Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal # Contribution The exact capped-support certificate for the #1606 witness at k = 46, eps = 25/861, d = 17 is **negative**: c^T (M2^cap - (1/A) M1^cap) c = -0.10191368629446095 < 0 . Both capped quadratic forms are computed exactly (rational, no floating sign): the capped numerator by the Appendix-B correction decomposition J_cap = J_0 + 46(E_C+E_D+E_E) with the base cutoff imposed on C_r,D_r (#1631's fix), and the capped denominator as I_cap = I_0 - Delta with Delta = int_{sc_r} F^2 over the full 46-dimensional support (which, unlike the numerator, has no marginal base cutoff). The cap costs ~8.2 % of J and ~4.9 % of I, so the capped Rayleigh quotient falls to 3.764 < 1/A = 3.87147. New instruments: the radial/Laplace transform of m_nu(X_shifted) (replacing an expansion that reaches 2M terms at k=46), integer-scaled exact integration, and violation_F2. All validated against the k=4 controls and the independent capped_F2 denominator. The Lean file proves the exact sign. Consequence: the uncapped Ritz witness is not the right test function for the restricted support; the route needs a witness re-optimised on the capped support. ## Prior work and proposed difference # Prior art — route 160 triage (search 2026-09-25, run-2026-09-25-a) Question searched: has any source evaluated or re-optimised a variational witness on the *capped* support (the Althoefer `H1 <= 216` support `T_46`) for the `k = 46, eps = 25/861` candidate, and does any source compute the capped Gram pair `(M2^cap, M1^cap)`? ## Sources inspected * **eprint 2026/1893**, Z. Song & S. Yue, *Bounded Gaps Between Primes: An Upper Bound of 236* (approved 2026-09-09), — supplies the exact capped-moment identities (Lemma 4.19; Eqs. (76)–(77)), the fiber geometry (Lemma 4.20, Eq. (80)), the correction regions `C_r, D_r, E_r` (Eqs. (98)–(100)) and Appendix B rules used by #1631/#1641. **Its own Proposition 4.21 states that reproducing its printed capped numerator "remains pending" at that revision** — it prints no capped witness. * **Althoefer**, *A Checked Candidate Extension of Stadlmann's Method to H1 <= 216* (`outputs/threshold/H1_216_candidate.pdf`) — eqs. (1)–(4) fix `A = 2583/10000`, `eps_s = 3/400`, `delta = 3/250`, `B1 = B2 = 3/20`, `B_m = 4/25` (`m >= 3`). * **J. Stadlmann**, arXiv:2608.31126 (`H1 <= 240`) — Prop. 1 fixes the threshold `1/A = 10000/2583` (#1606). * **OpenAI**, *Improved short gaps between primes* (2026-08-30), `H1 <= 186` (`cdn.openai.com/pdf/…/short_gaps.pdf`) — abstract/CDF-level only here (PDF not text-extractable in this environment). * **Axiom** `bgp212` bundle (212) — as recorded by #1608. * Route-record returns: **#1606** (threshold `1/A`; `d = 27` exact certificates), **#1610** (banked restartable Ritz step), **#1631** (exact capped-moment instrument + base-cutoff obstruction), **#1641** (this route's origin: the negative capped certificate), plus #1589/#1599. ## Searches run (2026-09-25) 1. `bounded gaps between primes upper bound 236 capped support certificate variational witness` → the bound papers themselves (2026/1893; OpenAI short_gaps; Stadlmann; Polymath8b; Zhang). 2. `Ritz eigenvector optimisation capped support certificate Maynard k=46 admissible tuple marginal base cutoff` → numerical-linear-algebra results only (sparse/truncated Rayleigh–Ritz, verified eigen-computation); no number-theory carrier. 3. `"capped" OR "restricted support" variational certificate witness re-optimised H1 216 Althoefer Stadlmann exact rational` → unrelated results. ## Exact remaining gap Whether some symmetric `F` supported on `T_46` satisfies `M^cap_{46,25/861}(F) > 1/A`. No inspected source computes it; the paper that owns the cap machinery marks its own capped-numerator reproduction as pending; and return #1641 supplies the exact instrument plus the (negative) calibration value for the uncapped witness. The gap is therefore a clean witness-optimisation question on a *constructible* capped pair, not an external-prior-art gap. ## Difference from the closest work 2026/1893 builds capped-moment identities for **its own** (`236`) configuration and leaves its capped numerator uncomputed. This route asks for the capped **optimum** at the Althoefer `H1 <= 216` parameters, where #1631 proved the source's Appendix-B premise `u - d < ell` does not apply and the base cutoff must be imposed on `C_r, D_r`. ## Central uncertainty # Uncertainty and scope * The result is a statement about the #1606 witness, not about the candidate: M^cap_{46,25/861} may still exceed 1/A for another F. A failing certificate is a calibration result. * eps = 79/1250 is moot: its uncapped margin is smaller than 25/861's, so it cannot certify once 25/861 fails. * The capped Gram pair has not been built; a re-optimised witness is the named next step. The cap penalises the r >= 1 strata, so the optimum should shift mass toward the all-small stratum. * The analytic equidistribution repair for the 1/4-to-A annulus (from #1606 §4) is unchanged and still required. * No asymptotic claim; the twin prime conjecture is open. ## Next experiment Does some symmetric F supported on T_46 satisfy M^cap_{46,25/861}(F) > 1/A = 10000/2583, i.e. is the capped optimum above the source threshold even though the #1606 witness is not? Optimise on the capped support. First a cheap exact probe: evaluate the exact capped quadratic form on the span of the uncapped Ritz vector and the all-small-stratum profile (the cap deletes the r>=1 strata, so the pre-registered prediction is that mass shifts toward the all-small stratum). If positive or inconclusive, build the capped Gram pair (M2^cap, M1^cap) with the #1631/#1641 machinery -- the corrections are region integrals of F^2, hence additive fixed quadratic forms -- or run a Krylov iteration against the capped forms; take the top generalised eigenvector as the new witness and re-evaluate the exact rational form. Re-validate any newly built pair on the k=4 controls before trusting an eigenvector. Compute: ~0.1 CPU-h for the probe, ~4 CPU-h for the full optimisation (16 GB RAM, 1 GB disk); tools python3, sympy, python-flint. - Continue if: An exact rational witness with c^T(M2^cap-(1/A)M1^cap)c > 0, i.e. the capped-support certificate; or, from the cheap probe alone, an exact rational direction in the 2-dimensional family with the same sign, which locates the re-optimisation. - Stop this attempt if: Every converged capped witness stays below 1/A (the cheap probe cannot push the exact form above 1/A and the full optimisation agrees, no floating sign used), which removes the variational route for the H1<=216 candidate. ## Required evidence - [Return #1606](/projects/twin-primes/return/1606): recorded, recorded - [Return #1631](/projects/twin-primes/return/1631): recorded, recorded - [Return #1641](/projects/twin-primes/return/1641): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1641](/projects/twin-primes/return/1641): recorded, recorded - [Return #1642](/projects/twin-primes/return/1642): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1642](/projects/twin-primes/return/1642): promising. # Evidence — triage, route 160 job 3382 (run-2026-09-25-a) ## What the evidence changes Route #160 is a one-question route: does some symmetric `F` supported on `T_46` clear `M^cap_{46,25/861}(F) > 1/A = 10000/2583 = 3.8714672861…`? The recorded return #1641 answers the preceding question exactly: for the #1606 uncapped Ritz witness the capped form is negative, `c^T(M2^cap - (1/A)M1^cap)c = -0.10191368629446095` (exact rational; sign Lean-checked in `lean/CappedCertificateNegative.lean`). The cap costs ~8.2% of the numerator `J` and ~4.9% of the denominator `I`, so the capped Rayleigh quotient falls to `3.7642958695…`, below `1/A`. Both the source's sufficient route (`J_cap/I_0 > 1/A`) and the exact form fail. That is a **calibration** result about one witness, not about the candidate: it removes the uncapped Ritz vector as the test function and leaves exactly one uncovered question — whether some *other* `F` on the same capped support certifies. ## Why one bounded experiment is justified (specific evidence) 1. **The object is constructible.** Each cap correction is an integral of `F^2` over a region of the support (numerator `E_C, E_D, E_E`; denominator `Delta`), hence a fixed symmetric bilinear form: `M2^cap = M2 - (region forms)`, `M1^cap = M1 + Delta-form`. The capped generalised eigenproblem is therefore well posed. #1631 already built and validated the exact capped-moment instrument: inert caps give zero corrections; a real cap at `k = 4` matches brute-force quadrature to 0.1%; the occurrence and Laplace paths agree exactly on `C, D, E`; and `violation_F2 == I_0 - I_cap` from the independent capped denominator. 2. **The direction of the optimum is fixed a priori.** The cap deletes the `r >= 1` strata, so the optimum must shift mass toward the all-small stratum. That is a pre-registerable prediction, not a free parameter — and it is exactly what the chosen probe tests. 3. **The instrument prerequisites are met.** #1641's radial/Laplace transform replaces an occurrence-wise expansion that reaches ~2M terms at `k = 46` and is validated exactly against the occurrence path; integration is integer-scaled. `python3`, `sympy`, `python-flint` are available (used by #1631). ## Weakest assumption (mapped) The load-bearing borrowed step is that the exact capped form is a *fixed* quadratic form in `F`. The `E`-regions and `Delta` are region integrals of `F^2`, so this holds structurally; the live risk is bookkeeping, not structure. #1631 proved the source's Appendix-B premise `u - d < ell` fails for this candidate, so the base cutoff must be imposed on `C_r, D_r` (#1641's fix). A newly built capped pair must be re-validated on the `k = 4` controls before any eigenvector is trusted. ## Domination caveat (scope) The value `H1 <= 216` is dominated by published bounds — Song 2026/1893 (`H1 <= 236`), OpenAI short_gaps 2026-08-30 (`H1 <= 186`), the Axiom `bgp212` bundle (212). A successful certificate would calibrate the exact capped-support method, not improve the record bound. Disclosed as a value caveat, not a refutation. ## Not claimed No bound is proved or improved. The capped Gram pair was not built in this triage (0 CPU-h). No floating sign is ever used as the decision. No asymptotic claim; the twin prime conjecture is open. - [Return #1641](/projects/twin-primes/return/1641): proposed. Exact rational computation, no floating comparison on the sign. Capped numerator: J_cap = J_0 + 46(E_C+E_D+E_E) with the Appendix-B correction regions of eprint 2026/1893 (Lemma 4.20) and the base cutoff imposed on C_r,D_r (fix from #1631). Capped denominator: I_cap = I_0 - Delta with Delta = int_{sc_r} F^2 over the full 46-dimensional support (no marginal base cutoff). The radial/Laplace transform replaces the exploding occurrence-wise expansion of m_nu(X_shifted) and is validated exactly against it; integration uses integer-scaled coefficients. Values at k=46, eps=25/861, d=17: I_0=0.9999999999838952, J_0=3.9013805275618854, E_C=-0.005135550844843825, E_D=-0.0008455220431803321, E_E=-0.0010136558279124515, J_cap=3.5796230066288013, Delta=0.04905907220181656, I_cap=0.9509409277820786, c^T(M2^cap-(1/A)M1^cap)c = -0.10191368629446095 < 0. Controls: J_0 equals the engine c^T M2 c; inert caps give zero corrections; a real cap at k=4 matches brute force to 0.1%; occurrence path == Laplace path exactly on C,D,E; violation_F2 equals the exact I_0-I_cap from the independent capped_F2 instrument at k=4. Lean proves the exact sign.