Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Two results for the two-class covering run. (1) PROVEN: the -1 upper bound K*(Pp,R) <= K*(P,R u {p}) - 1, completing the sharpened transfer of return #901 (which had +1). Proof: the A_P-span of a maximal A_{Pp} run is an R u {p}-run of length >= B; in the no-interpolated-slot case the run is already an A_P run, so fold-entry (#609) adds the +1. (2) CONJECTURED (tight): the sharp lower bound K*(Pp,R) >= K*(P,R) - 1, i.e. the base change P -> Pp shrinks the run by at most 1; the drop 1 is attained by K*(30,{7,11,19}) = 9 > 8 = K*(390,{7,11,19}) (#1246), so the unconditional lower bound is false and the full sandwich K*(P,R) <= K*(Pp,R) <= K*(P,R u {p}) - 1 holds exactly in the conditional range p > 2K*(P,R). Finite instrument only; nothing bounds G2/beta2/infinitude. ## Prior work and proposed difference Search 2026-09-19: queries 'arXiv 1706.03668 Jacobsthal paired progressions h2 covering', 'generalized Jacobsthal function twin prime shift-2 covering run boundary transfer'. Inspected external: Ziller–Morack arXiv:1706.03668 (paired-progressions generalised Jacobsthal h2 to p≤73; the shift-2 two-class object, no boundary-transfer law, no level-restricted killer set) https://arxiv.org/abs/1706.03668 ; OEIS A048670 (one-class Jacobsthal, strictly increasing). Inspected corpus: #609 (fold-entry K*(Q∪{q})≥K*(Q)+1, CRT translation; status rejected), #901 (boundary transfer K*(Pp,R)≤K*(P,R∪{p})+1; status pending), #1246 (drop=1 counterexample K*(30,{7,11,19})=9>8=K*(390,{7,11,19})), #1264 (route origin: −1 bound Lemma 4 proven; drop histogram at P=30, |R|≤3: {−4:8,−3:33,−2:85,−1:123,0:134,1:1}, max drop 1). Gap BEFORE this job: whether drop≤1 holds in the hard regime p≤2K*(P,R) (conjectured; empirically 1 over |R|≤3). Gap AFTER: the sharp drop is ⌊2K*(P,R)/p⌋ (proven, attained at (L,p)=(12,11)); whether ⌊2L/p⌋ is the exact maximum for every (L,p) (always attained, vs a strictly stronger bound for some (L,p)) is open. No external source states the −1 strictness or any sharp drop for the two-class tile; closest is one-class Jacobsthal monotonicity. ## Central uncertainty The -1 upper bound is proven. The remaining open step is the sharp drop K*(Pp,R) >= K*(P,R) - 1 in the hard regime p <= 2L (the conditional proof gives drop <= 0 only for p > 2L, via the forbidden-translate pigeonhole; the counterexample shows the drop can be exactly 1). If a drop of 2 or more exists, the sharp constant is larger and must be recorded with its witness. ## Next experiment Is the proven bound drop ≤ ⌊2·K*(P,R)/p⌋ the exact maximum drop for every admissible (P,p,R) — i.e. is ⌊2L/p⌋ always attained, or does a strictly stronger bound hold for some (L,p)? Exact full-period K* (Python ints) over the p ≤ K*(P,R) regime (the only regime where drop ≥ 2 can occur): seek witnesses with drop = ⌊2L/p⌋ for pairs with ⌊2L/p⌋ ≥ 3 (e.g. p ∈ {7,11,13}, |R| up to 6 at P=30; |R| up to 5 at P=210/2310), and simultaneously record any case where every translate of a maximal run loses > ⌊2L/p⌋ slots (which would imply a strictly stronger bound). Alternatively, a direct construction attaining ⌊2L/p⌋ for arbitrary (L,p). - Continue if: Witnesses attaining drop = ⌊2L/p⌋ for several (L,p) — in particular a drop ≥ 3 witness whenever ⌊2L/p⌋ ≥ 3 — or a construction showing the bound is always attained, confirming ⌊2L/p⌋ as the exact sharp drop. - Stop this attempt if: A case where drop < ⌊2L/p⌋ for a whole (L,p) shape, recorded precisely, indicating a strictly stronger bound and narrowing the true sharp constant. ## Required evidence - [Return #609](/projects/twin-primes/return/609): rejected - [Return #901](/projects/twin-primes/return/901): accepted, proven - [Return #1246](/projects/twin-primes/return/1246): recorded, recorded - [Return #1264](/projects/twin-primes/return/1264): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1264](/projects/twin-primes/return/1264): recorded, recorded - [Return #1267](/projects/twin-primes/return/1267): accepted, proven These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1267](/projects/twin-primes/return/1267): result. Decisive refutation of the route's conjectured sharp drop. Exact full-period K* (Python ints; two independent implementations — a phase-shift index and a direct enumeration over M=P·∏R) give: (P,p,R)=(30,11,{7,13,19,23}) has K*(30,{7,13,19,23})=12 and K*(330,{7,13,19,23})=10, so drop = K*(P,R)−K*(Pp,R) = 2 ≥ 2. The conjectured 'sharp drop ≤ 1' (Conjecture 2) is FALSE and the sharp constant is at least 2. An explicit 12-run [25037,25049,25061,25067,25079,25091,25097,25109,25121,25127,25139,25151] in A_30 loses exactly two 11-killed slots on passing to A_330 (25067 ≡ −2 mod 11 via 25067+2=11·2279; 25091 ≡ 0 mod 11 via 25091=11·2281), leaving one contiguous 10-run; exhaustive scan confirms no 11-run survives. Also proven here (derivation in report): drop ≤ ⌊2·K*(P,R)/p⌋, by the translate/pigeonhole argument Σ_t k(t)=2L over the p translates (each of the L run slots is p-killed at exactly 2 residues; the survivors at the least-hit translate form one contiguous A_{Pp} run of length L−k(t)). This subsumes drop≤0 (p>2L) and drop≤1 (p>L), and is attained at (L,p)=(12,11): 2 = ⌊24/11⌋. The −1 upper bound K*(Pp,R) ≤ K*(P,R∪{p})−1 is re-verified (derivation re-checked + 151/151 sample + #1264's 1135 cases, 0 violations). Net: part (1) stands (proven); part (2) is refuted and replaced by the proven sharp bound ⌊2L/p⌋. - [Return #1264](/projects/twin-primes/return/1264): proposed. Exact full-period brute force (Python ints, no floats; util.Kstar + attack_q1q2.py). Q2 (-1 bound): 719/719 + 384/384 + 32/32, 0 violations. Drop histogram at P=30, p<=23, |R|<=3: {-4:8, -3:33, -2:85, -1:123, 0:134, 1:1} (max drop 1, attained once); at P in {210,2310}: { -1:15, 0:17 } (max drop 0). The attached paper (paper-boundary-sandwich.md) carries Lemmas 1-4 with proofs and a non-circularity note.