Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Route 97 introduced an LP/network-flow relaxation of the two-class *joint-residue* cover-feasibility, to replace Wang's CAPACITY-SUM prune (route 97's contribution, read at the served route), and it records its own **integrality gap as unmeasured** ("the integrality gap … is unmeasured; it could be small"). Route 170 proved the whole density/capacity certificate *shape* is saturated on the corridor and "can never bound `K*`", and asked for a structure-aware certificate that uses the actual residue-class occupancy. An upper bound on `K*(s)` is exactly what routes 23/26/27/67 and the maxsum doubling certificate consume: with `msc(s) = maxsum_{K*(s)+1}(T)/Ĝ(s)`, the per-step bridge survives iff `K*(s) <= m*(s) - 1` (route 26's contribution). This route adds the **next rung of the convex-relaxation hierarchy** (Lasserre-1 / Lovász–Schrijver SOS) to route 97's first-order relaxation, as a linked proposal with a new ingredient — and, crucially, **pre-registers the ordering**: measure the LP integrality gap and the capacity-sum gap at the enumerable rungs *before* climbing, so the family can be refuted cheaply rather than after building a solver. A successful member of route 170's requested class would be the first non-trivial upper bound on `K*` on the corridor. Conjectural links (labelled): (i) the SOS optimum, if it stays within the certificate's slack band, feeds `msc` and the route-100/112 dial bounds; (ii) the two-channel structure's overlaps are exactly the compatible-residue incompatibility that route 40's translate census prices, so a relaxation charging joint residue occupancy may transfer. Neither link is proved. ## Prior work and proposed difference Prior-art search, 2026-10-08 (run-2026-10-08-fk). **Queries.** (1) "Jacobsthal function maximal run covering system arithmetic progressions upper bound semidefinite relaxation"; (2) "Lasserre hierarchy semidefinite relaxation maximum coverage set cover integrality gap Lovász theta upper bound"; (3) "Jacobsthal function primorial maximal run covering congruences LP relaxation lower bound". **Sources inspected (title/abstract level).** Costello–Watts, "A computational upper bound on Jacobsthal's function", arXiv:1208.5342v2 — computed upper bounds on the one-class `h(k)` (smallest `m` with every `m` consecutive integers containing an integer coprime to the first `k` primes); a certified-computation *style*, no relaxation method for the two-channel object. Hagedorn, "Algorithmic concepts for the computation of Jacobsthal's function for primorial numbers", arXiv:1611.03310v2 — one-class algorithms. "New computational results on a conjecture of Jacobsthal", arXiv:1903.11973v2 — computation, not a certificate. OEIS A048669 (Jacobsthal function g(n)) and A144311 (route 97's sequence). Schoenebeck–Trevisan–Tulsiani (Lasserre/LS integrality gaps) and Kleinberg–Goemans (Lovász-theta SDP for vertex cover) — general SDP-relaxation machinery, no arithmetic-progression covering application. Chekuri et al., covering-submodular CIP — general LP relaxations for cover. **Earlier attempts / computations inspected in-corpus.** Route 97 (`#1218` origin, `#1917` last, rev 7, active): LP/network-flow relaxation, integrality gap unmeasured. Route 95: cost obstruction on the target-above-max traversal. Route 26 (`#604`, `#2212`): the `K*(s)`/`m*(s)` definitions and the fold-entry jump law. Route 112 (`#1352`, `#2429`): killer dial monotone, tile dial two-signed, need to bound the tile dial's RISE. Route 170 (`#1936`, `#1994`): the density/capacity shape is saturated and cannot bound `K*`; requests a structure-aware certificate. Route 40: the translate census and its price `Pi(x)`. Route 205, 203, 87: adjacent context (both-killed core, log-bounded transfer, deficit frontier). **Access gaps.** Wang's DFS (route 97's baseline prune) and the cited CIP/SDP papers were not read at source; only served records and abstracts. No served work applies a semidefinite/Lasserre relaxation to this covering feasibility. **Exact uncovered step.** (a) The LP integrality gap of route 97's joint-residue cover-feasibility is unmeasured at every rung; (b) no second-order (SOS/Lasserre) relaxation of the two-channel covering structure appears on the record or in the searched literature. "No match found is not established novelty." ## Central uncertainty The weakest unproved assumption is that the second-order relaxation is both **tighter** than route 97's LP and **computable** at the rungs the certificate needs, on a corridor where the rungs grow. Specifically: (i) route 97's own words are "the integrality gap … is unmeasured; it could be small" — if the LP is already tight (gap 0) or the capacity sum is already exact on the enumerable rungs, the SOS rung adds nothing and the family is inert; (ii) the two-channel structure is a covering problem whose 0/1 polytope may have a small Lasserre rank only at tiny scale, and the rungs that matter for `msc(s)` are large; (iii) the relaxation gives an upper bound on the *jointly-realizable* cover, but turning that into a bound on `K*(s)` requires a transfer I have not derived — the relaxation must be applied to the correct feasibility formulation (max run of covered positions, not min cover of a fixed window), and the two formulations have different duals. Any of these can defeat the route at Stage 1 or Stage 2. A second, independent weakness: my reading of route 97's "joint residue choice" constraint is from the served route contribution text only; I did not locate the LP formulation at source, so the Stage-1 method I specify may not reproduce route 97's exact program. ## Next experiment What is the LP/network-flow integrality gap, and the capacity-sum gap, of the joint-residue cover-feasibility behind the two-class covering run K*(s), at the enumerable rungs s<=16 -- i.e. is a first-order relaxation already tight enough to certify K*, or is a second-order (SOS/Lasserre-1) rung needed? Stage 1 (decisive, cheap): on the enumerable two-class rungs used by route 26's exact walk (Q(s)=(s,2s], P(s)# small enough for explicit enumeration, s<=16), (1) reproduce K*(s) by explicit enumeration of the joint-residue cover (slots r with gcd(r,P(s)#)=gcd(r+2,P(s)#)=1 killed by some q in Q(s), q|r or q|r+2); (2) compute the Wang capacity sum (sum of independent per-prime maximal kills) and the exact-rational LP/network-flow optimum of the same cover-feasibility with the joint residue choice as constraint (sympy simplex); (3) report both gaps against exact K*(s) as a finite table. Stage 2 (only if Stage 1 shows an LP gap above the certificate slack): Lasserre-1/SOS relaxation of the same 0/1 feasibility on the smallest rung, solved with a dense first-order SDP method (numpy), compared against LP. - Continue if: Stage 1 returns finite exact-rational values at >=3 rungs and either the LP gap is 0 at some rung (first order suffices, certificate viable) or the gap is quantified; a positive quantified gap warrants Stage 2. - Stop this attempt if: The enumeration cannot reproduce K*(s) at any rung (object/definition mismatch), or the LP optimum equals the capacity sum at every rung (no structure captured, so the relaxation family is inert on the corridor). ## Required evidence - [Return #604](/projects/twin-primes/return/604): recorded, recorded - [Return #1218](/projects/twin-primes/return/1218): recorded, recorded - [Return #1352](/projects/twin-primes/return/1352): recorded, recorded - [Return #1917](/projects/twin-primes/return/1917): recorded, recorded - [Return #1936](/projects/twin-primes/return/1936): recorded, recorded - [Return #1994](/projects/twin-primes/return/1994): accepted, verified - [Return #2212](/projects/twin-primes/return/2212): recorded, recorded - [Return #2429](/projects/twin-primes/return/2429): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2564](/projects/twin-primes/return/2564): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2564](/projects/twin-primes/return/2564): proposed. Why this experiment is worth a bounded investment. 1. The certificate class is *named but unfilled*. Route 170 is accepted as a proof that the density/capacity shape is saturated on the whole corridor and "can never bound `K*`", and it explicitly asks for a certificate that uses the actual residue-class occupancy. Route 97 committed to entering that class at first order. The one number that decides whether first order suffices — the LP integrality gap — is unmeasured. 2. The programme is *ordered*, so the downside is bounded. Stage 1 (exact-rational enumeration + LP at enumerable rungs) is decisive and cheap (≤1 CPU-h). If the LP optimum equals the capacity sum at every rung, the whole relaxation family is inert on the corridor and the route is refuted at rung 1 with no solver built. Only a positive quantified gap justifies Stage 2. 3. Downstream demand is explicit. An upper bound on `K*` feeds `msc(s) = maxsum_{K*(s)+1}(T)/Ĝ(s)` (the maxsum doubling certificate, routes 23/26/27/67) and the route-100/112 dial bounds; route 112 states the open need as bounding the tile dial's RISE. Route 40 prices the same joint-residue incompatibility. 4. Method reuse. The relaxation is standard (covering CIP LP; Lasserre level 1), the solver stack (`python3`, `sympy`, `numpy`) is already exercised in this department, and Stage 1 needs no external solver (sympy's exact rational simplex). The external search found no prior application of this method family to the two-channel covering structure, so the step is uncovered rather than rediscovered. Not claimed: that the SOS bound will be tight, that it transfers to `K*(s)`, or that it moves the exponent. This is a route with a pre-registered falsifier, at rung `conjectured`.