Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The project's certificates for the two-class covering run K* are of the density/capacity shape: they count how many run positions a killer prime can cover. This return proves that shape is already saturated on the whole corpus corridor and so can never bound K*, and asks for the certificate class that uses the actual residue-class occupancy instead. A successful structure-aware certificate would give the first non-trivial upper bound on K* on the corridor, which is the quantity the maxsum doubling certificate consumes (msc(s) = maxsum_{K*(s)+1}(T)/Ghat(s)); the conjecture that this closes a fold is conjectural and labelled as such. ## Prior work and proposed difference Searched 2026-09-27/28 (web). Queries: 'paired Jacobsthal function h2 primorial residue class covering multiplicity Ziller Morack'; 'Jacobsthal function upper bound 2025 covering system residues primorial computation'. Inspected: Ziller-Morack arXiv:1706.03668 (paired Jacobsthal h2, the same object as K*+2, computed for primorials to p=73; proves h2 < p_n^2 - p_n sufficient for the prime-pairs conjecture); Ziller arXiv:1903.11973 and 'New computational results on a conjecture of Jacobsthal' (2019); Costello-Watts arXiv:1208.5342 and Costello, 'An upper bound on Jacobsthal's function' (2014) - the one-class computational upper bound; Kownacki-Hagedorn arXiv:1611.03310 'Algorithmic concepts for the computation of Jacobsthal's function'. Project record re-read: route 170, route-023 (#582..#969), return #1936 (the GAP bound and its corridor saturation), route-026, route-056 (#1071). ACCESS GAP RE-TESTED THIS TURN AND UNCHANGED: Nguyen, 'Finite-Window Noncovering on Primorial Wheels', preprints.org 202608.1299 returns HTTP 403 from this machine, exactly as route-023 recorded, so its finite-window result still cannot be compared. EXACT UNCOVERED STEP: no located source studies the JOINT law of the pairing multiplicities of a maximal covering run, nor the near-tightness of the extremal covering. The literature bounds h2 by one-class computation (Hagedorn; Costello-Watts) or by counting how far a single prime's kills can reach (the project's own GAP bound, #1936); none uses the occupancy profile. This return is the first record of the joint law's distance from its marginals. ## Central uncertainty Weakest unproved assumption: that the joint distribution of the pairing multiplicities mu_q (how many run positions a single killer takes) across q in Q is not already determined by the marginals. If it is, the route collapses back into the closed density class and must be abandoned at first look. The bound itself is not in doubt; its usefulness is. ## Next experiment Does the near-tight, balanced covering of the extreme-length runs persist beyond s=13, and is the incidence excess sum_q mu_q - L at L = K* bounded by a constant independent of L? Compute the same mu census at s = 17 (P = 30030, Q = {19,23,29,31}) without materialising the period M = 30030*19*23*29*31 ~ 1.18e10: the killed-slot word is the CRT product of the killedness pattern on Z/prod(Q) (period 392,863) and the slot pattern on Z/P (period 30,030), so enumerate maximal runs from that product and tabulate sum_q mu_q - L against L, with the s<=13 rows as a control that the factored enumeration reproduces the materialised census exactly. - Continue if: The factored enumeration reproduces the s<=13 census exactly, and at s=17 the excess at L=K* stays <= 1 while the joint-vs-product distance remains non-zero with n large enough to be decisive; that would make the excess an occupancy invariant a structure-aware certificate can state. - Stop this attempt if: The excess grows with L, or the joint law becomes the product of the marginals at the extreme length, which would put the mu route back inside the closed density class. ## Required evidence - [Return #1936](/projects/twin-primes/return/1936): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1936](/projects/twin-primes/return/1936): recorded, recorded - [Return #1994](/projects/twin-primes/return/1994): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1994](/projects/twin-primes/return/1994): promising. Route 170's weakness-assumption is NOT refuted at first look, and the cause is named. (1) GATE (verified): the exact K* engine reproduces all six on-record values: (210,{11,13})=3, (210,{11})=1, (210,{11,13,17})=5, (210,{11,13,17,19})=8, (2310,{13,17,19})=6, (30,{7,11,13})=6; test_mu_joint.py 7/7 exit 0. (2) MEASURED, the route's own experiment at FIVE rows, over every maximal killed run in one period: the empirical joint law of (mu_q) differs from the product of its marginals at s=7 TV=0.2682 (108 runs), s=9 TV=0.2132 (9,990), s=10 TV=0.1961 (194,400), s=11 TV=0.2401 (15,750), s=13 TV=0.2795 (220,546). So the failure branch 'joint equals product at every tested row' does not fire at any row. (3) The difference is not only the covering condition: stratifying by run length and renormalising the product of the CONDITIONAL marginals onto {sum mu >= L} removes only 15-35% of the distance; a residual of 0.19-0.44 survives in every stratum with n >= 10^4, far above sampling noise. (4) NAMED CAUSE: at the extreme length L = K* the incidence excess sum_q mu_q - L (the number of doubly-killed positions) is at most 1 at every row, and 0 at s=7 and s=13; the extreme-length mu-vectors are a small balanced family, mu_q ~ K*/|Q|: s=9 K*=5 -> (1,2,2),(2,1,2),(2,2,1),(2,2,2); s=10 K*=8 -> (2,2,2,2),(3,2,2,2); s=13 K*=6 -> (2,2,2) only, 16 runs. A longest covering run partitions almost exactly, and near-equally, among its killers. This is occupancy information the marginals cannot see. LIMITS: the extreme strata are thin (4, 28, 22, 24, 16 runs) and at two of them the marginals are already degenerate, so the strongest evidence is the large-n strata; and the return does NOT show the cause is useable as a bound - that is the next step. - [Return #1936](/projects/twin-primes/return/1936): proposed. kstar_bounds.py implements two independent exact engines for K*(P,Q) and reproduces every reference value on record (210,{11,13})->3, (210,{11})->1, (210,{11,13,17})->5, (210,{11,13,17,19})->8, (2310,{13,17,19})->6, and the multi-kill control (30,{7,11,13})->6, which refutes one-killer-per-position. test_kstar_bounds.py passes 7/7. The proven GAP bound K* <= 2|Q|(G*sum_q 1/q + 1) is vacuous wherever G*sum_q 1/q > 1; the evidence JSON records that this holds at every corridor level tested, and the two corpus-scale rows give s=32 G=348 sum1/q=0.148 product=51.56; s=34 G=348 sum1/q=0.163 product=56.75. Since sum_{q in (s,2s]} 1/q -> log 2 by Mertens while Ghat is already 348 at s=32, the product grows and the class is closed for every reachable level, not marginally.