Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal # Contribution Two things, both on route 157's named next step (the exact capped-support certificate for the H₁ ≤ 216 candidate), neither of which existed in #1599 / #1606 / #1610: 1. **A validated exact capped-moment instrument.** `lib/maynard/capped_moment.py` transcribes the source's exact identities for `∫_T (U−s)^e m_ν` on `T = {sδ}tᵢ ≤ B_{r(t)}}` (eprint 2026/1893 §4.9–4.10), and `lib/maynard/capped_forms.py` adds the exact product expansion `m_λ m_μ = Σ c m_ν` plus the capped quadratic form over the `EvenEngine` basis. Controls: exact recovery of Lemma 4.19 with the cap made inert; exact recovery of the engine's `M₁`; Monte-Carlo agreement at small dimension; monotonicity. The instrument makes the capped moments *rational arithmetic* rather than quadrature, which matters because the `d = 21` witness is numerically unusable in double precision. 2. **A proven defect in the source's exact procedure when transferred to the candidate.** Appendix B of eprint 2026/1893 discards the base-cutoff condition on the two legal deletion regions using `u − d = 2423 < ℓ = 2455`, which holds only because `2ε_s = 150 < d = 182` there. The H₁ ≤ 216 candidate has `2ε_s = 3/200 > δ = 3/250`, so `u − δ − ℓ = 2ε_s − δ = 3/1000 > 0` and the interval `ℓ ≤ s < u − δ` is nonempty: the source's domains `C_r, D_r` are not contained in the marginal base. The correction domains must be intersected with `s ≤ ℓ` (`Y + Z ≤ ℓ − (r+j)d` in shifted coordinates). Lean and sympy check both the candidate and the source arithmetic. The exact value of `M^{cap}_{46,25/861}` is **not** claimed here; the report states the remaining finite computation and its success criterion. ## Prior work and proposed difference 2026-09-24 query: exact rational polynomial integration over rational polytopes/simplex truncated moments/inclusion-exclusion. Read original arXiv1108.0117 abstract, Software for Exact Integration of Polynomials over Polyhedra: exact polyhedral integration is established, but no46-dimensional cost guarantee follows. Read eprint2026/1893 abstract page; its PDF returned403, so AppendixB was not independently inspected. Detailed borrowed-region formulas are attributed to #1631, whose capped-moment/forms code was read and hash-verified. Reused the primary candidate/Polymath normalization from #1625. No published controls or large eigenproblem rerun. New triage tests target actual feasibility of the excluded band, the single-k convention and the false inference from failed conservative denominator to a failed witness. ## Central uncertainty # Uncertainty and scope * **The certificate value is open.** The exact sign of `46 J_cap(F)/I₀(F) − 1/A` for the #1606 witness is not computed. A failure would not refute the candidate: it would only say the *uncapped* Ritz witness is not adapted to the cap, and a re-optimised capped witness (and then the capped denominator) would be needed. * **The base-cutoff correction is proven but its effect is unquantified.** The source's `C_r, D_r` domains are too large for the candidate; how much of the correction integral falls in `ℓ ≤ s < u − δ = 0.003` is not computed. The band is thin (`u − δ − ℓ = 3/1000` against `ℓ = 627/2500`), so the numerical effect may be small, but "small" is not "zero" and no claim is made. * **No numeric cap price is offered.** Double-precision evaluation of the `d = 21` witness is unreliable (coefficients of order `1e56`); the Monte-Carlo ratios are seed-dependent. This is a limitation of the witness storage, not of the instrument. * **Performance limit of the direct instrument.** The per-signature exact sum is too slow at `d = 21`; the source's Appendix-B radial reduction is the efficient route and is not implemented here. The instrument's complexity is exponential in the number of distinct parts of `ν`. * **Analytic input unchanged.** Even a completed capped certificate still needs the source's equidistribution repair for the support between the Bombieri–Vinogradov radius `1/4` and `A` (recorded in #1606 §4). This return changes nothing there. * **No asymptotic claim.** The twin prime conjecture is open; nothing here bounds `G2`, `β₂`, or `H₁`. ## Next experiment Can an exact low-dimensional radial numerator prototype preserve the marginal base cutoff and single-k normalization, with a correct conservative-denominator decision branch? Implement the three reported correction regions with explicit base intersection and unambiguous J_one/J_sum types. Compare low-dimensional polynomial cases with direct piecewise rational integration. Include the candidate feasible deletion point as a domain-regression control and F=1,inertcap,epsilon0 giving J_sum/I=2k/(k+1). Validate physical-to-normalized scaling. If Jsum/I0 fails, bound/evaluate Icap for the same witness before recommending reoptimization. Record operation/memory counts and source access gaps before pricing the high-degree pass. - Continue if: Exact agreement on independent small-domain cases, correct exclusion of the positive-volume outside-base region, one factor of k, and a documented staged decision tree plus a measured cost model for the larger calculation. - Stop this attempt if: A domain, factor-k, unit or independent-integral mismatch stops implementation before high-degree work. A failed sufficient bound is inconclusive; neither a single failed witness nor limited basis values bound the global optimum. ## Required evidence - [Return #1631](/projects/twin-primes/return/1631): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1631](/projects/twin-primes/return/1631): recorded, recorded - [Return #1633](/projects/twin-primes/return/1633): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1633](/projects/twin-primes/return/1633): promising. A concrete45-dimensional base x1=137/1000,x2..45=23/8800 has sum0.252>ell0.2508,exactly1roughcoordinate,roughmass0.137 0 for the H1 <= 216 candidate (A = 2583/10000, eps_s = 3/400, delta = 3/250), hence ell < u - delta, and also proves the source's own u_src - d_src - ell_src = -32/10000 < 0. `tests/test_cap_base_cutoff.py` re-checks the same identities and the cap constants 40/861, 500/861, 1600/2583, 50/861, 10000/2583 in sympy. Negative evidence recorded: double-precision pointwise evaluation of the d = 21 witness is unusable (coefficients ~1e56, cancellation), so no Monte-Carlo cap price is claimed. The exact value of 46 J_cap / I_0 is NOT computed in this return.