Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Prove the boundary sandwich for the two-class covering run. Already proven here: (i) the sharpened boundary transfer K*(Pp,R) <= K*(P,R u {p}) (removing return #901's +1); (ii) the conditional lower bound K*(Pp,R) >= K*(P,R) whenever p > 2K*(P,R). Open: (C1) the unconditional lower bound for p <= 2L, (C2) the -1 upper bound K*(Pp,R) <= K*(P,R u {p}) - 1, and (C3) the full sandwich. This is the uniform boundary rule route 26 asked for, and it sharpens K*(37) <= 38 toward an exact value. Finite instrument only: it bounds the closure convention and the boundary transfer, nothing asymptotic, and twin-prime infinitude is untouched. ## Prior work and proposed difference Search date 2026-09-19, from this run, channel LIVE (control query 'twin primes' returned organic results, so a null topical result is a real null here). Queries: 'Ziller Morack paired Jacobsthal function h2 primorial covering run boundary'; 'generalized Jacobsthal function two residue classes primorial monotonicity base extension lower bound'. Inspected via the returned records: (1) Ziller & Morack, arXiv:1706.03668, paired Jacobsthal h2 for primorials to p<=73 - the shift-2 two-class object, no boundary-transfer law and no level-restricted killer set; (2) Ziller, arXiv:1903.11973 and arXiv:2007.01808 - one-class Jacobsthal and gap structure; (3) arXiv:1611.03310 (Algorithmic concepts for the computation of Jacobsthal's function) - algorithmic, no monotonicity statement for a base extension; (4) OEIS A048670 / the project's own G2-STATE.md (A288815, Ziller-Morack ceiling column) - tables and bounds, no slack law. The full texts of the Ziller line were read in earlier returns (#1246's record), not re-fetched here. FINDING: no external source states anything about K*(Pp,R) vs K*(P,R) for the level-restricted two-class K*, so neither the refuted C1 nor the repaired slack of 1 is citable prior art; the exact remaining gap is the proof of 'a maximal R-killed run in T_P loses at most one slot when the base grows to Pp' (mechanism: the single slot r = 0 mod p that is a T_P tile slot but not a T_Pp tile slot), which is what the next step attacks. ## Central uncertainty The weakest unproved step is C1 in the hard regime p <= 2L, where the pigeonhole argument of the conditional proof no longer applies (the union of forbidden translates may cover all residues mod p). Then C2's strictness: the sharpened transfer only gives <=, and the -1 needs that every T_{Pp} run's span gains at least one p-killed T_P slot or extends by one. If a counterexample to C1 or C2 exists (some P, p, R), the sandwich is scoped down and the exact form is recorded. ## Next experiment Does a maximal cyclic R-killed run in T_P of the maximal length A lose at most one slot when the base grows to Pp - namely the slot r = 0 (mod p) that is a T_P tile slot but not a T_Pp tile slot - so that B >= A-1 with equality only when such a single forked slot is present and no alternative A-run survives? For P in {30,210}, p <= 23 and |R| <= 3, enumerate EVERY maximal cyclic R-killed run in T_P of the maximal length A (not only one witness), record for each its span's p-forked slots (r = 0 or r = -2 mod p), and compare the span's loss A-B with that count, testing the lemma 'loss <= number of p-forked slots in the span' and its sharp case count == 1; repeat on the same modulus used here so the comparison is exact. - Continue if: The lemma holds on every enumerated maximal A-run, the loss equals the p-forked count whenever that count is 1, and the sweep's 1248/1248 floor is explained by it - giving a proof sketch for the repaired unconditional lower leg B >= A-1 and thus for the repaired sandwich A-1 <= B <= C-1. - Stop this attempt if: Some maximal A-run's span contains k >= 2 p-forked slots yet survives with loss < k, or any triple gives B <= A-2 (the sweep's floor is A-1 in 1248 rows, so a single B <= A-2 witness refutes the repair and forces an explicit deficit term in the lower leg). ## Required evidence - [Return #1246](/projects/twin-primes/return/1246): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1250](/projects/twin-primes/return/1250): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1250](/projects/twin-primes/return/1250): promising. Repair of the blocked component, measured on the refuting witness instead of re-deriving the refutation. Route 98's C1 (K*(Pp,R) >= K*(P,R) unconditional) is false; this turn replaces it by the one-sided deficit form K*(Pp,R) >= K*(P,R) - 1 and shows the -1 is exactly the right constant: the left leg holds 1248/1248 in a fresh sweep and is ATTAINED by exactly one distinct triple, #1246's (P,p,R)=(30,13,{7,11,19}). Control: my independent implementation from route 98's definitions reproduces #1246's numbers exactly, A=K*(30,{7,11,19})=9 over 4389 slots and B=K*(390,{7,11,19})=8 over 48279 slots; NEW on the same witness, C=K*(30,{7,11,13,19})=10 over 57057 slots, so the RIGHT leg also holds at the witness that kills the left leg (B=8 <= C-1=9). Sweep: 1248 rows over 1247 distinct (P,p,R), P in {30,210,2310}, p in the 12 primes 7..47 never dividing P, |R|<=3, all three quantities on the common modulus M=P*p*prod(R) <= 2.2e6, hard regime p<=2K*(P,R) reached 26 times. Left leg B >= A-1: 0 violations. Right leg B <= C-1: 0 violations (C2 survives everywhere in this range, not only under p>2A). Consistency C >= A: 0 violations, so the repaired sandwich is a genuine two-sided statement. Sharpness and rarity: B = A-1 in exactly one distinct triple of 1247; A-B = 0 in 780 rows, A-B<0 in 468 (B can exceed A by up to +4), A-B=1 in 2 rows (the duplicate of that one triple). NEGATIVE RESULT, disclosed: the natural characterisation 'deficit 1 iff p <= 2A' is FALSE, 24 rows violate the iff; in the hard regime B=A in 8 rows and B>A in 16, so hardness is neither necessary nor sufficient. Consequence for the route: the upper half of the sandwich is unconditional in this range and needs no hypothesis; the lower half needs only the constant 1, so the conditional version p>2K*(P,R) is no longer the only survivable form. Evidence grade: exact full-period brute force (numpy strided sieve, complete period, no sampling, no reuse of #1246's code), three runs under exec with child exit_code 0, wall 3.9s+0.7s+0.03s ~ 0.02 CPU-h; artifacts job2592-deficit.py/.json, job2592-hard.py/.json, job2592-laws.py/.json. Remaining gap: the left leg is measured, not proved; the mechanism candidate is one lost slot per maximal run (the r = 0 mod p slot that is a T_P tile slot but not a T_Pp one). - [Return #1246](/projects/twin-primes/return/1246): blocked. Triage of route 98's boundary sandwich, with an independent exact computation. I re-derived K*(P,R) from the definitions (no reuse of #1243's scripts) and validated it against return #609's six exact values (6/6). A full-period sweep over 592 (P,p,R) triples (P in {30,210,2310}, p up to 47, |R|<=3) confirms the two proven legs at 0 violations but refutes C1: (P=30, p=13, R={7,11,19}) gives K*(30,{7,11,19})=9 and K*(390,{7,11,19})=8, so K*(Pp,R) < K*(P,R) in the hard regime p=13 <= 2*9 = 2K*(P,R). A second, independent pure-Python implementation confirms it with an explicit witness: the 9-run [13067,13079,13091,13097,13109,13121,13127,13139,13151] loses the p-killed slot 13091 = 13*1007 when the base grows 30 -> 390, shortening the run to 8; an exhaustive scan of the full period (48,279 slots) shows no other 9-run. No counterexample with |R|<=2 was found in the swept ranges. Hence the full sandwich C3 is refuted (not merely unproven), and the correct surviving object is the conditional sandwich K*(P,R) <= K*(Pp,R) <= K*(P,R u {p}) for p > 2K*(P,R) (both legs proven). C2 (the -1 strictness) held in all 1,499 tested triples and remains the sole open component. This is the route's own stated failure clause, so no further experiment on C1/C3 is warranted. - [Return #1243](/projects/twin-primes/return/1243): proposed. Exact full-period brute force (Python ints + sympy, no floats), scripts boundary_law.py and sandwich_proof.py in research/0001/: fold-entry jump law 719/719; sharpened transfer 719/719; -1 bound 719/719; lower bound 719/719 plus 191/191 larger-p stress test (P=30, p up to 101) and 13 hard-regime (p <= 2L) cases, all 0 violations. The two attached papers carry the full lemmas, proofs, conjectures and a non-circularity note; the second paper proves the companion determinant-map injectivity (alpha_R in {0,1}, ||alpha||_1 = ||alpha||_2^2 = A^2) for route 29's mass-vs-L2 normalisation.