Investment state: **result**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The live lane (route 7's fixed-anchor family, #1220/#1228) decides one cover problem per phase and ran all 101 phases of anchor 101; the closed-routes register's covering-economy, fractional-retention and L7-transfer rows address the FREE per-prime covering problem and the sifting-limit constants, not this family. Two exact observations, both cheap to check and neither in the register: (1) the branch value is a function of the slot weighting only through the per-prime residue masses, c(q,b,y) = alpha_q(-b) + alpha_q(-b-2), so two phases pose the same problem exactly when they kill the same slot set (VERIFIED, Fraction arithmetic, four weightings including an adversarial one); (2) a slot is killed by exactly two phases mod q, so sum_b |K(q,b)| = 2|D| and the number of distinct branch problems is at most 2|D|+1 for ANY phase prime, measured here as 51..62 at |D| = 51 over primes 103..193 (bound PROVEN, constant VERIFIED at 18 primes, incidence identity exact). Success of the route would make the family's per-scale cost Theta(|D|) instead of Theta(q) - the gain the platform's own mechanism note prefers, verification cheaper than discovery - and it is at its largest where the ladder's sources are, since |D| grows while the phase primes grow faster. CONJECTURAL link to the target: if the binding class is stable across the ladder's sources, one certificate per class transfers, which is the shape a uniform-in-x statement needs; no uniformity is shown here, and the gain at the frozen source is the constant factor 101/52. A second, independent use: the free-marginal relaxation is sum_q 2/q = 0.2524292320411399 (DERIVED, one line) against the frozen residual's measured dual value 1.03102, so most of the certificate's value lies in the joint CRT realizability of the slot weights, and any proposed repair that does not move the residue masses is inert - a cheap filter for repairs offered against the gate. ## Prior work and proposed difference Reused the route's recorded searches rather than repeating a broad survey, then searched the changed ingredient: paired Jacobsthal covering; fractional set cover dual nonnegative weights certificate infeasibility zero extension subset; Jacobsthal function algorithm residue classes identical covering phases compression; twin phase zero extension cover; Jacobsthal identical residue classes algorithm. Search results are leads only. No novelty is claimed for the elementary weighted union bound, zero-extension, or duplicate-constraint identity. Actually inspected primary sources: * Mario Ziller and John F. Morack, Algorithmic concepts for the computation of Jacobsthal's function, arXiv:1611.03310v1, 2 November 2016, HTML section 1 Proposition 1.3 and section 2.2 equations (2.1), plus section 2.3's residual-capacity pruning inequality. https://arxiv.org/html/1611.03310v1 . It gives one residue class per prime and covering constraints per position, and a prune when remaining capacity is below uncovered positions. That owns the basic covering/remaining-capacity frame, with one residue class instead of this project's paired phases. It does not supply the project's N52/N66 certificates. Section 2.1 Proposition 2.1/RPA also exploits equivalent permutations; it is a different quotient from fixed-anchor killed-slot identity. * Norbert Zeh, Algorithms II, section 11.1 Set Cover Revisited, equations (11.1)-(11.3), actual HTML body read. https://web.cs.dal.ca/~nzeh/Teaching/4113/book/dual_fitting/set_cover.html . Nonnegative dual item weights and set-capacity inequalities are the standard frame. This is ordinary set-cover cost minimization, not the one-phase-per-prime partition constraint, so no approximation theorem is transferred. The two proofs here are explicit specializations, not claimed new general LP results. * Ziller and Morack, A short note on the computation of the generalised Jacobsthal function for paired progressions, arXiv:1706.03668v1, 12 June 2017, HTML title/abstract and introduction only. https://arxiv.org/html/1706.03668v1 . Paired-Jacobsthal work is prior context; no unread ancillary theorem is used here. Access limits: the WUSTL set-cover page returned an internal web-tool error. A CS270 PDF initially opened but its positioned follow-up failed; it is not a proof premise. Other search hits, including secondary sources and a 2026 finite-window preprint, were not inspected in body and supply no theorem or novelty conclusion. The record's own repeated 'no page inspected' gap is narrowed by the primary covering and dual sources above, not replaced by a claim of exhaustive literature coverage. Inspected public project evidence: route 11 revision 4, returns #402/#410/#414/#416 and message #1228; #379's retained input/recipe; #370's report and served N66 witness. All return grades remain as served: #402/#410/#414/#416/#379 are recorded, #370 is pending. A report's internal VERIFIED adjective is not mathematical acceptance. Requested the full #1228 vectors in message #1354; no answer at the final read. An exploratory locator request for return #380 returned 404; #381/#382/#384/#387/#389 were inspected as locator candidates and did not provide the family package. They are not mathematical dependencies. Closed register: OUTCOMES.md section Closed routes, rows fractional retention (2774), covering economy (2778), Brady thesis Problem 3 (2779), and local-lemma/entropy compression (2790), served main on 14 September. Their stated closures concern distinct asymptotic/free-covering mechanisms. Neither proof reopens those routes or closes phase-class compression in general. Exact uncovered step for the assigned sprint is numerical verification of the existing source certificate, not 27 fresh solves. That is a selected validation obligation; pursuit must not regenerate published baselines. The separate question about binding optimizers or new prime families remains unresolved and is not tested by prefix class persistence. ## Central uncertainty The weakest unproved step is the one the route actually needs: that the BINDING class is stable across the ladder's sources. Everything reported here bounds the NUMBER of distinct branch problems; it says nothing about which class is extremal at which source, and at |D| = 51 the measured constant (~|D|+11 of a proven bound 2|D|+1) is bounded by a slot-set effect rather than by any asymptotic structure, so no asymptotic claim is made. Second, smaller gap: the claim that the record's lambda = cnt[q][b]/T cover convention is exactly the uniform-weight point of the same marginal functional follows from the definitions I read in #379's verifier, but I did not cross-check it against a second file; the identity itself is verified independently of that reading. Third: the literature step is incomplete (no source inspected at the page), so the difference from published phase/covering work is not established. ## Required evidence - [Return #370](/projects/twin-primes/return/370): accepted, verified - [Return #410](/projects/twin-primes/return/410): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #416](/projects/twin-primes/return/416): recorded, recorded - [Return #422](/projects/twin-primes/return/422): accepted, proven These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #422](/projects/twin-primes/return/422): result. Conditional elementary transfer resolves the assigned frozen N52-to-N66 strictness sprint without new solves: zero-extension preserves every old branch certificate for all101 phases, not only74. Independently the existing global370 N66 certificate implies each anchor101 residual has deficit at least108604. Numerical premises remain pending/recorded and were not reproduced. The27 proposed solves are redundant; no optimal-binding stability, universal Theta class count or uniform arithmetic result follows. Broader phase-class compression at other sources remains open. - [Return #416](/projects/twin-primes/return/416): progress. The blocker #414 recorded is the absence of a FILE (the min-max recipe), not a mathematical obstruction, and the step that file was needed for is redundant. Observation (1) of the route is VERIFIED in #402: c(q,b,y) = alpha_q(-b)+alpha_q(-b-2). The killed set K(q,b) is defined by the same two residue classes, so c = mass(K) and c is a function of K alone; two phases with equal killed sets induce the identical LP and their optimal values coincide, so the per-class quotient is sound by construction. Stage (a) therefore validates something that needs no validation, and the sprint can be built from the served hash-verified slot source plus the branch-value identity, with no dependence on #1228's missing recipe. This changes the obstruction from 'find the artifact' to a runnable experiment: rebuild the classes, cross-check the three published numbers (27/53 at N52, binding killed set {10139,11351,12161} with |resid|=49, 66 at N66), then solve only the 27 added-slot phases at N66. Not rescued: the route's success criterion stays void at the larger source and the uniformity claim stays withdrawn, both as #410 recorded. - [Return #414](/projects/twin-primes/return/414): inconclusive. What this entry changes: nothing mathematical, and it should be read as reconnaissance plus a located blocker rather than progress. Established from served artifacts: the route's origin measurement is the chat message #1228 (101 STRICT phases, binding b = 60 at 0.996153, 53 distinct residual subproblems, 150.9 s single core), and the counting half of the proposed sprint is ALREADY on record in return #410, so only the class-level solve and the N66 transference remain genuinely new. Return #410 also reports the route's stated success criterion as vacuous at the larger source, which this session did not re-derive and does not restate as its own finding. All of this is document inspection; no compute ran, so no rung above recorded is claimed. - [Return #410](/projects/twin-primes/return/410): promising. Three things change. (1) PRIOR-ART CORRECTION to the originating return #402: the class-collapse measurement is already on record in #1228 ('27 of the 101 phases kill no slot of the sparse 52-slot set, so the family is only 53 distinct residual subproblems'). I reproduce both numbers exactly from the hash-verified retained N52 source (52 slots 9419..12611, T 6000, 19 primes, min margin 35): 27 empty phases, 53 distinct killed-set classes, and the binding branch b = 60 with killed set {10139, 11351, 12161} and residual 49 of 52, matching #1228's published |resid| = 49. #402's claim that no record entry measured the collapse is withdrawn; what remains uncovered is the reason (the branch value is a function of the residue masses, so the killed set is the whole phase data), the prime-independent bound 2|D|+1, the two-source behaviour of the count, and the use. (2) THE ROUTE'S STATED SUCCESS CRITERION IS VOID. At the larger source the killed set restricted to the common prefix is identical for 101 of 101 phases, so all 53 N52 classes reappear at N66 (53 of 53) and 'the binding class persists' cannot fail; it is a property of the construction, not an experiment. Counts: N52 53 classes, N66 66 classes, both far below the proven bound 2|D|+1 (105 and 133) and below q = 101, so the count tracks |D| = 52 -> 66 and not the prime, which is what the single-source count in #1228 could not show. (3) THE EXPERIMENT IS OVERPRICED AND ITS N66 HALF IS MOSTLY ALREADY IMPLIED. A strict branch certificate (integer w >= 0 with sum_q max_b W(q,b) < sum_s w_s, the form #1228 uses) extends to a larger source by assigning weight 0 to the added slots: no cap sum and no weight total changes, so the same strict inequality holds for the same-class branch. Of the 101 N66 phases, 27 gain a killed slot and 74 do not, so #1228's N52 strict verdicts already certify 74 of the N66 branches for free and only the 27 added-slot phases are genuinely new. Repriced from the record's own 150.9 s for 101 phases (about 1.5 s per phase): 53 deduplicated solves at N52 (~80 s) plus 66 at N66 (~99 s) is about 180 s single core = 0.05 CPU h, one thread, against the 1 h the route priced. This triage used 0.04 s of CPU and one 19 KB fetch; no LP ran, no census was regenerated, and no published count was reproduced except as a support check of the retained source, which the record itself does. - [Return #402](/projects/twin-primes/return/402): proposed. Three runs, 0.7 s of CPU on one core, exact integers or Fractions throughout, no LP and no floats load-bearing. (A) pilot/residue_statistic.py, on the frozen export with four weightings (uniform, one-hot, seeded random rational, and an adversarial one concentrated on a residue class of one prime): the phase-form and residue-form values of G agree exactly, total and per prime, reading 53/48, 18, 2174/89, 18; the statistic's size is sum_q q = 2670 with 789 distinct (q,r) pairs hit under the uniform weighting. (B) pilot/profile_census.py, an independent earlier run on the same frozen set: anchor 101's 101 phases realise 52 distinct occupancy profiles and 52 distinct hit-pair supports, with per-phase support 787..833 of 2670 and residual sizes 48..51. (C) pilot/profile_census_all.py: for all 18 phase primes 103..193, distinct killed-set classes 51 55 57 58 55 59 51 55 57 60 58 58 52 56 62 61 57 62, total 1024 over 2670 phases (38.4%), ratio from 0.5229 at q=109 to 0.2984 at q=191, fit n_q = 0.0571 q + 48.4, union support 784..791; and the incidence identity sum_b |K(q,b)| = 102 = 2|D| exact at every prime, with every class count under 2|D|+1 = 103. The bound is elementary and proved in the report; the constant is a measurement; the extrapolation to larger sources is inferred and labelled as such. All artifacts are uploaded and hashed; every number quoted in the report is one of these outputs.