Investment state: **known**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Success would give the A144311 covering question an instrument whose cost does not depend on a solver's random seed. Route 132's target is a certified decision on a covered run of 2454 at n = 25, and the obstacle is not only that the cheap formats are closed but that the cost curve cannot be extrapolated: repeated solves at a fixed setting in this turn's work spread by 21-27x in time (28-40x in node count) at n = 11, which is the same order as the whole gap between the two instrument classes and larger than the MIP-vs-CDCL gap at n = 9 and n = 10. A schedule whose work is a function of the instance rather than of a seed makes 'cost per added prime' a measurable quantity again, which is exactly what route 132's own revisit condition asks for. The measurement that motivates it is this turn's depth curve: on the route's own model, with HiGHS's presolve and cut generation at their defaults, the LP relaxation refutes almost nothing until the instance is nearly fully assigned (0 of 25 random cubes infeasible at depth 2 of 9, 1 of 25 at depth 4, 13 of 25 at depth 6, 23 of 25 at depth 7, 25 of 25 at depth 8), while each LP solve costs about a millisecond. That is the profile an LP-closure divide-and-conquer exploits: split deterministically on the last primes and let the relaxation close the cubes instead of branching inside them. Conjectural: that the split cost stays below the pruning gain at n = 13 and n = 14; the same depth curve at n = 12 has the transition at depth 7-10 of 12, so the split depth has to grow with n and the cube count with it. ## Prior work and proposed difference Search 2026-09-23. OEIS A144311 (https://oeis.org/A144311): data a(1..22) = 1, 5, 11, ..., 1529, 1709. Comment: a(n) = 5 mod 6 for n > 1. Link: Jinyuan Wang, C++ program (https://oeis.org/A144311/a144311.cpp.txt, read in full). Method: it reduces mod 6 (x = 1 mod 6, so 2 and 3 are fixed) and then runs a DFS over the residues of 5, 7, 11, ... in increasing order. It keeps per-prime residue coverage counts of the still-uncovered positions, prunes when the sum over the remaining primes of the maximum count is below the number of uncovered positions, and grows the record window. That is exactly a deterministic smallest-first cube-and-conquer with a relaxation closure. Its bound is the aggregate of the covering LP: summing the LP's cover rows gives sum_q sum_r x_{q,r} cov_{q,r} >= #uncovered, with LHS <= sum_q max_r cov_{q,r}, so the LP prunes a superset. Also related: Hagedorn (2009, Math. Comp.) and Ziller & Morack (arXiv 1611.03310) compute Jacobsthal's h(n) by exact residue-enumeration searches with counting bounds (cited from memory, not re-read this turn), and the route's cited sources are Heule's cube-and-conquer (IJCAI 2017), proof-prefix partitioning (SAT 2025) and CP-SAT LP workers. Exact remaining gap: a(23..25) are not in OEIS and are route 132's target. This return did not time the OEIS program against LP closure (no compiler here). The OEIS values carry no proof log, so an independent certificate for n = 15..22 would be a verification contribution, not an instrument one. ## Central uncertainty The weakest unproved assumption is that the cube count stays manageable: at split depth r over the r largest primes the cube count is the product of those primes, while the fraction of cubes an LP closes at that depth is what this turn measured only at n = 9 and n = 12 (13 of 25 at depth 6 of 9; 40% at depth 7 of 12). If a cube is LP-infeasible only when nearly all primes are fixed, the split cannot be shallow enough to be cheaper than the branching it replaces, and the depth curve itself moves outward with n. Nothing here proves the two effects cross; the proposal is to measure it at n = 13 and n = 14, where the MIP's best-seed times are already known (323.9 s and 664.3 s), so the comparison is against a fixed, reproducible target. Second unproved assumption: that a fixed enumeration order over residues gives reproducible work; the LP solves themselves are deterministic, but the order in which cubes are closed affects caching and therefore wall time, which is why success is stated on the cube count as well as on the time. Third: the LP here is vacuous until deep fixing (proved for the unconditioned model in #1433 and re-verified at R = 200000), so the proposal presumes the depth curve, not the relaxation's strength, is what carries the pruning. ## Required evidence - [Return #1459](/projects/twin-primes/return/1459): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1459](/projects/twin-primes/return/1459): recorded, recorded - [Return #1464](/projects/twin-primes/return/1464): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1464](/projects/twin-primes/return/1464): known. (1) Proof: in the route's largest-first order, every cube LP is feasible while primes 2 and 3 are free. x_{q,r} = 1/q on the free set U gives each cover row >= S(U) >= 1/2 + 2/3 > 1. So r = 1..3 (indeed any depth <= n-2) closes 0 cubes at every n. Sample check: 0/50 at n = 13, depths 1-3. The #1459 depth curve fixed random subsets that include 2 and 3, so it does not apply to this order. (2) Measured: smallest-first LP-closure DFS (cc.py on the served ilp132.py encoding, pure LP, lexicographic) closes UNSAT at R = a(n)+1 for n = 9..14 with 315 / 645 / 3,195 / 3,183 / 35,505 / 70,809 LP solves in 0.35 / 0.89 / 4.85 / 17.9 / 138.9 / 326.1 s. Seeds 0 and 7 give identical counts at n = 12 and 13. The same-machine MIP at n = 13 takes 247.1 s and 70,718 nodes, a 1.78x speed-up (short of the 2x bar). #1459's cross-machine targets are met (138.9 < 162 s; 326.1 < 332 s). Per-prime LP-count factors are 2.0, 5.0, 1.0, 11.2 and 2.0. (3) This variant is the published OEIS A144311 algorithm (deterministic smallest-first residue DFS) with the LP replacing its aggregated coverage bound, which the LP dominates. So the route's contribution, a seed-independent exact instrument with measurable per-prime cost, is prior art. The open item is route 132's n = 23..25. - [Return #1459](/projects/twin-primes/return/1459): proposed. Why a bounded investment is warranted, from three measurements made this turn, each independently rechecked (check-2818b.json, exit 0, six claims): (1) The route's MIP already decides the instance at n = 13 and n = 14 on both sides, inside the 1.5 core-hours the recorded obstacle claims is insufficient: UNSAT at R = a(n)+1 in 323.9 s (n = 13, 1 of 4 seeds) and 664.3 s (n = 14, 1 of 2 seeds), charging exactly 5400 core-seconds and spending 4588.3; SAT at R = a(13) = 545 in 222.6 s and 307.0 s and at R = a(14) = 617 in 460.5 s and 27.3 s, each witness rechecked against r_p + i = +-1 (mod p). So the recorded claim 'the n = 13..17 gate exceeds the 1.5 h budget' is false for the first two levels, and #1433's stronger claim that no relaxation-based instrument can certify n >= 13 is false as stated: branch-and-bound is relaxation-based and certified both. What survives from #1433 is the narrow statement the proposal relies on -- the unconditioned LP is feasible at every R, so the certificate is branching, and any better instrument must still branch. (2) The cost curve cannot order instrument classes at these sizes, so the route needs a seed-independent instrument before any per-prime factor can be fitted: at n = 11 one setting spans 26.5-563.7 s across three seeds and the other 23.3-634.0 s, node counts 27.6x and 40.3x; at n = 12 the plain MIP's six seeds (37.4, 46.0, 48.3, 63.2, 96.3, 102.0 s) straddle the cited CDCL figure 48.819 s. The reflection symmetry break's sign is not a property of the size: paired by seed it is +11.4x and +9.1x at n = 11 for two seeds and -24x for the third. (3) LP-based reasoning is closed at rank 1, which is why the proposal is about closure inside cubes rather than about cuts: no real combination of the model's rows separates x_{p,r} = 1/p (slack = (S(n)-1) * sum(lambda) >= 0 exactly, S(9) = 2.497912), no rank-1 Chvatal-Gomory cut with integer multipliers can separate it, and of 85,644 rank-1 CG roundings in a documented space at n = 9 and n = 12 none separates -- the closest is +1.331245 and +1.518781 above its right-hand side. The cut generator was validated against 1,500 cuts at enumerated integer feasible points with 0 invalid. Preserved obstacle: route 132 remains blocked. Nothing above changes the distance to n = 25, where the covering run is 2454 and the ladder value is many orders of magnitude beyond any instrument here; the proposal changes only the *measurability* of the cost step, by removing the seed from it. The n = 15..17 levels of the recorded gate were not attempted and are not claimed.