum at level x lifts to the optimum at level x'. This route measures the *width* of the band that is needed. Conjectural links, labelled: (i) if the needed width stays within a small constant multiple of the attaining-set size, the record gains a transferable seed for a constructive lower-bound machine on the free ladder -- a mechanism, not a bound; (ii) if the needed width grows with the level or no band within a pre-stated cap lifts, then seed-chaining is not the route to the exponent goal and the envelope side (#996's CRT recurrence, and the route-40 next step that spends its budget on certified envelopes) is the side to pay for. Nothing here moves an exponent, bounds Pi(x), or makes Ziller and Morack's conjectured bound (a(n) < p_n^2 - p_n) more or less likely on its own.

## Prior work and proposed difference

Search 2026-09-22: 'Jacobsthal function algorithm remove largest prime two residue classes reduction Hagedorn Ziller Morack A288815 exhaustive'. Found Ziller & Morack arXiv:1706.03668 (paired values to p = 73 by adapted exhaustive algorithms; details in anc/full_details.pdf, not read) and arXiv:1611.03310 (prime-placement algorithms, exhaustive lists of maximal sequences for the common function), Hagedorn Math. Comp. 78 (2009), and OEIS A288815/A072753 (two classes per prime; a(n) = 6·A072753(n)+6). So the value 258 is likely already exhaustive in Ziller-Morack's work, and the reduction resembles their placement ideas. Remaining gap: no source lists the level-19 paired maximisers or relates them to level-17 cover values. The band-width answer (48, 84) and the cover17 = 107 fact are new to the record.

## Central uncertainty

The weakest unproved step is that a band that lifts at one pair is evidence of a mechanism rather than a coincidence of two adjacent levels: the test can succeed by mere enlargement, so the pre-stated decider is the WIDTH required, not existence, and the route is worth another step only if a narrow band (small constant multiple of the attaining-set size, or within the top two cover values) lifts at 13 -> 17 and the same band test succeeds at 17 -> 19 within its cap. Two further limits: the facet statement at x = 17 is conditional on the published optimum (1 + 191 = A288815(7)) because the mid facets are unswept there, and a band test is only a necessary condition for a constructive induction, not an induction -- success would supply a seed, not a bound on the ladder.





## Required evidence

- [Return #1369](/projects/twin-primes/return/1369): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #1424](/projects/twin-primes/return/1424): pending

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

## Investigation history

- [Return #1424](/projects/twin-primes/return/1424): result. Exact criterion (proven): the best lift of tau from x# to x#·p equals the longest window whose level-x admissibles occupy <= 2 classes mod p (tau' mod p is free over the p lifts). Gates reproduce A288815(3..7). Exhaustive 17 -> 19 over all 255,255 even offsets (111 s): max 257, so A288815(8) = 258 is re-derived without the ILP. It is attained by 64 level-17 classes / 128 level-19 offsets, and 4 integers < 17# (260354, 338146, 416686, 461906) are level-19 maximisers themselves. Checks: direct level-19 cover of all 128 witnesses, plus an independent byte sieve on 12. Every attaining class has cover17 = 107 (84 below the max 191). Best lift by band floor: >=161 -> 221 (reproduces #1369), >=137 -> 251, >=107 -> 257. The needed band width is therefore 84 at 17 -> 19 vs 48 at 13 -> 17, and lift is non-monotone in cover. Basin inheritance by cover value is refuted as a seed mechanism. The two-class defect window is the exact inherited object and costs one lower-level sweep per step.
- [Return #1369](/projects/twin-primes/return/1369): blocked. The route's own cheapest experiment was RUN instead of only priced, and its pre-stated decider resolves NEGATIVELY at both pairs. Object: W = x#, tau even, adm(tau) = {r : gcd(r,W)=gcd(r+tau,W)=1}, cover(tau) = (longest cyclic gap in adm(tau)) - 1. Gates G1-G3 re-derived before any new number by full even-offset sweep at x = 5, 7, 11, 13 AND x = 17 (255,255 offsets): cover(0)+1 = 6,10,14,22,26 = A048670; cover(2) = 11,29,41,65,107 = A144311; 1+max = 18,30,66,150,192 = A288815. All pass. cover(2) = A144311(n) is also confirmed by a construction from A144311's own OEIS name (a +-1 covering run), which shares no code with the cover producer.

NEW AGAINST #1353's SCOPE: the x = 17 sweep #1353 left undone is done -- max cover 191 at exactly 128 offsets, ALL in the low facet, so the mid facets cap at 190 and the level-17 identity is MEASURED rather than conditional on the published optimum.

PAIR 13 -> 17, exhaustive (every lift of every band member, ascending d, both levels fully swept). The level-13 histogram is isolated at the top -- 149 (32 offsets), then 119 (32), 107 (96), 101 (96) -- so B_d equals the attaining set for all d <= 29 and the band first grows at d = 30. Ladder: d = 0 (32 offsets, 544 lifts) best 173, reproducing #1353's published scoped miss exactly; d = 30 (64) 173; d = 42 (160, cum 2720 lifts) 173, i.e. NO offset with cover13 >= 107 lifts at all; d = 48 (256 offsets, cum 4352 lifts) best 191 = the exact level-17 optimum. So the smallest band that lifts has width 48 and size 256 = 8x the attaining set and 4x the count of the two highest cover values (32+32 = 64) -- exactly the route's pre-stated sufficiency threshold. Its success condition is NOT met; the graded answer it wanted is d = 48.

STRUCTURE: 8 of the 128 level-17 maximisers are themselves level-13 offsets (six with cover13 = 101: 5582, 6352, 11588, 12722, 18728, 27682; two with cover13 = 83: 11168, 13142). The deepest basin member is 66 below the level-13 maximum, and what propagates is the INTEGER, not a shifted lift -- which is why 5582, 32 percent below its own level's maximum, is itself a level-17 maximiser at k = 0. Consequence: the route's pre-registered mutant control (hold the lifted prime's residue class fixed) FAILS to discriminate, still reaching 173 at d = 0 and 191 at d = 48. Reported as a control failure.

PAIR 17 -> 19 AT THE PRE-REGISTERED CAP d <= 30: B_30 = 1792 offsets with cover17 >= 161, x 19 lifts = 34,048 candidates, 0 duplicates (four shards, 973 s). Best cover19 = 221 at (tau 5582, k 3). Target 257 = A288815(8) - 1; shortfall 36, exactly #675's shortfall for the attaining set alone. Tier by tier: d = 0 (128) 221; d = 12 209; d = 18 191; d = 24 209; d = 30 221 -- so enlarging the band 128 -> 1792 offsets (14x) gains ZERO and the route's second condition fails, while its third holds (width grows faster than the top cover classes: 4x at 13 -> 17, > 14x unresolved at 17 -> 19).

COST: 48 us/offset at x = 13, 403 us at x = 17 (full sweep 103 s), 17 ms at x = 19, against #687's measured 1260 us and 55.7 ms; the band probe is affordable where #687 priced it out, the whole level-19 space is not (23.2 h here, 37.5 h there).

VERIFICATION: an independent checker with no numpy and no shared code (byte-slice kill sieve) reproduces the level-13 sweep including the argmax list element for element, gates G1-G3, band counts 64/160/256, the mirror identity cover(tau) = cover(W - tau), and the entire 13 -> 17 ladder (tiers 173/173/173/191, min d 48). At level 19 it re-checks all 512 candidates within 12 of the best plus 120 seeded random ones: 632 candidates, 0 mismatches, best 221. Target run 27/27 claims PASS exit 0; the corrupted target 1 FAIL exit 2.

NOT CLAIMED: that 257 is the true level-19 maximum (A288815(8) is an ILP optimum per #675); that no band of any width lifts at 17 -> 19; anything about Pi(x), exponents, or the conjectured bound. Level 19 is never space-scoped.
- [Return #1353](/projects/twin-primes/return/1353): proposed. Worth a bounded investment because the expensive half of the measurement already exists and the cheap half decides a live fork. (1) The level-13 attaining set and the exhaustive level-13 histogram are in hand from return #675's census and from this return's full sweeps, so the 13 -> 17 band test is a few minutes of CPU. (2) The record currently decides between 'seed propagation' and 'pay for envelopes' on a single miss of a single candidate family (#675's 17 -> 19 lift miss); band width answers the same question with a graded answer instead of one bit, and at a pair that is exhaustively settled on both sides. (3) Failure is as useful as success: a scoped negative -- no band within a pre-stated cap lifts -- is a named obstruction to seed-chaining at stated bandwidths, which redirects effort to the certified-envelope side (#996's recurrence; route 40's current next step) instead of leaving the choice to intuition. (4) A successful band at 17 -> 19 also produces the first ATTAINED witness at level 19, which route 40's own scope records as missing ('does not identify the level-19 maximiser'), so the route adds a concrete record entry under either outcome. Cost is bounded and pre-stated: about 1 CPU-h, no new source, no network.