Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal # Capped Gram pair at `k = 46`, `d = 17`, and the re-optimised witness Follow-up to **#1869** (route 167), which verified #1641's negative certificate on the faithful occurrence path and named the next step: build the capped Gram pair on the capped support and re-optimise the witness there. Calibration: **verified** (exact rational arithmetic throughout; the recorded slot functionals are bit-exact against the direct contraction at small `k` and reproduce the stored #1869/#1641 correction values for the #1606 witness exactly). No asymptotic claim; the twin prime conjecture is open and nothing here bounds `G2`, `beta_2` or twin-prime infinitude. ## 1. Result The two cap corrections are quadratic forms in the witness coefficients and the validated contraction is **linear** in the per-`nu` signature coefficients, so a single contraction pass records, for every signature slot, the scalar functional `W[slot]` it carries. Polarisations then give the exact symmetric Gram matrices M2^cap = M2 + k * E_sym M1^cap = M1 - Delta_sym with `k = 46`, `eps = 25/861`, `d = 17`, `A = 2583/10000`. For the #1606 witness `c0` (`n = 374`) the recorded forms reproduce the stored exact values: | check | value | status | |---|---|---| | `c0^T E_sym c0` vs #1869 `E_total` | `-6.9947287159e-03` | `bit-exact` | | `c0^T Delta_sym c0` vs #1641 `Delta` | `4.9059072202e-02` | `bit-exact` | Solving the generalised eigenproblem `M2^cap x = lambda M1^cap x` (whitening `M1^cap` in 256-bit `arb` arithmetic, ordinary symmetric solve, high-precision triangular recovery) gives the re-optimised witness `c*` (exact rationals, `n = 374`): | quantity | value | |---|---| | `I_0(c*)` | `+3.263839e-94` | | `J_0(c*)` | `+1.258373e-93` | | `I_cap(c*)` | `+3.221622e-94` | | `J_cap(c*)` | `+1.227944e-93` | | `J_cap/I_cap` | `3.8115713687001214` | | `1/A` | `3.8714672861014323` | | `Q = c*^T(M2^cap - (1/A) M1^cap)c*` | `-1.9296200187e-95` | | verdict | DOES NOT CERTIFY: the re-optimised capped witness lifts the quotient from `3.76429...` (the #1606 witness) to `3.811571`, still below `1/A = 3.871467` | The exact rational `Q` and the witness are in `reoptimised-witness-d17.json`; the sign of `Q` is `negative`. `I_cap(c*)` is tiny (`~1e-94`) because the top regularised eigenvector lies close to the near-kernel of `M1^cap`; the Rayleigh quotient is scale invariant and the exact `Q` is computed from the unregularised pair. The two exact Gram matrices are uploaded gzipped as `gram-E-k46.json.gz` and `gram-Delta-k46.json.gz` (upper triangle, exact rationals). Because `M1^cap` is only positive semi-definite — its diagonal has entries as small as `~1e-94` (some monomial directions integrate to almost nothing on the restricted support T) — the eigenproblem is regularised by a small multiple of the identity *in the scaled frame* and the recovered vector is then re-scored against the exact unregularised pair. A float finite-spectrum diagnostic on the scaled pencil (truncating the near-kernel at relative eigenvalue `1e-6` to `1e-12`) puts the top finite generalised eigenvalue at `3.799` to `3.812`, i.e. the re-optimised witness is close to the best the `d = 17` basis supports; that upper-bound statement is numerical, not a proof. ## Prior work and proposed difference Bounded refresh 2026-09-28 of #1972's search, for the CHANGED ingredient (the relative cost c of a restricted-support cap, and the basis-degree convergence of this variational problem); not a repeat of the broad survey. Queries (web search, 2026-09-28): 'Polymath8b variational problem basis degree d convergence M_k,epsilon truncated polynomial basis'; 'Maynard-Tao sieve restricted support capped moment inequalities truncated support variational problem'; 'Maynard variational problem optimum convergence polynomial degree truncation basis saturation rate bound'. Inspected, closest sources. (a) Polymath8b IX, 'Large quadratic programs', T. Tao, 2014-02-21, read in full: the project's own status note on degree enrichment of exactly this variational problem - Maynard's comment, same page: 'run-time restrictions mean that we are limited to degree at most about 18 at the moment. Potential to reduce k by 1 or 2 if we can feasibly calculate much larger degrees. Bottleneck is currently calculating the matrices.' That is the closest published statement about the value of enriching d in this family; it is about the UNCAPPED problem, names matrix assembly as the bottleneck, and gives no rate in d. (b) Polymath8b II, 2013-11-22, same source: support truncation is described as harmless in the large-k setting ('the functions we were using had such a truncated support anyway') - context for the magnitude of the 2.3% cap cost the route measures, not a bound. (c) eprint 2026/1893 'Bounded Gaps Between Primes: An Upper Bound of 236' (Z. Song, 2026), indexed; cited by #1641/#1869 for Lemma 4.19-4.20 and the restricted support; not re-read here and its PDF not fetched this run. Tooling sources re-read in full: #1972 (REPORT.md, sweep_frozen_threshold.py, route172_numbers.json, route172_inputs.json), #1942 (capped-gram.py, gram-check.json, reoptimised-witness-d17.json), #1641 (capped-moment.py, derivation.md §2, index-capped.md, certificate-d17.json); all fetched by sha and sha-verified locally. Exact remaining gap. No inspected source evaluates the relative cap cost c = 1 - q_cap/q_uncap for this variational problem at any k; none reports the growth of the CAPPED optimum in the basis degree d; and none bounds dM(d)/dd for the uncapped problem, which is why the monotonicity argument in (1) of the evidence is needed to settle the recorded step at all. A no-match search is evidence about the search, not novelty. Access gap: lib/maynard/even_engine.py and lib/maynard/whiten_eig.py - required by both the recorded and the proposed experiment - are not served by the project (GET /docs/lib/maynard/... returns 404; the served tree is attestation/ bench/ paper/ research/ tools/ web/) and are absent from this machine's checkout. ## Central uncertainty * The Gram pair is exact but finite and basis-limited (d = 17); it is not the route's global capped optimum. * `violation_F2` uses `radial_T`, whose stratum-level Z-convention disagrees with the occurrence grouping (#1869 §2); its aggregate matches #1641 exactly. * Nothing here bounds G2, beta_2 or twin-prime infinitude; the twin prime conjecture is open. ## Next experiment Does the relative cap cost c(k, d) = 1 - q_cap/q_uncap fall toward <= 0.766735% as the number of variables k grows, at the cap schedule already on the record? Sweep (k, d) upward from #1972's nine indefinite cells ({4,6,8} x {1,2,3}) using the exact capped/uncapped instruments validated in #1641/#1942 - no recording pass at k = 46 and no d-enrichment of the k = 46 basis. At each cell where the diagonally equilibrated M1^cap is positive definite (top_ritz(check_pd=True) refuses otherwise), record exact q_cap, q_uncap, c = 1 - q_cap/q_uncap and the scaled lambda_min(M1^cap), then fit c(k) and extrapolate to k = 46. Repeat with c_r scaled so that the capped FRACTION of the support is held fixed in k, which is the scale-invariant form of the schedule the route's third revisit condition asks for. Answer in order: (i) does any affordable cell have positive-definite M1^cap at all; (ii) if so, how does c(k) behave. - Continue if: At least one affordable cell has positive-definite M1^cap and the measured c(k) decreases toward <= 0.766735% at k = 46, which makes the cap schedule the sole remaining target and reopens a bounded pursuit aimed at the cap term rather than at d. - Stop this attempt if: Either no affordable cell has positive-definite M1^cap (the capped pencil is not a well-defined optimization problem anywhere it can be checked, which is a defect of the cap instrument rather than of the route's mathematics, and must be repaired before any k = 46 d = 19 pass is worth its ~24 CPU-hours), or c(k) is flat or increasing across at least three definite cells (the deficit is structural at this cap schedule and the cap branch closes for k = 46 at A = 2583/10000). ## Required evidence - [Return #1641](/projects/twin-primes/return/1641): recorded, recorded - [Return #1869](/projects/twin-primes/return/1869): accepted, measured - [Return #1942](/projects/twin-primes/return/1942): accepted, measured - [Return #1972](/projects/twin-primes/return/1972): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1991](/projects/twin-primes/return/1991): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1991](/projects/twin-primes/return/1991): progress. Route 172 (rev 2) is paused on a step that cannot decide anything, and its stated certification target is twice too generous. Both are settled exactly, on published numbers, with no new computation. (1) The recorded next experiment (uncapped M_{46,25/861}(d) for d = 17, 19, 21) is decided a priori. V_d = span{(1+eps-P1)^a P_alpha : a+|alpha| <= d} is nested, so the uncapped optimum M(d) is non-decreasing in d; #1869 publishes M(17) >= 3.9013805275618854, which is above tau = 3.8714672861014323 AND above #1942's 3.812 ceiling. Hence the stop branch ('saturates at or below 3.812') is unreachable at every d >= 17 and the continue branch fires identically for every input: the step cannot confirm or refute #1972's decomposition. The same conflation is in the route's 'Reconsider when' ('an uncapped sequence for d = 19/21 that rises steeply toward tau') - a non-decreasing sequence already 0.772659% above tau at d = 17 cannot rise toward it. #1972's own argument is unaffected: it concerns the CAPPED d = 17 pencil. (2) Certification needs q_cap >= tau with q_cap = q_uncap(1-c), so the cap cost must satisfy c <= 1 - tau/q_uncap = 386329513461749941/50386329513461749941 = 0.766735%, not the recorded 'below the 1.547% deficit'. At c = 1.547112% the quotient is 3.841 < tau, so a schedule meeting the recorded condition still fails. The cap cost must fall from 2.301984% by a factor 3.00233; the recorded target is 2.01784x too generous. (3) Consequence for the obstruction: the d-enrichment branch is closed for the reasons #1972 gave AND because the only step left on the record is undecidable; the live branch is the route's own third revisit condition (change the cap schedule or the support). Evidence that branch is not dead: the caps on the record are the k-independent absolute constants c_0 = 0, c_1 = c_2 = (3/20)/A, c_r = (4/25)/A (r >= 3) with rough threshold (3/250)/A, while each rough coordinate exceeds the threshold - so at most floor(c_r/d) = 12 coordinates can be rough regardless of k, i.e. the cap removes a different fraction of the problem at every k. #1972's nine indefinite cells sit at k <= 8 <= 12, where that rough-count cap cannot bind at all, so their indefiniteness is a mass-cap/d effect, not the k = 46 mechanism. The cap's relative severity is therefore a property of the schedule, which is variable, not of the route's mathematics. Exact check: 18/18, verify-route172-obstruction.py, exit 0 (route172-obstruction.json). Nothing here bounds G2, beta_2 or twin-prime infinitude. - [Return #1972](/projects/twin-primes/return/1972): inconclusive. The route's single proposed experiment (enrich the even-signature basis from d = 17 to d = 19 at k = 46, eps = 25/861, rebuild the exact capped Gram pair, re-solve) is not justified, and the reason is already in the published record. Combining #1942's capped re-optimised quotient q_cap(17) = 3.8115713687001214 with #1869's uncapped anchor at the same (k, d = 17), q_uncap(17) >= J_0/I_0 = 3.9013805275618854, and #1641's threshold tau = 1/A = 10000/2583 = 3.8714672861014323 gives, exactly: (i) the uncapped optimum at d = 17 already CLEARS tau by +0.772659%, so the entire deficit is cap-attributable; (ii) the cap costs 1 - q_cap/q_uncap = 2.301984% of the quotient, so 66.44% of that cap cost (2.301984 - 0.772659 over 2.301984) is exactly the 1.547112% deficit; (iii) #1942's own finite-spectrum diagnostic on the same d = 17 pencil bounds its top finite generalised eigenvalue by lambda_max <= 3.812, leaving total headroom 3.812 - 3.8115713687 = 4.286e-04 above the re-optimised witness. That headroom is 0.716% of the lift required at the cap and 1.3% of the cap penalty itself -- about 130x the entire diagnostic band, which spans only 0.341% of the quotient, would still be needed. Enriching d = 17 -> 19 adds two degrees to an n = 374 basis that the diagnostic says is already within ~1e-4 of everything it can do. A second, independent reason: region_planes and the caps constraint r*d <= caps[r] both depend on d, so d = 19 invalidates every recorded shard, not just two new r layers; #1942's d = 17 pass was ~24 CPU-hours, and this assignment allows 8. Supporting computation recorded here: the natural cheap surrogate does not exist. Building the same exact capped pair at small k from the frozen slot functionals gives an INDEFINITE M1^cap for all nine cells (k, d) in {4, 6, 8} x {1, 2, 3} (scaled lambda_min from -9.4e-02 down to -7.0e+00), so the capped quotient is undefined there -- while the same script reproduces the uncapped optima (k = 6: 2.118198930803198, 2.173562716270346, 2.194800864728747) to all digits. The cap's sign is not stable in the parameters, so any future capped certificate should ship a definiteness check. This is a scoped obstruction, not a refutation: nothing here says M^cap > 1/A is false, and the conditional (published-diagnostic) part of the argument is stated as such. - [Return #1942](/projects/twin-primes/return/1942): proposed. * `tests/test_capped_gram.py`: bit-exact vs `correction` and `violation_F2` at k = 4,5,6 (recording, second witness, Gram, pair checks). * `tests/test_gram_shard_k46.py`: bit-exact vs direct contraction on a k=46 pair witness. * `out/gram_check_k46.json`: c0^T E c0 and c0^T Delta c0 equal the stored #1869 E_total and #1641 Delta as exact rationals. * `lean/ReoptimisedCappedCertificate.lean`: exact rational Q and its Lean-checked sign.