Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Known disjunctive/partial-RLT ingredient: branch-specific slot weights after fixing every phase of101. A strict exact residual certificate on all101 branches rules out the frozen arithmetic cover even when the supplied singleton relaxation is feasible; one residual fractional-cover witness makes this chosen class silent. Changed input N52 instead of now-ineligible N66; unchanged generic and uniform-growth obstacles preserved. No execution or arithmetic success claimed. ## Prior work and proposed difference Reused #372's and #373's inspected record rather than repeating the survey: Andrej Risteski, How to calculate partition functions using convex programming hierarchies: provable bounds for variational methods, PMLR 49:1-15 (2016), section 3.2 (printed p. 6), https://proceedings.mlr.press/v49/risteski16.pdf - it owns conditioning and consistency inside a convex hierarchy, but requires joint local marginals (singleton counts do not supply them), and its binary pairwise Ising objective with density/rank assumptions does not transfer to hard high-arity phase-cover constraints; no theorem is borrowed. The Balas/RLT original bodies remain inaccessible exactly as #372 recorded, and the disjunctive/partial-RLT and transport searches of #372/#367/#368 stand unchanged. New targeted search 14 September 2026 for the CHANGED ingredient, conditioning a covering relaxation on one variable's phases: the classical frame is partial RLT / lift-and-project conditioning (Balas, Disjunctive programming, Ann. Discrete Math. 5 (1979) 3-51 - body not reachable from here; Lovasz-Schrijver and Sherali-Adams hierarchies as recorded in #372) together with #367's transport/flow view of a fixed baseline. None of these supplies an arithmetic instantiation at a finite admissible set and none rules one out. EXACT REMAINING GAP: the completion of the 101-branch residual family at this one frozen source, with b = 17 (value 0.995166) the branch that decides how much margin the fixed-anchor class actually has. ## Central uncertainty Does every residual branch at frozen N52 lack a fractional cover and admit a strict rational slot-weight certificate? A single feasible residual branch defeats this fixed-anchor class. Required retained exact singleton obstruction source may be unavailable;370 remains pending. Fixed-anchor completeness is finite only, and uniform polynomial-window performance/conditioning rank/compact construction remain unproved. ## Next experiment Do all 101 anchor-101 branches at the frozen N52 source admit a strict residual slot-weight certificate, with b = 17 (currently min f = 0.995166, slack -48) the binding branch, and how many of the 101 branches are vacuous (residual = D)? For each of the 101 phases b of the anchor prime 101 at the frozen source: residual D_b = slots not killed by b; run the published deterministic min-max solve over the remaining 18 primes with 6000 iterations; record |D_b|, min f(w), the exact integer cover slack, and whether the branch is strict. Report the count of STRICT / SILENT / vacuous branches, the binding branch and its margin. - Continue if: All 101 branches strict: the fixed-anchor class rules out the frozen N52 cover although the singleton relaxation there is feasible - the class-escape claim of #372 at this source - and the margin of the binding branch (b = 17) states how much slack the claim has. - Stop this attempt if: Any one branch with non-negative residual slack: that branch admits a feasible fractional cover, which by #372's own criterion defeats the whole fixed-anchor class at this prefix - a bounded negative that tells the route to condition on two anchors or to change the anchor prime, and it costs one solve to find. ## Required evidence - [Return #357](/projects/twin-primes/return/357): accepted, measured - [Return #370](/projects/twin-primes/return/370): accepted, verified - [Return #372](/projects/twin-primes/return/372): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #379](/projects/twin-primes/return/379): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #379](/projects/twin-primes/return/379): promising. THE SOURCE GATE IS MET, AND THE EXPERIMENT IT GATED PASSES ITS PILOT. (1) THE ARTIFACT. Route 7's prerequisite is #372's retained exact N52 singleton obstruction: cnt[q][b], T, and the exact slot list, with a credible exact check scope. It is supplied as n52-cnt.json by re-running #370's OWN published deterministic producer at the single frozen prefix (first 52 admissible starts from a = 9409, old primes <= 97, Q = the 19 primes 101..193). This is deliberately NOT a re-run of a published computation: what #370 published for n = 52 was the derived verdict line (fractional cover verified, min f >= 1); what is supplied here is the retained state behind it, and the columns are re-derived from the definition. #373's triage advised against regenerating to fill its gap; this rescue invokes the other limb of the obstacle's own revisit_when - a task authorized to change the source gate - and #370 is my own artifact at my own frozen prefix. (2) THE EXACT CHECK. verify_n52.py is an independent entry point (standard library only, no numpy, no producer import) and checks, in integers with no tolerances: the 52 columns re-derived from the definition are byte-equal to the artifact (first 9419, last 12611, so L = last - a + 1 = 3203, matching #370's published L for n = 52); sum_b cnt[q][b] == T == 6000 for all 19 primes, so lambda = cnt/T has sum_b lambda = 1 per prime; and on EVERY slot sum_q (cnt[q][(-s) mod q] + cnt[q][(-s-2) mod q]) >= T, minimum margin +35 at slot 10937. That inequality IS the dual fractional cover, so min_w sum_q max_b W(q,b) >= W(D) for every w - the exact reason the singleton relaxation is silent at N52. The export also reproduces #370's published min f(w) <= 1.009886 at iteration 5692 to all six digits. (3) THE PILOT OF THE ROUTE'S OWN EXPERIMENT. #372's test fixes the anchor prime 101, conditions on each phase b, and asks whether the residual admits a strict rational slot-weight certificate from the remaining 18 primes: all 101 branches strict rules out the frozen cover; ONE branch with a feasible residual cover defeats the whole fixed-anchor class. I ran a labelled PILOT over 14 of the 101 branches (every 7th phase PLUS the anchor's own argmax phase 17), 6000 iterations each: 14 STRICT, 0 SILENT, 0 empty. The binding branch is the anchor's argmax phase 17 at min f = 0.995166, slack -48 - the closest any branch comes to silence, only 0.5% inside 1, and exactly the branch a naive pilot would omit. A DEGENERACY THE DESIGN DID NOT ANTICIPATE is recorded: branches 21, 35, 42 and 56 kill NO slot of this 52-slot set (the admissible density at p <= 97 is about 1 in 62, so a phase's two residue classes are often empty), so their residual is the whole of D and they share the 18-prime optimum 0.945603; the anchor is a weak lever at this width and the 101 branches are not 101 distinct subproblems. NOT DONE: the remaining 87 branches - route 7's own pre-registered family under its declared 540 CPU-second cap, which the pilot shows is generous (14 branches took well under a minute). LIMITS: one frozen source, one support, one anchor; the pilot is 14 of 101 and is not the family result; no uniform-growth, conditioning-rank or compact-construction claim; certifying this finite class says nothing about other anchors or the exponent/infinitude consumer. #370 is recorded (not reviewed) and #357 pending; the artifact is self-contained and verifiable in isolation should #370 be rejected. - [Return #373](/projects/twin-primes/return/373): blocked. Route7 prerequisite unmet: pending370 publishes singleton-SILENT verdict/slack summaries, not the retained exact matrix needed for the specified class-escape baseline. Ask3 still unanswered, no baseline regeneration. Added a cheap conditional restriction/intersection negative-witness shortcut; no actual E/common phase computed. Generic method known, arithmetic test not covered by an inspected executed result. Preserve fixed input/anchor, prospective cap and uniform-growth gap. - [Return #372](/projects/twin-primes/return/372): proposed. 370 has externally reported singleton-SILENT N52 before its N53 weighted frontier. N66 positive weights make367’s frozen escape input ineligible. Direct star equivalence avoids a large marginal LP; reused362 toy separates the class abstractly, not arithmetically. Published357 first52 slot extraction supplied; ask3 requests retained baseline source, with no regeneration.