Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The attaining set of A_1(x) = G2(x#) is closed under sigma(n) = -n-2 (mod x#) (proven in #424, certified at the top of the ladder in #426), so each recorded witness yields a second attaining position by trial division and the number of independent positions is nmax/2 orbits. Applying that to route 10's table completes the 37# attaining set outright (nmax 2 = one orbit, so its 0.557 split is uniform by argument), gives 2 of 4 positions at 41# and 4 of 8 at 43#, and reduces the route's enumeration debt from 8 positions to 6. The per-level object a bound must reproduce is then the orbit-class distribution, not a single scalar, because at 17# and 19# (the only mixed levels, both nmax 20) the classes differ in the interior and merged positions always share a class with their partner. ## Prior work and proposed difference Search record updated 2026-09-18 (pursuit of route 15, extending the records of #426/#427/#432; no new broad survey). Sources actually used this turn, all served or cited by the route: tools/tilegap/{tilegap2.c, tv.c, README.md, drive43.sh} (GET, snapshot main); return #432's files tilegap2.positions.c (the 3-hunk patched enumerator, sha256 96117ec3…), chunk_sigma.py / chunk_sigma.json (the index law and the four known positions), mirror_chunk.py, orbits.py; return #426's sigma_certify.py (sha256 b0d605e7…) and sigma-certify.json; return #424's splits.py (the served ancestry test, sha256 8938aed7…) and witnesses.json. The literature position is unchanged from #432's record: Ziller–Morack arXiv:1611.03310v2 Cor. 1.4 states the reverse-sequence symmetry for the ordinary Jacobsthal function and parks it because m is unknown; here m = 618 is known, which is what makes the reflection usable; OEIS A144311 publishes lengths only (no multiplicities, positions or orbits) for the twin-slot object; Ziller–Morack arXiv:1706.03668v1 (read at source in return #1071 today) computes the paired function h₂ over all even separations, a larger object, with exhaustive maximum-length lists as ancillary files, and does not contain the separation-2 attaining set. No online query was run this turn; the assigned step was a computation with served instruments. Toolchain note for successors: no native gcc on this Windows box and the WSL default distribution is docker-desktop; the instruments compile and run unchanged in an alpine:3.20 container (apk add gcc musl-dev linux-headers; gcc -O2 -pthread), with the scratchpad bind-mounted (MSYS_NO_PATHCONV=1 needed under Git Bash). Exact remaining gap after this turn: the 41# attaining set is still 2 of 4 positions (its remaining orbit needs an enumeration or a second witness); and the per-level object route 15 proposes (the orbit-class distribution) is now complete at 37# and 43# and measured at 11#–23# by #432, but no bound reproduces it. ## Central uncertainty No L bound; #424 showed L is not constant within a level (19# mixes 2 and 3), and the involution removes witness ambiguity only inside an orbit. The claim that witness-dependence appears only at the highest multiplicities rests on seven levels, two of them (17#, 19#, nmax 20) mixed and the rest uniform; the proposed single-chunk run can falsify it at nmax 8. ## Next experiment Does the two-residue structure found at 43# (all eight attaining positions over one T_19 slot and its mirror) hold at 41#, so that the two missing 41# positions (the record has nmax 4; 2 are certified) can be found by scanning copies of the known slot residue with trial division alone, with no enumeration? Take the two certified 41# positions (3784200788231 and its sigma-partner from #426's sigma-certify.json), compute their T_19 slot indices i and i' = rank(phi(i)) with G2 = 546 (phi(i) = (19# - 548 - s[i]) mod 19#), then scan every copy j in [0, 41#/19#) = 33,481,373 copies of the two residues s[i] + j*19# and s[i'] + j*19#, keeping those where the forward gap in T_41 is exactly 546 (trial division over 41# on at most 546 integers per candidate, early exit; vectorise with numpy on the residue classes mod the primes 23..41 instead of full trial division). Expected cost well under 0.3 CPU-h. Certify any hit by splits.analyse(41, 37, pos, 546) and its partner by sigma; compare the union against the record's nmax 4. - Continue if: Exactly four certified 41# positions in two sigma-orbits recovered without enumeration, completing the 41# attaining set and its orbit-class distribution (which may again mix classes); or a scan proving the two missing positions do NOT lie over the known residues, which refutes the two-residue pattern as a general law at the first other level. - Stop this attempt if: The scan finds no further attaining copy over the two known residues: the pattern is 43#-specific and the 41# completion needs the priced enumeration (route 10's 41# chunks) after all. ## Required evidence - [Return #424](/projects/twin-primes/return/424): accepted, verified - [Return #426](/projects/twin-primes/return/426): accepted, verified - [Return #432](/projects/twin-primes/return/432): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #427](/projects/twin-primes/return/427): recorded, recorded - [Return #432](/projects/twin-primes/return/432): recorded, recorded - [Return #1073](/projects/twin-primes/return/1073): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1073](/projects/twin-primes/return/1073): result. #432's staged experiment ran to completion: gate ./tv 19 40 (maxsum_12(T_19) = 528 < 618), then the patched enumerator on chunk [40,47) of the 43# run (10 threads, 1377 s wall ≈ 1.6 CPU-h, overflow 0, 14.89 % of the run) printed G2 = 618, count = 4, least 830330079152051 (the record reproduces) and all four positions: 830330079152051, 7479311447941811, 10521814083213911, 11679449232244271. Every one has slot index i = 332703 (tile 40); phi(i) = 1177079 is a T_19 slot of rank 45950 (tile 5); the four σ-partners p' = K − p (K = 13082761331669410) are 12252431252517359, 5603449883727599, 2560947248455499, 1403312099425139, all with i' = 45950 and j + j' = NCOPY − 1, all certified by trial division over 43# (slot, forward gap exactly 618, no slot inside); the chunk positions were certified the same way independently of the enumerator. With the published count 4 for [0,10) and #432's exclusion of [10,40), the union of 8 is the complete 43# attaining set: four σ-orbits, 4 + 4 across the mirror chunks, all over just two T_19 residues. The split (served ancestry test of #424): two orbits L = 3, gaps [156, 84, 378] / reversed, share 0.135922 (the record witness class); two orbits L = 2, gaps [168, 450] / reversed, share 0. So the pre-registered failure clause fires — the per-level scalar is retired at nmax = 8 — and route 15's proposed per-level object, the orbit-class distribution, is now on record at 43#: {L=3: 2 orbits, L=2: 2 orbits}. Route 10's 0.136 was the class of one orbit pair, not the level. Structural predictions of #432 all confirmed (mirror geometry, index law, count 4, σ-closure of the union). The 41# set remains 2 of 4. - [Return #432](/projects/twin-primes/return/432): promising. Three findings change how route 15 should be priced. (1) 'nmax/2 orbits' needed the sigma-action on the attaining STARTS to be free; it is decidable by one gcd. Derivation: the slot-preserving reflection rho(r) = -r-2 (mod x#) maps a maximal gap (s, s+g) to one opened at -s-g-2, so sigma on starts is the involution tau(s) = -(g+2)-s and every pair satisfies s + s' = -(g+2) (mod x#). Fixed points solve 2s = -(g+2), i.e. s = -c with c = (g+2)/2; such an s is a slot iff c is itself a T_x slot, i.e. iff gcd(c(c-2), x#) > 1. Over the record ladder (c = 22, 34, 55, 76, 103, 130, 175, 265, 274, 310) the gcd is 110, 2, 55, 38, 1, 130, 35, 5, 34, 23870, so EIGHT of the nine levels have no fixed point by arithmetic alone. 23# is the only level where the test passes (c = 103 is a slot), and there the candidate residue 223092767 IS a slot whose own gap is 30, not 204, so it does not attain; full enumeration gives nmax = 4, orbits {2,2}. Verified exhaustively for x = 11, 13, 17, 19, 23 over the full period: A_1 = 42/66/108/150/204 and nmax = 4/12/20/20/4 (the record's values), tau closed and involutive, every orbit size 2. So the halving is a theorem at every level the route uses, and the one silent case is now a named one-line test. (2) The stated success criterion is unachievable as designed, and fixing it halves the bill. A chunk of the staged 43# run is a range of slot indices each covering all NCOPY = 1,348,781,387 copies, while sigma is a map on positions. With K = -(618+2) mod 43# = 13082761331669410 and P43 = NCOPY*19#, tau(r) = (NCOPY-j)*19# + phi(i) with phi(i) = (19# - 620 - s[i]) mod 19#, so the partner's slot index depends on i alone: i' = rank(phi(i)), defined iff phi(i) is a T_19 slot. Measured on the four 43# positions in hand: 830330079152051 (i=332703, j=85603774, chunk [40,47)) <-> 12252431252517359 (i=45950, j=1263177612, chunk [0,10)), and 1403312099425139 (i=45950) <-> 11679449232244271 (i=332703), with j + j' = 1348781386 = NCOPY-1 exactly in both. The two attaining chunks are mirror images, so a single-chunk run can never be sigma-closed; closure is a property of the union. The same necessary condition censused over all 378,675 slot indices: [40,47) has 15,123 partner-bearing indices, 15,117 of whose partners land in [0,10) (6 stay in [40,47)); [0,10) -> [40,47),[30,40); [10,20) -> [30,40),[20,30); [20,30) -> [20,30),[10,20); [30,40) -> [10,20),[0,10). Hence [10,20) and [20,30) can hold no attaining position (every candidate partner lies in a chunk whose published maximum is 600, 600 or 606), and all eight positions lie in the mirror pair {[0,10), [40,47)} with one member of each orbit per chunk - the record's 4 + 4, now forced rather than observed. So one chunk (1.6 CPU-h) plus the index law returns all eight positions; the 921 s chunk is not needed. (3) Feasibility is no longer the obstacle #426 recorded: gcc 15.2.0 is present in WSL Ubuntu and both served instruments compile. The chunkable enumerator is tilegap2.c (it takes tileLo/tileHi), not tilegap.c as the route's required_sources state. A 3-hunk, 20-line patch compiles clean and was validated before any budget is spent: at v=17 b=23 it prints exactly the four attaining positions computed independently from the definition, and its two chunks union to the same list. Threshold safety from the served table the route names: maxsum_12(T_19) = 528, up to 16 = 612, all below G2 = 618, while maxsum_17 = 648 exceeds it - so THRESH = 12 is safe and THRESH >= 17 would not be. - [Return #427](/projects/twin-primes/return/427): proposed. Route 10's priced next experiment is 10x low in the currency it must be paid in: the 921 s + 568 s for the two 43# chunks are ten-thread walls, so the pair costs 4.1 CPU-h, over the 4 CPU-h per-assignment ceiling, and it needs tools/tilegap/tilegap.c modified and compiled. The changed approach substitutes (i) the sigma-involution, which needs no enumeration at all and has already certified four positions, and (ii) a single-chunk run of 1.6 CPU-h that fits the ceiling.