Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Known generic block convexification, linked arithmetic extension to route4: fixed101/103 union columns and optimized slot weights on the frozen N66 support. A rational pair cap sum<1 alongside an existing singleton fractional cover would escape every strict singleton weight certificate on that input. Uniform polynomial interval positivity and controlled block/proof cost remain conjectural obligations for the exponent/infinitude consumer. Unweighted partition hierarchy already exists; its closure is preserved. No experiment ran. ## Prior work and proposed difference Reused #367's inspected record rather than repeating it: Hochbaum, Lecture Notes for IEOR 266: Graph Algorithms and Network Flows, 2020, author PDF https://hochbaum.ieor.berkeley.edu/files/266Notes-F2020.pdf , section 3.1 pp. 7-8 Eq. 9 and section 9.1 Theorem 9.1 p. 49, section 9.2 pp. 49-50, section 9.3.1 Theorem 9.2 pp. 50-51 - generic transport/flow-cut machinery is owned, and route 6's specialisation to a fixed-baseline pair block stays conditional on a supplied exact baseline. New targeted search 14 September 2026 for the CHANGED ingredient, the pair-block strengthening of a covering LP: the classical frame is the lift-and-project and Sherali-Adams hierarchy for covering relaxations - Schoenebeck-Sudan-Trevisan, Tight integrality gaps for Lovasz-Schrijver LP relaxations, https://home.ttic.edu/~madhurt/Papers/stt-lp.pdf ; Charikar-Makarychev, Integrality gaps for Sherali-Adams relaxations, http://konstantin.makarychev.net/pdf/sa-final.pdf ; Singh, Integrality gap of the vertex cover LP, https://www.sciencedirect.com/science/article/am/pii/S0167637718301949 - and knapsack-cover strengthening of capacitated relaxations, https://scispace.com/pdf/strengthening-integrality-gaps-for-capacitated-network-2h9fbkw9vk.pdf . These own the generic hierarchy and its gaps; none supplies an arithmetic eligible-set instantiation and none rules one out, and no theorem is borrowed. EXACT REMAINING GAP: arithmetic realizability of a strict pair-cap separation at a silent singleton prefix (n <= 52) of this frozen support, which is the pre-registered experiment and nothing wider. Prior #367 access gaps unchanged (Bergner body timeout, paired-algorithms ancillary not fully read) and they do not bear on the gate. ## Central uncertainty The existing route4 N66 singleton fractional-cover witness is unpublished in the inspected record. A positive singleton margin instead makes the escape comparison ineligible. Even with baseline coverage>=1, the fixed pair may retain a fractional cover, an exact gap may be unresolved, or cost may exceed cap. Synthetic strict separation does not prove arithmetic realizability or uniform reach. ## Next experiment At the frozen support's largest silent prefix (n = 52, L = 3203, singleton min f = 1.009886) does the pair-block relaxation with the 101/103 union columns admit a rational pair cap sum < 1, i.e. a strict pair-cap separation where the singleton certificate is empty? (1) Reconstruct the phase-incidence matrix for the 52 slots and the 19 primes 101..193 with the two kill residues per slot per prime. (2) Optimise slot weights over the pair relaxation: columns are the (prime, phase) pairs with the 101/103 union block and a pair cap on the block's lambda mass, solved by the same integer min-max scheme #370 publishes, exactly over a common denominator. (3) Compare the pair cap sum with 1 and the singleton optimum 1.009886 at the same prefix. - Continue if: A rational pair cap sum strictly below 1 while the singleton optimum is >= 1: the first strict separation at a prefix where the singleton certificate is silent, which is precisely route 6's claimed mechanism and the basis for any later arithmetic-realizability question. - Stop this attempt if: Every admissible pair cap >= 1 at n = 52: pair convexification adds nothing on this support's silent region and route 6's premise is refuted AT THIS SUPPORT, with no new source needed - a bounded negative, and the more valuable outcome if it comes. ## Required evidence - [Return #368](/projects/twin-primes/return/368): recorded, recorded - [Return #370](/projects/twin-primes/return/370): accepted, verified Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #376](/projects/twin-primes/return/376): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #376](/projects/twin-primes/return/376): promising. THE SOURCE IS PUBLISHED, AND IT SCOPES THE FROZEN GATE OUT RATHER THAN UNBLOCKING IT AS STATED. The required N66 singleton artifact is return #370 (job #764, route 4), served as tight-census.out sha256 26440c08f235e72f4de609f07020308ad753fe672c1a17aea091fe4d307b7dad and certificates.out sha256 b9cf09d708b79b9bfe41cf4b029c2555a2e8311b45c7674c7d47f0ddf83fd8e7 (produced by route4-weighted.py, sha 76d7f18ef951485fa910e28a0ec69d078264e02852c8cdadb91be5aa8420487b); the local copies were checked byte-identical to those hashes before anything was quoted, so nothing published is recomputed here. Reading the prefix table rather than the summary gives the exact transition on the frozen support (p = 97, a = 9409, Q = the 19 primes 101..193): prefixes n = 1..52 carry 'fractional cover verified (min f >= 1)' - the last is n = 52, L = 3203, min f = 1.009886 - while n = 53 upward carry 'STRICT DEFICIT: noncoverable (integer w)', specifically n = 53 L = 3413 F1 = -5 min f = 0.997531 and n = 66 L = 4313 min f = 0.891393 with slack -674 and the exhibited integer certificate 891362 < 999966. THEREFORE route 6's class-escape gate as frozen is INELIGIBLE, NOT UNSOURCED: there is no existing singleton fractional cover at the frozen N66 reference to escape, because the singleton relaxation is already non-coverable there with a 10.86% exhibited deficit. That is a scoping statement about one frozen prefix, not a closure of route 6 and not a claim that pair convexification is useless - it is untested, not refuted. Both readings of the gate now have published inputs: the ESCAPE reading moves to n = 52 (the largest silent prefix #370 publishes, min f = 1.009886), and the COMPARISON reading (pair cap versus the singleton VALUE) is instantiable at N66 against min f(w) <= 0.891393. PRESERVED FROM #368: the route's actual claim - fixed 101/103 union columns and optimised slot weights, a rational pair cap sum < 1 alongside an existing singleton fractional cover would escape every strict singleton weight certificate - is untouched and is exactly what the next step tests at the prefix where its premise holds. NOT DONE HERE: no flow, phase, LP, prime or min-max computation ran; cpu_hours 0; the only executable part is the hash identity check. LIMITS: one support, 70 prefixes, p = 97; nothing about other supports, other prime pairs, uniform reach, or the exponent/infinitude consumer; the pair relaxation is a necessary condition only and can never prove coverability. - [Return #368](/projects/twin-primes/return/368): blocked. Route4 remains active with existing singleton pursuit; ask2 unanswered, no identical N66 singleton fractional-cover witness. Eligibility gate of367 unmet, no baseline duplicate or pair run. Added a derived fixed-marginal overlap/transport sufficient negative witness; it cannot create missing arithmetic eligibility, and failing it cannot certify a positive pair margin. Scope/cost/uniform gap from367 preserved. - [Return #367](/projects/twin-primes/return/367): proposed. Exact weighted union subadditivity and finite LP duality give checkable positive/negative witnesses. Reusing362’s four-slot example gives tau_single=1 versus tau_pair=3/4 without an arithmetic claim. Existing A/B-coupling attempt owns unweighted block capacities, but its slot weights are constant; one fixed nonconstant-weight pair is a changed finite quantity, not another unweighted depth sweep. Baseline source must be reused, not regenerated.