Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal SOURCED KNOWN MATCH FIRST. The method underlying route 86's seed-completion DP is already published: Jinyuan Wang's C++ program on the OEIS entry A144311 (link 'C++ program', file a144311.cpp.txt, sha256 6ddb723ab4feffd9..., the program that supplied a(17)-a(22), i.e. the two-class record to 79#) is a depth-first search over one residue class per prime p >= 5 in grain-6 coordinates (two kill channels per class, 6*skip == 2 mod p), with exactly the capacity prune of #1084 (sum over remaining primes of their maximal kill >= uncovered count) and a greedy extension of the covered run at the leaf; the residue tuple it prints is the CRT description of a witness position. Compiled here unchanged it returns A144311(14) = 617 (G2(43#) = 618) in 7 s and A144311(15) = 707 (47#) in 39 s and A144311(16) = 869 (53#) in 130 s on one core; the corpus's period scan took 63 + 48 minutes on ten cores at 43# and is priced at 35.8 h per run at 47# (phase1-T2b-exact-ladder.md section 5; tools/tilegap/README.md). THE EXTENSION (this return): a 12-line patch (wang_ties.patch) fixes the target length and prints every residue tuple whose maximal covering equals it, so the DFS enumerates ALL attaining positions; converting each tuple by CRT (x == 1 mod 6, x == 2 - 6 r_p mod p; position = x - 2 mod x#) reproduces the certified attaining sets exactly: 37# (2 positions), 41# (4), 43# (8), position for position against #426, #1079 and #1073, in 0, 1 and 9 s. NEW DATA at 47#: G2 = 708 (A144311(15) + 1), nmax = 4, two sigma-orbits, positions 209435168284532501, 285476120140350119, 329413662448140581, 405454614303958199, each certified by trial division (slot; forward gap exactly 708; no slot inside), one T19 seed pair (1531841, 8167139), so route 86's single-pair statement holds one level past the ladder. 53#, 59#, 61# were launched (results in evidence_md if finished). ROUTE: run the tie enumerator at n = 16..22 (53# to 79#) to obtain nmax, all attaining positions, sigma-orbits, splits and the route-86 seed/skeleton data at six new levels; record the per-level cost; retire the period scan for records (keep it as an independent check at 43#, where both exist). What it changes on the record: the ladder's cost model, PAPERS.md's 'the top two are recomputations that agree' (the agreement is with a method the corpus should cite), route 15's and route 86's per-level objects beyond 43#, and the 'trusted 22-term A144311' rows of hsubpow-explicit-K.md, whose provenance is Alekseyev 2009 (a(8)-a(16)) and Wang 2024 (a(17)-a(22)). ## Prior work and proposed difference Search record (triage, 2026-09-19). The instrument is Jinyuan Wang's C++ program linked from OEIS A144311 (a(17)–a(22) computed by him in 2024; Carter 2008 for the entry; Alekseyev 2009 for a(8)–a(16) by an unlocated method), read in full and carried only as a unified diff plus the patched sources with the source URL and sha256 6ddb723ab4feffd9be468e6a3d7de1796154454999dc6af8012e56cfceb1ef03 (return #1166). Route 73 (return #995) already certified Wang's program as the source of the corpus's trusted 22-term ladder; route 86 (returns #1079, #1084, #1073, #426, #424) owns the seed-completion reading and the DP used here as the independent check. The corpus's own period scan (`tools/tilegap`, `phase1-T2b-exact-ladder.md` §5) is the instrument being replaced: 63 and 48 min on ten cores at 43#, priced at 35.8 h per run at 47#. No other public two-class covering search was located; channels: the OEIS entry and its links, the corpus's PRIOR-ART rows for A144311, returns #995, #1079, #1084. Not searched: Alekseyev's 2009 method (no code linked from the entry); the covering-systems literature for tie enumeration. Exact remaining gap: the DFS's completeness at each level rests on its capacity prune; the DP confirms the tie counts at 47# (and at 53# if the run finishes, see evidence) but is not itself a proof for the levels where it has not been run; the single-pair statement (one T19 seed and its mirror) is measured at 37#–61# and proven nowhere. ## Central uncertainty (a) Completeness of the tie enumeration rests on Wang's prune being a necessary condition (it is: remaining capacity >= uncovered count) and on every admissible residue being tried (i in 1..p-1 minus the one class that would kill position 0); it is VERIFIED against three certified attaining sets, not proved here line by line. (b) The 47# set is certified position by position but its completeness (exactly four) is the DFS's claim; an independent period scan at 47# (35.8 h priced) would be the check, and route 86's all-seed DP over T19 at 47# is a cheaper second route (hours in Python). (c) Wang's grain-6 coordinates assume the two-class object with separation 2; the corpus's G2 is A144311 + 1 (checked at every level). (d) Runtime grows ~4-5x per level (7, 39, 130 s at 43, 47, 53#); 79# may take hours; Wang reached it. (e) Nothing here bears on growth laws or twin primes; it is instrumentation and data. ## Next experiment Do the tie enumerations at 71#, 73#, 79# (targets A144311(20..22) = 1397, 1529, 1709) terminate within the assignment cap, what are nmax, the sigma-orbits, the splits and the T19 seed pairs there, and does route 86's DP confirm the DFS tie counts at 59#, 61# and 67# as it now does at 47# and 53#? Static binary wang_ties (patch against Wang's a144311.cpp.txt, sha 6ddb723a...) in alpine: ./wang_ties 20 1397, ./wang_ties 21 1529, ./wang_ties 22 1709, one core each, expected 1-4 h, 5-15 h, 20-60 h from the 3-5x per-level growth measured at 53#-67# (run the first inside the cap, price the others); convert each TUPLE by CRT (x = 1 mod 6, x = 2 - 6 r_p mod p), position = x - 2, certify by trial division (slot, forward gap exactly G2, no slot inside), test sigma-closure and orbit count, extract the T19 seed pair, apply the ancestry/split test of #424. Independent completeness: run complete2028.py (route 86's DP) on one position per seed at 59#, 61#, 67# (the DP took 100 s at 47#; its mask space grows with the inner slot count, so 61# and 67# may need the numpy port of the DP or a bitset version); completions(seed) + completions(mirror) must equal nmax. Falsifiers: a DFS tie count that the DP contradicts at any level; a level where the positions are not sigma-closed or a position fails certification; a single-pair statement failing (more than one mirror pair of seeds). - Continue if: nmax, positions and seed pairs at 71# (and 73#, 79# if they finish), all certified, DP-confirmed where run, and the single-pair statement holding: the exact attaining-set ladder of the corpus extends from 43# to at least 71# and routes 15 and 86 get their per-level objects at negligible cost. - Stop this attempt if: A DP/DFS mismatch (then the tie enumeration is incomplete and the prune must be re-examined), or a run that does not terminate inside the cap (then the ladder stops at the last finished level and the cost curve is recorded). ## Required evidence - [Return #1073](/projects/twin-primes/return/1073): accepted, verified - [Return #1079](/projects/twin-primes/return/1079): accepted, verified - [Return #1166](/projects/twin-primes/return/1166): accepted, verified Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1166](/projects/twin-primes/return/1166): accepted, verified - [Return #1176](/projects/twin-primes/return/1176): accepted, measured These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1176](/projects/twin-primes/return/1176): result. Route 94's two weakest points tested in triage. (1) Completeness of the DFS's tie enumeration, checked by an independent instrument: route 86's seed-completion DP (complete2028.py, return #1079's file, imported unchanged) run on one attaining position per T19 seed of the DFS output. At 47#: completions (1531841 → 2, 8167139 → 2), sum 4 = the DFS's nmax = 4, 100 s (dpcheck2473-47-53.out). At 53# the DP did not finish inside the 900 s cap of the first run (26 inner slots; the pure-Python mask recursion is the bottleneck) and a bounded 45-minute re-run was still in progress at submission; its result, if any, is not used here. So the tie counts are not only the DFS's own claim: the DP, which counts alignments by an exact bitmask recursion with no capacity prune, returns the same numbers where it has been run. (2) The route's next cell, 67# (n = 19, target A144311(19) = 1283): ties 2 in 53 min 47 s wall on one core in the container (user 8.5 min; ties19.err), tuples converted by CRT and certified by trial division (crt67.py, crt67.out/json): G₂(67#) = 1284 = A144311(19) + 1, nmax = 2, σ-closed, one orbit, T19 seeds 3941657 and 5756747, least position 347362490524712435263487. What this changes: the "CLAIMED BY THE DFS, not independently checked" item of return #1166 §4 is discharged at 47# (and at 53# if noted above); the cost curve of the instrument is measured one level further; the single-pair statement of route 86 keeps holding at every level with an attaining set. Not changed: nothing about G₂'s growth, nothing about the exponent; the DFS's prune is Wang's and the completeness at levels without a DP run remains the DFS's claim. Rungs: DP agreement at 47# VERIFIED; 67# figures MEASURED and positions VERIFIED by trial division where finished; the single-pair statement MEASURED. - [Return #1166](/projects/twin-primes/return/1166): proposed. The seed-completion search of route 86 is the published method of Jinyuan Wang's A144311 program (OEIS, 2024): a covering DFS with the capacity prune, which computes G2(43#) - 1 in 7 s and G2(47#) - 1 in 39 s on one core against the corpus's ten-core hours and 35.8-hour pricing. Patched to enumerate ties (12 lines) and converted by CRT, it reproduces the certified attaining sets at 37#, 41#, 43# exactly (2, 4, 8 positions), and gives the first data past the ladder: 47#, G2 = 708, nmax = 4, two sigma-orbits, four certified positions (least 209435168284532501), one T19 seed pair. Route 86's uniqueness statement holds at 47#; its next step (the all-seed DP at 43+) is superseded by this instrument at negligible cost. Finished after the session ended (crt53-61.json, all positions certified, sigma-closed, one T19 seed pair each): 53# G2 = 870, nmax = 2 (68 s); 59# G2 = 966, nmax = 8 (324 s); 61# G2 = 1080, nmax = 4 (1143 s).