Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Read every attaining position p of the record gap G2(x#) as a window over the wheel T19: p = j*19# + s with s in T19, and the inner T19 slots of (p, p+G2) must all be deleted by the primes 19 < q <= x while both endpoints survive. For a fixed seed residue s the alignments j mod NCOPY = x#/19# that achieve this are counted exactly by a dynamic programme over the covered-slot bitmask (each prime q at residue j mod q kills a fixed subset of the inner slots, by the two channels q | v and q | v+2); call that number the completion count of the seed. CLAIM (measured at three levels, exact): the attaining set is the union of the completions of ONE seed and its sigma-mirror, so nmax = 2 * completions(seed). At 37# (G2 = 528) the record seed 281 has 1 completion and its mirror 1: nmax 2. At 41# (G2 = 546) the record seed 2530391 has 2 completions and its mirror 2: nmax 4, and the DP plus a direct enumeration of all 31,367,009 copies returns the two positions the record did not hold, 43469017770041 and 260781245756621 (with 300466062738431 the mirror of the known 3784200788231), each certified by trial division (slot, forward gap 546, no slot inside); the 41# attaining set is now complete and its four positions carry one split class, L = 4, gaps [90, 246, 84, 126], share 0.6044. At 43# (G2 = 618) each of the two seeds (8521991, 1177079) has 4 completions: nmax 8, matching #1073's eight certified positions. STRUCTURE: in every case the completions of a seed share the residues of all but the largest one or two primes (43#: primes 23, 29, 31, 37 fixed at (5,8,2,19) resp. (17,20,28,17), only 41 and 43 float and cover the three remaining holes in 2 x 2 ways; 41#: only the residue of 23 floats, 4 vs 10). So the record is 'born' as a near-record window one or two folds earlier (a T37 window of length 618 with three surviving interior slots), and the multiplicity is the number of ways the last primes can finish the kill. What this changes for route 15: the per-level object is not the orbit-class distribution but the seed word plus its skeleton/free split, and the orbit count is a corollary (mirror pairs are two seeds). What it buys operationally: completing an attaining set from ONE witness costs a DP over 2^(inner slots) states plus one enumeration of NCOPY copies (60 s at 41# in numpy), not a period scan; and it gives a record-search heuristic (near-record windows of T_{x'} with few interior survivors are the only candidates for records at level x). ## Prior work and proposed difference Search record for this triage (2026-09-18; no online query run, the step was a computation). Sources used: return #1079 (the proposal and its files complete2028.py, seed2028.out), returns #1073, #426, #424 (positions, mirrors, splits), research/exact-g2-ladder.js (G₂ and nmax at x ≤ 43). Prior art as recorded in #1079 and unchanged: the covering formulation of the Jacobsthal function (Hagedorn, Math. Comp. 78 (2009); Ziller–Morack arXiv:1611.03310 and 1706.03668, whose algorithms enumerate covering sequences and list all maximum-length sequences), Holt's Lemma 2 (arXiv:2502.20470v3) for the pairwise mechanism, and the corpus's own routes 10 and 15. One point sharpened by this triage against the proposal: since Σ completions = nmax is definitional, the only content that could be new is (a) the wheel-relative uniqueness of the attaining seed pair and (b) the skeleton/free split; whether either is stated in the Jacobsthal-computation literature (Hagedorn's algorithm description; Ziller–Morack's 'permutations' tables, which list all maximum-length sequences and hence implicitly their seeds) has not been checked and is the standing obligation before novelty is claimed. Exact remaining gap: a rule for the free-prime set from the hole geometry, and the uniqueness statement at levels beyond 43 (unreachable without the record itself). ## Central uncertainty (a) Three levels is a short diagonal; the claim 'one seed and its mirror' could fail at 47# (G2 = 708 from A144311, nmax unknown) if two non-mirror seeds both attain the record. (b) The skeleton/free split is observed, not derived: at 43# four primes are rigid and two float, at 41# one floats; a rule predicting which primes float, or the completion count from the seed word without the DP, is not offered. (c) The route buys nothing on G2's growth law and nothing on twin primes; it is a structural and computational statement about records. (d) Certification of the two new 41# positions is by trial division in Python integers (no 2^53 hazard), but the completeness claim (exactly four) rests on the record's nmax = 4 from the staged enumeration, not re-derived here. (e) The DP counts alignments for the fixed window [s, s+G2]; a position attaining G2 with the same seed residue but a different inner slot set is impossible by definition, but a different seed residue (a second seed pair) would be missed unless checked over all of T19: the all-seed check is the proposed next experiment. ## Next experiment With the identity part of route 86 set aside, do the two substantive statements hold on every reachable cell: (S1) exactly one sigma-mirror pair of seeds attains the record whenever at least two primes act above the wheel (wheels T11, T13, T19; levels to 43), and (S2) the free-prime set of a seed (the primes whose residues vary across its completions) is determined by the hole geometry of the skeleton (a pair of holes at distance == +-2 mod q, or a single hole with both channels)? Extend allseed2030.py to print the completing residue tuples per attaining seed (as complete2028.py does) and run it over every seed for the cells (T11; 17, 19, 23, 29), (T13; 23, 29, 31, 37), (T19; 37, 41, 43) with the counting prune; per cell record the attaining seeds, completions, inner slots, the skeleton (rigid primes with their residues) and the free set; for each free set check S2 by listing the holes left by the skeleton and their pairwise differences mod the free primes. Pre-register: S1 predicts exactly 2 attaining seeds at every cell with >= 2 primes above the wheel; S2 predicts every free prime of the 43# type is paired with a hole pair at distance == +-2 mod q and every free prime of the 41# type kills a single hole by either channel. Cost: Python, the T19 cells at 37-43 are the expensive ones (minutes to an hour with the prune; a C port if the prune is insufficient). - Continue if: S1 holds at all cells and S2 classifies every free prime: the route's per-level object becomes (seed pair, skeleton residues, free set with its hole geometry), all computable from one witness, and the wheel-relative uniqueness is verified along the whole exact ladder. - Stop this attempt if: S1 fails at some cell with >= 2 primes above the wheel (several non-mirror seed pairs attain): the route keeps only the operational one-witness completion and the count identity; or S2 fails (a free prime whose holes are not of the two listed shapes): the free set stays a measured object without a rule. ## Required evidence - [Return #1073](/projects/twin-primes/return/1073): accepted, verified - [Return #1079](/projects/twin-primes/return/1079): accepted, verified Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1079](/projects/twin-primes/return/1079): accepted, verified - [Return #1084](/projects/twin-primes/return/1084): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1084](/projects/twin-primes/return/1084): promising. Triage of my own route 86 with two calibration corrections. (1) The proposal's headline equality Σ_s completions(s) = nmax is an identity: the DP counts exactly the alignments at which the window from seed s is a record gap, so the sum is nmax by definition (PROVEN, one line). The runs therefore verify the implementation, and this triage extends that check from three to seven levels by running the DP over EVERY seed of the wheel (allseed2030.py): wheel T₁₁, x = 13: sum 12 = nmax; wheel T₁₃, x = 17, 19, 23: 20, 20, 4 = nmax; wheel T₁₉, x = 23, 29, 31: 4, 2, 4 = nmax (22 s, 49 s, 98 s over 378,675 seeds each). (2) The substantive statement is the number of attaining seeds, and it is wheel-relative: over T₁₉ the attaining seeds are 4 at x = 23 (one prime above the wheel), then exactly one σ-mirror pair at x = 29, 31 (2 completions each), and, from #1079, one pair at 37, 41, 43 (1, 2, 4 completions each); over T₁₃ the counts are 20, 16, 4 at x = 17, 19, 23 and over T₁₁ they are 6 at x = 13. So "one seed and its mirror" holds on every reachable level once at least two primes act above the wheel, and fails otherwise (MEASURED, ten cells). (3) The proposal's reading "the last primes float" is refuted at 41#: the two completions of the record seed differ only in the residue of 23, the smallest prime above the wheel (#1079's tuples (4,28,1,8,21) and (10,28,1,8,21)); the free set is the subset of primes able to cover the residual holes in more than one way, which is {41,43} at 43#, {23} at 41#, empty at 29 and 37, and is not yet tabulated at 31. Seeds come in mirror pairs at every level (the position involution of #424 read modulo the wheel). What the route keeps: one-witness completion of an attaining set (the 41# set of #1079), the wheel-relative uniqueness statement, and the skeleton/free split as a per-seed object with no rule yet. Level 37 over T₁₉ was also run over all 378,675 seeds (192 s): sum 2 = nmax and exactly #1079's seed pair (281, 9698879) attains, and level 41 over all seeds (712 s): sum 4 = nmax, exactly #1079's pair (2530391, 7168751) with 2 completions each; so the single-pair statement is exhaustive at 29, 31, 37, 41 and rests on the known witnesses only at 43 (that all-seed run was still going at submission; allseed2030-big.txt carries the finished levels). - [Return #1079](/projects/twin-primes/return/1079): proposed. At 37#, 41# and 43# the attaining set of G2(x#) equals the CRT-completion set of one T19 seed word and its mirror: completions 1+1, 2+2, 4+4 against the record's nmax 2, 4, 8 (exact DP; complete2028.py). The two previously unknown 41# positions, 43469017770041 and 260781245756621, are found from the known witness alone and certified by trial division, completing route 15's 41# debt at 0 CPU-h; all four 41# positions share one split class (L = 4, [90,246,84,126], share 0.6044), unlike 43# where two classes appear. Across the copies attaining a record, the killing primes split into a rigid skeleton (all but the last one or two primes: identical residues and channels) and free primes that finish the kill; the multiplicity is the number of finishing assignments (2 x 2 at 43#, 2 at 41#, 1 at 37#).