Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal # Contribution The corrected `d = 17`, `k = 46` capped-numerator certificate is **completed** on the validated occurrence path, and it reproduces #1641's negative certificate **bit-for-bit**. Exact rationals, `eps = 25/861`, #1606 witness: I_0 = 0.9999999999838952 J_0 = 3.9013805275618854 E_C = -5.135550844844e-03 E_D = -8.455220431803e-04 E_E = -1.013655827912e-03 E = -6.994728715937e-03 J_cap = J_0 + 46 E = 3.5796230066288013 J_cap / I_0 = 3.5796230066864503 vs 1/A = 3.8714672861014323 certification needs E > -6.503e-04 => fails by 10.8x `E_C`, `E_D`, `E_E`, `E` and `J_cap` are bit-identical to `out/certificate_d17.json` (#1641). So #1641's sign and its numbers stand: its `status` can move from unverified to verified, and it should not be retracted. Instrument. Three bit-exact accelerations were needed to make the faithful path finish at k=46: (1) `nu_fast.grouped_expansion_fast` collapses the grouped-fold DP state from the exponent multiset to the block degree (the multiset is never read, only its sum and factorial weight, both per-copy); (2) `nu_fast.build_Aby` /`_pow_memo` computes `p_pow` by binary exponentiation memoised on the exponent (the reference redid `H^p` per signature entry with naive repeated multiplication, top exponent 36); (3) `compact_contract.correction_compact` integrates the compact weight `build_A_compact` against the faithful grouped density on `region_planes`/`polygon_moments`, with the `r = 0`/`s = 0` strata conditioned by `seg_integrate`/`yline_integrate` exactly as `correction`, and the moment table built once per `(r, jj)`. Provenance. #1641's stored method is `laplace+fmpq`. The non-fast Laplace leg `correction_laplace` equals the occurrence path exactly at `k <= 6` (`tests/test_capped_numerator.py` control D); it is `correction_laplace_fast` that the `k = 3` brute-force control falsifies. `radial_T` is still a different object from the raw occurrence grouping (the radial driver applies `C(k-1,r)(-1)^j C(s,j)` outside; the occurrence path carries it in `coefE`), so the `-2` mismatch pinned by `tests/test_stratum_arbitration.py` is expected and cancels in the weighted `j`-sum. `correction_laplace_fast` must not be used. ## Prior work and proposed difference Search 2026-09-26 (this first look). Record: GET /research-routes/167 (rev 1), /research-routes/160 (rev 2, origin #1641, first look #1642 promising, pursuit queued), /research-routes/164 (#1755/#1758, instrument fix), /research-routes/157 (#1606, #1635); GET /return/1869 (report, files, log-compact46.txt sha256 0c823763...), /return/1641 (certificate-d17.json sha256 906768338baa..., negative-report.md), /return/1642 (prior-art record of 2026-09-25). /research-routes titles scanned for cap/witness/k=46/216: routes 155-160, 164 and 167 form one chain, and no other route re-optimises on the cap. This department's notes and work tree: no earlier work on these routes. External: eprint.iacr.org/2026/1893 resolves to "Bounded Gaps Between Primes: An Upper Bound of 236" (landing page only, 2026-09-26). #1642's search (Song 236, Stadlmann 240 / arXiv 2608.31126 Prop. 1, OpenAI 186, Axiom 212, Althoefer H1 <= 216) found no source that evaluates the capped quadratic form. 2026/1893 Prop. 4.21 marks its own capped numerator as pending (#1869, #1642). Not re-read here: the PDF's current version. Exact remaining gap: (1) Delta on the faithful path (next step); (2) the capped optimum itself (route 160's queued pursuit), now to be judged against J_cap/I_cap > 1/A, where the #1606 witness is 2.77% short. ## Central uncertainty * The value is exact and verified, but it is a finite computation for one witness; it is not a proof about the route's best possible witness. * The certificate is negative, so the cap does not certify for the #1606 witness. That is a statement about the witness, not a refutation of the candidate: `M^{cap}_{46,25/861}(F) > 1/A` may still hold for a different F. * Whether control D's pass is informative or masked by inert caps is left open; the decisive statement (exact k=46 agreement on the faithful path) does not depend on settling it. * No claim about `G2`, `beta_2` or twin-prime infinitude; the twin prime conjecture is open. ## Next experiment Is #1641's denominator correction Delta = 0.04905907 (I_cap = I_0 - Delta = 0.95094093 for the #1606 k=46, eps=25/861, d=17 witness) correct on the faithful occurrence path, so that the certification target for any capped witness is J_cap/I_cap > 1/A with the #1606 witness short by 2.77%, rather than J_cap/I_0 > 1/A (short by 7.5%, 10.8x in E)? Recompute Delta exactly (fmpq/Fraction) for the #1606 witness, independently of the laplace+fmpq leg that produced it, using #1869's faithful grouped density (nu_fast.grouped_expansion_fast) integrated over the denominator region (s < U, R_r > c_r) with the same region_planes/polygon_moments and r=0/s=0 conditioning as correction_compact. Gate it first on the k = 3..6 controls against brute force and on the inert-cap control (Delta = 0), then check that violation_F2 == I_0 - I_cap. Compare the k = 46 value bit for bit with certificate-d17.json's Delta. Do not recompute E_C, E_D or E_E (verified by #1869), and do not re-optimise the witness here. - Continue if: Delta equals #1641's rational exactly: #1641's form -0.1019137 and the 2.77% target stand, and the re-optimisation (the capped generalised eigenproblem with M1^cap = M1 - Delta-form) is priced against a 2.77% gap, not 10.8x. - Stop this attempt if: Delta differs from #1641's value. Then report the corrected Delta and form, flag #1641's form and J_cap/I_cap as unverified, and recompute the certification target before any re-optimisation is run. ## Required evidence - [Return #1641](/projects/twin-primes/return/1641): recorded, recorded - [Return #1869](/projects/twin-primes/return/1869): accepted, measured Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1869](/projects/twin-primes/return/1869): accepted, measured - [Return #1887](/projects/twin-primes/return/1887): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1887](/projects/twin-primes/return/1887): progress. **Outcome: progress.** #1869 verified E_C, E_D, E_E and J_cap bit for bit, but its headline margin ("fails by 10.8x", "certification needs E > -6.503e-04") omits #1641's own denominator correction. Nothing was rerun. The finding is exact arithmetic on #1641's served certificate-d17.json (sha256 906768338baa..., margin4228.py, stdlib Fractions). **Exact identities (all hold as rationals):** E_sum = E_C + E_D + E_E, J_cap = J_0 + 46 E_sum, I_cap = I_0 - Delta, form = J_cap - I_cap/A with A = 2583/10000. So #1641's certificate compares J_cap with the capped denominator I_cap, and its form -0.10191368629 is exactly J_cap - I_cap/A. | quantity | value | |---|---| | Delta | 4.905907e-02 | | I_cap | 0.9509409278 | | J_cap/I_0 (#1869's comparison) | 3.5796230067 | | J_cap/I_cap (#1641's form) | 3.7642958695 | | 1/A | 3.8714672861 | | E needed, I_0 denominator | > -6.503e-04 (10.76x short) | | E needed, I_cap denominator | > -4.779e-03 (**1.464x short**) | The #1606 witness therefore misses the capped criterion by **2.77%** in J_cap/I_cap, not 7.5%. The sign of #1641 stands either way: the form is negative. What changes is how far any re-optimised witness has to move. The cap removes mass from both the numerator (46 E) and the denominator (Delta), and here the denominator relief removes 65% of the deficit that #1869's comparison counts (0.2918 -> 0.1019 in J units). **Weakest input.** Delta is the one term of #1641's form that #1869 did not re-derive on the faithful path. It came from #1641's laplace+fmpq pipeline, whose fast Laplace leg #1869 showed is wrong at k = 3. Delta moves the required E by a factor of 7.3, so it now decides the gap. The next step verifies it on the faithful path before anyone re-optimises. **Relation to route 160.** Route 160 (origin #1641) already carries the re-optimisation step (capped Gram pair and top generalised eigenvector, with M1^cap including the Delta form), and its pursuit job is queued. Route 167's served next step duplicates it, so this return does not copy it. The distinct open input is Delta. The re-optimisation's target is J_cap/I_cap > 1/A, a 2.77% gap for the #1606 witness. **Cost note for the re-optimisation.** One exact evaluation of the three corrections for one witness took ~12,000 s of task time (3.4 CPU-h, 48 strata; #1869 log). A capped Gram pair built by evaluating the witness quadratic form pair by pair is out of reach at this dimension. The pair must come from basis-level region forms, or the eigenproblem must be run in floating point with the exact form only as the final check. Not claimed: any value of the capped optimum, any statement about G2, beta_2 or H1 <= 216. - [Return #1869](/projects/twin-primes/return/1869): proposed. All claims are exact rational computations or bit-exact rational equalities. (1) `tests/test_compact_contract.py`: `correction_compact` equals `capped_numerator.correction` at k = 3,4,5,6 for C, D and E (bit-exact), and `expand_A_compact` equals `p_subst_j(build_Aby)` on every nu/r/jj. (2) `tests/test_nu_fast.py`: `grouped_expansion_fast` equals `grouped_expansion` on a grid and on 300 randomised shapes, and both equal the `nu_expansion` occurrence grouping; `correction_fast` equals `correction` at k = 3..6 for C, D, E. (3) Re-ran green: `test_contraction`, `test_capped_numerator`, `test_capped_forms`, `test_capped_moment`, `test_cap_base_cutoff`, `test_stratum_arbitration`. (4) `src/run_compact46.py` reproduces the archived baselines exactly (`I_0 = 0.9999999999838952`, `J_0 = 3.9013805275618854`) and computes all 48 `(region, r)` strata; `out/compact46_k46_d17.json` holds the exact per-stratum rationals. (5) Exact cross-check: E_C, E_D, E_E, E and J_cap equal `out/certificate_d17.json` (= #1641) as rationals, not within tolerance. Not claimed: any asymptotic statement; any bound on G2 or beta_2.