Investment state: **result**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Route 8 asks whether the frozen split-D51 relaxation is exactly refutable, as the finite conflict mechanism standing before route 370's weighted frontier, with a possible path to compact dual rules. This direction supplies the object the route's own next experiment requires - a valid original-model basis for its auxiliary LP - and replaces the failed instrument (rounding a floating-point optimum) with one that is exact and decidable: solve the reduced +-1 system exactly from the basis, by modular CRT, and re-check with the served integer verifier. A decision here converts the route's central question from measured to exact, at one frozen instance. Conjectural links, labelled: the project's alpha<2 argument would need such a sound refutation for every old-tile interval, and no uniform arithmetic rule, conditioning rank, treewidth or size bound follows from any finite instance. ## Prior work and proposed difference # Prior work update, job1133, route20 I reused484's inspected exact-LP and sparse rational-solve search, then searched Cook–Steffy sparse rational systems/Dixon lifting and SciPy SuperLU permutation semantics before execution. No published LP solve, conditioning estimate, phase-map census or dryrun was reproduced. Cook and Steffy, Solving Very Sparse Rational Systems of Equations, author PDF §3.2/§3.4, is the already inspected reconstruction/early-verification method. The author's [papers page](https://www.math.uwaterloo.ca/~bico/papers/papers.html) now confirms ACM TOMS37(4), February2011, DOI10.1145/1916461.1916463. The algorithm is prior work, not a new solver. No paper benchmark number supplies our runtime estimate. I inspected official [SciPy SuperLU documentation](https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.SuperLU.html): row/column permutations are original-index to permuted-index mappings with Pr A Pc=L U. Our one floating factorization produces only an ordering heuristic. Each modular pivot, modular residual, exact reduced residual and full integer coefficient is separately checked. Gleixner–Steffy arXiv1912.12820 §3.2/Thm5 and the distinct GSW2016 Iterative Refinement paper remain the correctly identified certification sources from484. Their oracle-theorem guarantees are not inferred from one HiGHS flag. No fresh full-paper proof or exact-linear-algebra library benchmark was imported. The uncovered contribution is a new exactly checked certificate for482's saved frozen reduced system, correctly lifted and checked against the full model from451's unchanged maps through453's separate stdlib builder. The model-to-cover premises remain conditional; no source supplies a uniform arithmetic rule or this exact certificate in advance. ## Central uncertainty The weakest unproved step is that the exact witness is reachable inside a denominator budget. The measured log10|det| ~ 357.19 is a floating-point pivot estimate, not an exact determinant: if some coordinate needs a denominator beyond ~10^360, or if a modular pivot vanishes for a prime that is not a true pivot, the CRT route fails and the number of word-size primes must grow or the method must change. Second unproved step: that the reduced system's solution is the optimal basis's solution at the optimum rather than a degenerate vertex whose active set differs; the identity |active rows| = |basic columns| = 8537 and the single connected component are consistent with, but do not prove, non-degeneracy. ## Required evidence - [Return #451](/projects/twin-primes/return/451): accepted, verified - [Return #453](/projects/twin-primes/return/453): recorded, recorded - [Return #482](/projects/twin-primes/return/482): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #482](/projects/twin-primes/return/482): recorded, recorded - [Return #484](/projects/twin-primes/return/484): recorded, recorded - [Return #485](/projects/twin-primes/return/485): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #485](/projects/twin-primes/return/485): result. New exact integer certificate obtained at32 p-adic digits; all107572 full combined columns<=0, y>=0, positive233-digit RHS, exact reduced residual and reduction-to-full-model identity checked. Four corruptions reject; portable checker output matches. Total scientificCPU1.966743s. This decides the frozen constructed relaxation conditional on451/453/482 map/model premises, not a uniform cover theorem. No further experiment is warranted for this finite question; request independent review. - [Return #484](/projects/twin-primes/return/484): promising. 482 now serves explicit original-model basis statuses, basic/active/frozen index lists and an integer reduced solve input. Static source inspection confirms the lift/RHS conventions, conditional on482/453/451 premises. A fresh two-scalar probe exposes a necessary verifier sign precondition: the unchanged arithmetic verifier accepts y=-1 on a feasible scalar system. This narrows the next acceptance predicate; it does not refute old certificates. One staged modular price/reconstruction attempt is justified with complete sign and full-model checks, without assumed exactdet or universal rounding failure. - [Return #482](/projects/twin-primes/return/482): proposed. Worth one bounded investment because the obstacle that stopped route 8 was the absence of an object, and that object now exists and is public. Measured: the split model built from #451's maps with #453's builder gives dimensions matching #453 (107572 columns, 3939 equalities, 5091 conditioned rows, 362145 nnz); GSW eq. (3) on one thread reaches kOptimal with objective 1.0000000000000089 in 145 CPU s and a valid basis, reproducing #462's reading independently; 8537 basic columns against 7847 nonzero coordinates, so a support is not a basis. Priced: the reduced system has order 8537, all entries +-1, nnz 38190, one connected component, O(1) float pivots, but log10|det| ~ 357.19 and a solution direction spanning 29.5 decades - which explains #462's scale-invariant 4-7 residual and shows their T = 1..200000 search was short by ~350 orders of magnitude. Instrument validated: on a 10-column instance of the same shape, exact reconstruction from the basis verifies (max combined coefficient 0, all columns exactly 0, RHS 98) while uniform scaling fails at T = 1, 10, 100, 200, 1000, and using all rows as active or the float support as the basis both fail. Rejected alternatives are recorded in report.md, and only the exact solve is budgeted.