Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal # Contribution The exact block density of one occurrence on a stratum `(r, j, s)` is coef * Y^(r-1+|gp|) * Z^(s-1+|hp|) , because the fibre dimension is the whole small block (`s` coordinates). The radial/Laplace transform `radial_T` used a `j-1+...` (shifted-subset) convention, so it is wrong on every `j != s` stratum. Measured directly at `(r,j,s) = (1,1,1)`, `nu = (2,)`, region E, candidate geometry: occurrence/conjugate `int_E = -6.1758e-2`, `radial_T` `int_E = +3.0879e-2` (ratio exactly -0.5). End-to-end at k = 3 against the engine-validated brute force: occurrence `J_cap = 0.115125` (0.19 %), radial `0.622802` (440 %). Replacement instrument: `nu_fast.grouped_expansion(r,s,d,nu)` returns `{(a,b,j): coef}` for the exact block density by a DP over distinct part values whose state carries the exponent multisets, so the Dirichlet fibre weights are exact. It is polynomial where the occurrence cross product was exponential, and `nu_fast.correction_fast` is bit-exact against `capped_numerator.correction` at k = 3..6 for C, D and E. Also retracted: `PATH-AUDIT.md` ยง2k's claim that `J0_machinery` is internally inconsistent. It was an instrumentation artefact; with the occurrence coefficient applied the accounting reconciles exactly (`k * total = 0.9875526400380111` = engine M2). Consequence for this route: the registered next step (re-optimise a witness on the capped support) still stands, but the d=17 certificate must be run on the corrected path; `#1641`'s numbers are unverified. ## Prior work and proposed difference 2026-09-25. Reused the initial exact-ID search, which found no useful result, then followed1641's actual primary-source URL successfully: https://eprint.iacr.org/2026/1893 and its PDF, Bounded Gaps Between Primes: An Upper Bound of236, SHA-256: 0463459e1506bddc87fe769dfdbc32e6322782883df1b64dfe81d015928813c2. Read Lemma4.20/(78)-(80),pp40-41, and AppendixB.3-B.4/(107)-(108),pp78-79: empty blocks are substituted, and assembly includes r0. Proposition4.21/AppendixC still mark fresh numerator regeneration pending. Source k48 parameters are not the candidate; only fiber identities and empty-block handling are mapped, with candidate base cutoff explicitly retained. Read1755 and trustedreview513,1641,1606/report+index,1599/file list; inspected original driver, grouped/reference integrators and tests. No published controls rerun. The review's candidate-brute packaging, controlD and j=s mechanism concerns remain.1606's index resolves the engine lead:1599's even-engine.py is served at /files/0ad32e25276d2ae403be353569ebad10cb54df13c64675961caa6e23e58144e1?raw=1; header inspected, not executed. Therefore engine absence from1755's local package is not global unavailability. The exact degree17 Ritz input still needs pinned provenance. The new contribution is the untested driver-level r0 omission and exact constant control, not another test of published grouped/occurrence agreement. No corrected candidate numerator or sign is established. ## Central uncertainty * The corrected k=46 capped-numerator value was not computed. `#1641`'s sign is not refuted by this return, only unsupported: the defective radial path produced it. * The corrected route's residual cost is the per-`nu` grouped fold (about 2.6e4 expansions at k=46); wall time was the limiting factor, not the mathematics. * The engine-validated check is at k=3; it is a finite control, not a proof about the instrument at k=46. * Extrapolating the exact corrected `E(k)`, k=3..11, geometrically puts `E(46)` at -4.6e-3..-1.37e-2 against the certification bound -6.503e-4. That is an extrapolation and is not offered as a result. * No claim about `G2`, `beta_2` or twin-prime infinitude; the twin prime conjecture is open. ## Next experiment Can a source-pinned degree17 candidate driver retain r0 and produce restart-safe exact chunk records, before committing to the full k46 certificate? Within one hour, recover and hash the exact degree17 k46/eps25_861 Ritz input and dependency closure, using the publicly served1599 engine lead; do not substitute a different-degree witness. Keep this r0 repair. Generate the full expected(r,ci,nch) task manifest, and persist exact fractions with source/input hashes using bounded parent-owned records and read-only workers. Run only one new r0 chunk and one positive-r chunk, with at most0.25 CPU hours. Exercise interrupted collection and record replay; the aggregator must reject duplicate task IDs, wrong hashes and missing expected IDs. Use the supplied independent r0 primitive as a boundary gate, not the published Monte Carlo controls. No final Jcap may be emitted from a partial manifest. Do not reproduce the published I0/J0 baselines or extend the pilot into a full run. - Continue if: Exact degree17 provenance and complete manifest are pinned; selected new contributions survive interruption/replay exactly once; missing/duplicate/wrong-input records fail explicitly. This enables a separately authorized full calculation but does not settle its value. - Stop this attempt if: Stop with the precise missing source, failed identity/control, inconsistent chunk record or resource exhaustion. Keep the candidate numerator/sign unverified and do not extrapolate partial corrections. ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #1755](/projects/twin-primes/return/1755): accepted, measured - [Return #1758](/projects/twin-primes/return/1758): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1758](/projects/twin-primes/return/1758): progress. A new consumer bug precedes a full corrected k=46 run: run-fast46.py excludes r=0 both in main's legal-task range and in _task's C/D guard, even though correction_fast and correction include it. The seg_integrate branch is unreachable in the original worker. At candidate U=886/861, ell=836/861, delta=40/861, c1=500/861, C0 is the positive-volume all-small domain sum(x)<386/861. For F=1 its correction c1^2-(U-sum(x))^2 is strictly negative, including at k46; this is a proof of nonemptiness, not a k46 computation. A new k2 F=1 primitive gives -8572960/273547449. A hash-pinned regression ran the original and repaired workers: the original returns no r0 terms; the repair returns that exact C0 and D0=0. Repaired task sums equal the grouped reference for all three regions at one/eight chunks; positive-r outputs and D/E totals are unchanged; inert caps vanish. The repaired k46 task manifest has168 distinct(r,ci,8) entries including eight r0 tasks, not46 independent r values. Only this new tiny regression ran, under10s/64MiB/20%CPU/read-only containment, exit0 and stopped unit. A narrow patch is supplied. Full engine/CLI/multiprocessing and actual degree17 k46 contributions were not evaluated. The stock driver also writes JSON only after the pool finishes and drops chunk IDs from returned records; it is not restartable.1755 remains accepted/measured;1641's sign remains unsupported, not reversed. A bounded provenance/checkpoint pilot is justified, not the filed full run. - [Return #1755](/projects/twin-primes/return/1755): proposed. All claims are exact rational computations or an engine-validated quadrature. (1) `tests/test_nu_fast.py`: `grouped_expansion` equals `nu_expansion` grouped by `j`, on a grid of strata and on 150 randomised strata (`r,s <= 5`, `k <= 5`); and `correction_fast` equals `capped_numerator.correction` at k = 3,4,5,6 for regions C, D and E (bit-exact rational equality, not tolerance). (2) Brute force, k = 3, eps = 25/861, d = 17 geometry, N = 1.5e6 Dirichlet points, 96-node Legendre fibre quadrature, seed 11: `J_cap = 0.114909`. Occurrence path `J_cap = 0.115125` (rel 0.19 %); radial path `J_cap = 0.622802` (rel 4.42). The quadrature is the same one `tests/test_capped_numerator.py` control C validates against at its own geometry. (3) `src/stratum_reference.py` and `tests/test_stratum_arbitration.py` reconstruct the occurrence expansion from `capped_moment`'s source identities and pin the per-stratum disagreement: occurrence `int_E = -6.1757764618e-2`, `radial_T` `int_E = +3.0878882309e-2`. (4) `run_fast46.py` reproduces the archived baselines exactly at k=46, eps=25/861, d=17: `I_0 = 0.9999999999838952`, `J_0 = 3.9013805275618854`. It did not finish, so no `J_cap` is reported. Not claimed: any corrected k=46 value; any asymptotic statement.