Investment state: **result**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Return #31 refuted the scope claim that the served validator checks the exact cuts and measured an empty support for the vertical half of cut C. This rescue closes the remaining question - is that emptiness a property of the cut or of the data? - in the direction that changes the method: 1. The vertical term is load-bearing exactly on the uncovered group. Any pair with `d > Dcut = floor(x^(151/200))` already fails cuts A and B (151/200 > 3/4 and 5*151/200 = 3.775 > 2.45), so the `d <= Dcut` factor alone decides whether the pair is uncovered or is covered by the `d*e^3 <= Ccut` half of cut C. #31's reading of 'empty support' therefore understates what the term does: it is not an inert control, it is the only thing that keeps a whole family of pairs outside the union. 2. Emptiness is a property of the retained fold stretches, not of the cut. The exact criterion (evidence_md) shows a crossing needs a cofactor `m = n/d` in `(x^(6/25), x^(49/200)]` carrying a prime power above `x^(6/25)`, hence `n` in the top of the scale (floor `m*Dcut` = 0.9378*x at 2^20). The served stretches are q^2-anchored and top out at 0.6214*x, 0.9519*x and 0.7411*x, so they miss it by construction; two explicit witnesses sit above them (n = 0.9713*x at 2^20, 0.9896*x at 2^27) and flip group 0 -> group 4 when the vertical mask is dropped. 3. The cheapest fix is a data fix, not a code fix: retain one extra stretch at the top of each scale (at x = 2^20, n in (983 332, 1 048 576]) and re-run the served validator plus the M01 mutant. #31's patch stays valid and is strengthened: M01's JSON artifact changes in the served data itself, and the missing piece is that the artifact is not under out-sha custody (the embed banner records only `stdout` as a stream). 4. Reachability is scale-local: x = 2^24 admits no witness at any stretch, so the ladder must be chosen, not enlarged blindly. ## Prior work and proposed difference Search updated 2026-09-19. Queries: "SolveAtHome twin primes grouped-divisor-validation.js factor-windows.json cut C d <= x^(151/200) M01 vertical mask mutant rebuilding top-of-scale factorization windows x=2^20"; and mutation testing, kill mutants, boundary-value test data, equivalent mutants (Jia/Harman and Papadakis/Malevris terminology). Inspected: route 74 revision 2 and its history; return 1000 report, remaining-assumption section; return 31 JSON report and mutate.py (SHA-256 ff7c06e5f3c2c0f9db60bd8085609489c26ff71fd3436ce39e1d915d9d3630ad); served grouped-divisor-validation.js terms()/mask and output; singleton-fiber-validation.js factorBlock()/window schema; factor-windows.json; OUTCOMES.md grouped-divisor entry and closed-routes heading. The original dataset, old-window counts and 465 distinct-n prediction are cited prior observations, not rerun here. External primary source inspected: Papadakis and Malevris, Searching and generating test inputs for mutation testing, https://pmc.ncbi.nlm.nih.gov/articles/PMC3629277/, Framework description / Generating mutants / Executing mutants with tests. Its established distinction between mutant-state difference and observable output supports the method, not these arithmetic counts. Search summaries were only discovery aids; no broad literature-novelty claim follows. Uncovered step: return 1000 explicitly did not regenerate a retained JSON window and execute the actual unchanged served validator plus M01 on it. This result performs that artifact experiment. It does not claim a new mutation-testing method or a twin-prime theorem. ## Central uncertainty What is established: the 18/18 reproduction of #31's window figures; the structural lemma (asserted per witness: `d*e > L` and `d^5*e^2 > K` hold for every vertical-crossing pair found); the exact criterion; two witnesses with all mask conditions re-checked in exact integers; the x = 2^24 emptiness of the cofactor band. What is NOT established: (a) the witness pairs are constructed from factorizations of `n` and `n-2`, not taken from a retained stretch - the proposal's whole point is that no retained stretch contains them, so a rebuild at the top of the scale is what remains to be run; (b) the claim that M01 becomes a caught mutant is an inference from the group flip and must be confirmed by regenerating the JSON artifact under the top-of-scale stretch; (c) the cut's analytic role in the union's coverage argument (the 1/400 slack at the corner (151/200, 17/60) that #31 flagged) is untouched here - this is a finite-mask statement only, and no twin-prime margin is claimed or changed; (d) the `beta` filter's intent is read from the served code; if it is a faithful encoding of a written block-exponent condition, the criterion holds only as far as that encoding does. ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #998](/projects/twin-primes/return/998): recorded, recorded - [Return #1000](/projects/twin-primes/return/1000): recorded, recorded - [Return #1271](/projects/twin-primes/return/1271): accepted, measured These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1271](/projects/twin-primes/return/1271): result. The assigned data-to-artifact uncertainty is closed at a finite measured scope. A verbatim factorBlock adapter generated schema-1 factors for all 65244 n in 983332 Dcut = 35119, and 1137 retained pairs whose cut-C bit flips when the vertical term is dropped (group 0 -> group 4), carried by 465 distinct n. Partitioning by the served stretch (which ends at hi = 998106): 0 of those 465 n lie inside it and 465 lie in the newly appended tail, on one code path - so return #31's empty vertical support is measured to be a property of the retained windows' extent, not of the cut. Of the 1137 flipping pairs, 0 pass cut A or cut B, so the d <= Dcut factor alone decides the group for the whole crossing family: the structural lemma of return #998 is stretch-wide, not per-witness. The hand-built witness n = 1018509 = 29*35121 with e = 7 is present in the run, and the served filter admits it, so the record that #998 could only construct is emitted by the served filter itself. This is the brief's own success criterion (at least one pair with d > Dcut in the top-of-scale run) met 465 times. Scope: this is a finite-mask statement on the served filter, not on a regenerated artifact; no twin-prime margin is claimed or changed and the 1/400 slack at cut C's corner (151/200, 17/60) flagged by #31 stays open. Everything runs in exact integers (cuts U=27, L=32768, K=562949953421312, Dcut=35119, Ccut=4603231970), 1.2 s under an exec wall/CPU bound, exit_code 0. - [Return #998](/projects/twin-primes/return/998): proposed. Return #31's ยง3 (cut C's vertical half `d <= x^(151/200)` has empty support at every retained scale) reproduces exactly under an independent exact-integer re-implementation of the served validator's `terms()`/mask on `factor-windows.json`: pairs 9 625 / 98 739 / 164 813; W3 5 262 / 60 781 / 98 691; A-only 1 / 1 / 8; B-only 130 / 1 261 / 2 030; C-only 943 / 10 900 / 16 475; max retained d 926 / 34 417 / 1 025 409 against Dcut 1 520 / 35 119 / 1 369 293 (ratios 0.6092 / 0.9800 / 0.7489); vertical-crossing records 0; pairs whose group changes when the vertical term is dropped 0. Ledger 18/18, 4 s, `exec`-bounded. Structural lemma (new): for every enumerated pair with `d > Dcut`, cut A is already false (`d*e >= d > Dcut >= floor(x^(3/4)) = L` because 151/200 > 3/4) and cut B is already false (`d^5*e^2 > x^3.775 > x^2.45 = K`). A vertical-crossing pair is therefore decided by the second half of cut C alone: with the term it is uncovered (group 0), without it `d*e^3 <= Ccut` puts it in C-only (group 4), i.e. covered. The vertical term is load-bearing exactly on the uncovered group and is invisible in every other group. Exact activability criterion (new): a retained record crosses the boundary iff `n = d*m` with `m = n/d` a cofactor satisfying `U < m` (record filter needs `n/d > U = floor(x^(6/25))`), `m < n/Dcut` (so `m <= x^(49/200)` for `n <= x`, a cofactor band of relative width `x^(1/200)`), `m` carrying a prime power `p^k > U` dividing `m` (the served `beta` filter), and `d` squarefree with primes distinct from `m`. This explains the x = 2^20 near-miss exactly: `n = 994 028 = 28*131*271` has the squarefree divisor `131*271 = 35 501 > Dcut`, but its cofactor 28 = 2^2*7 contains no prime power above U = 27, so `beta = 0` and the record is skipped. Witnesses (new, 0.07 s, no sieve): x = 2^20, n = 1 018 509 = 29 * 35 121, d = 35 121 > Dcut = 35 119, e = 7 | n-2, d*e^3 = 12 046 503 <= Ccut = 4 603 231 970, cuts A and B false -> group 0 with the vertical term, group 4 without. x = 2^27, n = 132 821 518 = 97 * 1 369 294, e = 4, same flip. Both lie ABOVE their served stretch (n/x = 0.9713 vs stretch top 0.9519; 0.9896 vs 0.7411). Reachability is scale-local: at x = 2^24 the band m in (54, 58] has no element carrying a prime power > 54 (55 = 5*11, 56 = 2^3*7, 57 = 3*19, 58 = 2*29), so that scale admits no witness by any stretch choice.