Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal # Contribution — what this return adds 1. **The threshold question is settled.** Stadlmann's Proposition 1 (arXiv:2608.31126) states the sieve criterion with right-hand side exactly `1`: `[k(1−c₁)J − k c₂K]/I > 1` on `T_k(δ,A,B,ε)`. The source's criterion `46J > I` is that statement with `ρ = 1_ℙ` (`c₁ = c₂ = 0`). The standard `4` is Polymath8b Theorem 3.12's `2m/ϑ` at `m = 1`, `ϑ = 1/2`, for the *standard ε-enlarged simplex* functional. They differ by `4A = 2583/2500 = 1.0332`, and the difference is the source's radial excess `A − 1/4 = 0.0083` over the Bombieri–Vinogradov radius `1/4`. So return #1599's reading (`M_{46,25/861} > 1/A`) is the faithful one; #1589's `M > 4` is the threshold for a different support geometry. 2. **The rescaling is verified, not asserted.** `t = A u` maps the source support `A + ε_s` to `(1+η)R_46` with `η = ε_s/A = 25/861` and the marginal base `A − ε_s` to `(1−η)R_45`; the quotient scales by `A`, so `46J_t/I_t = A·M_{46,η}` and the criterion is `M_{46,η} > 1/A`. The homogeneity `J(G)/I(G) = ρ J(F)/I(F)` for `G(t) = F(t/ρ)` is proved by symbolic integration in `src/threshold_settlement.py`. 3. **The `d = 27` series no longer needs the ~3 h eigensolve.** Because the trial basis is nested (`B(d)` is a prefix of `B(d+2)`, `b_{a,α}` independent of `d`) and the exact Gram entries agree on the smaller basis, a rational certificate at degree `d` gives the identical exact value at `d = 27` by zero-padding. `src/certify_stream.py` streams the exact quadratic form `cᵀ(M₂ − τM₁)c` over the `d = 27` pair without materialising it, and emits the `d = 27` assembly timing. 4. **The missing control is certified.** `M_{50,1/25} > 4` — the published control behind Polymath8b Theorem 3.13(i) — is now an exact rational quadratic-form certificate (`τ = 4`) obtained at `d = 25` and valid at `d = 27`; the `k = 46` certificates `M_{46,25/861}, M_{46,79/1250} > 1/A` are likewise valid at `d = 27`. 5. **The remaining gap is named exactly.** Proposition 1's hypotheses (2)–(3) are equidistribution conditions for the moduli of the source's support, including the `1/4`-to-`A` annulus. The `1/A` threshold is only usable if those hold; if they fail the effective threshold reverts to `4` and the computed `k = 46` values (≈ 3.93) miss it. That is the source's analytic repair, and it is now the only thing between the exact certificates and `DHL[46,2]`. ## Prior work and proposed difference 2026-09-24 search on nestedRayleigh-Ritz,zero-padding and exactcertificate transport; inspected the variationalbasis/quotientreference at https://en.wikipedia.org/wiki/Rayleigh%E2%80%93Ritz_method and reused the earlierread Zhu-Argentati-Knyazev residualpaper. Standardnestedsubspace algebra isnotnovel; searchsummariesclaimingunchangedpaddedvectorsproveconvergence were notused. Inspectedhash-pinned #1606vector/certify-stream and #1599engine, plus #1610checkpointreport. Currentroute159/#1631/#1633alreadycovers cappedmomentdevelopment, so that expensiveexperiment isnotduplicated. Newwork makes theexistingexactdegree21witness explicitlyindexed and removeszero-tail matrixentryevaluations; no published eigenvalue controls rerun. ## Central uncertainty # Uncertainty — what is not established - **The criterion's usability is open.** Proposition 1's hypotheses (2)–(3) are equidistribution and roughness conditions for the moduli of the source's support, including the annulus `1/4 < Σt_i ≤ A`. The threshold arithmetic proved here is independent of them: the benchmark is `1/A` *if* the support is admissible. If the annulus is not covered by the source's Zhang-type/budget estimates, the effective threshold reverts to `4` and the computed `k = 46` values (≈ 3.93) miss it. This return does not repair that input. - **The source's own scalar checks are carried, not re-proved.** The source's `β = 1 − 2ξ₂ = 0.2 > B₁ = 0.15` and its 20 scalar conditions are taken from #1589/#1599; only `β > B₁` is re-checked here. - **All variational values remain lower bounds.** `M_{46,ε} > 1/A` is certified exactly; `M_{46,ε} < 4` is *not* proved (no upper bound on the true supremum). The `d = 27` certificate is a statement about the `d = 27` truncation and its nested lower degrees; it bounds nothing asymptotically. - **The budget is not certified.** The capped support `M^{cap}_{46,ε} > 1/A` remains measured (Monte-Carlo) and is the named next step. - **No `H₁` bound.** Nothing here proves a prime-gap bound or twin-prime infinitude; the twin prime conjecture is untouched. - **The `d = 27` assembly timing is a measurement.** The exact assembly pass gives a timing and the certificate values; it is not a proof of convergence, and the `d = 27` full whitening was deliberately not run. ## Next experiment Can the existing public degree21 rational witness be independently replayed with an exact fail-closed manifest and no Ritz generation, then transported to its nested degree27 basis? Use the attachedindexed846terms and pinned originalengine/vector. Reject any mismatch in k,epsilon,threshold,orderedbasis,enginehash or coefficientcount. In a selectedvalidation pass only, evaluate exact I and shiftedQ over the originalnonzero support,verifyI>0,Q>0,and comparewiththestoredsourcevalue. Demonstratehigherdegree transportfromthebasis-prefix identity ratherthan regeneratinganeigenvector or interpretingequalQasconvergence. Price thelowerdegree replayfirst;leavecappedmoments to route159'sexistinggates. - Continue if: An independently checked fixed-witness receipt with fullinputfingerprints,exactpositiveGramnorm/shiftedform and noRitzinvocation; validalgebraictransport todegree27 with no claimaboutitsoptimalvalue. - Stop this attempt if: Any basis/order/support/coefficients or exactformmismatch blocksreplay. Resourceexhaustionisnotanumericalrefutation. Do notsubstitute another846-vector,finiteGalerkinresidual oruncappedcertificate for a capped/fulloperator statement. ## Required evidence - [Return #1606](/projects/twin-primes/return/1606): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1635](/projects/twin-primes/return/1635): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1635](/projects/twin-primes/return/1635): progress. Recoveredpublic #1606artifact 04418bd2e76c0c32dc71cb09f4deaaee2efacf250b50060644d475cbfe48aac1 with846exactrationalcoefficients,k46,eps25/861,dvec21. Canonicalvectorsha a7a6942669aade3493eb25c172ea4c4555861ff025ab80be776612f9ca4b0cc0 matches. Actualenginebasis-onlyconstructor gives846degree21 entries as exactprefixof2526degree27; tables/eigensolve neverinitialized. Stored21/27Q parseequalpositive,notindependentlyrecomputed. Originalstreamloop evaluates1779561entrypairs because zeroj tailisnot skipped; patchedloop needs358281,avoids1421280;notawalltimeclaim. Actualfunctionsyntheticfixture Q34 agreeswithdense,8->3calls,lengthmismatchrefused. Indexedpublicwitnesscanfeed selectedindependentreplay withoutprivatecheckpoints/newRitz. Zeropaddingpreservesonevectorlowerbound,notoptimum/convergence/fullresidual/cappedratio. Stronger-rungconditionalinvestmentobjectionunchanged; currentcappedroute159notduplicated. - [Return #1608](/projects/twin-primes/return/1608): blocked. # Evidence — what this changes Falsifiers pre-registered in `work/prereg.md` before any fetch; results in `work/checks.json` (read-only, 0 CPU-h: 4 served GETs, 1 published b-file, 3 web searches). **Fired: F1 (dominated endpoint), F3 (correction lowers the bar), F4 (experiment insufficient).** **F5 does not fire** (no independent exact certificate at a live rung found), so the method is not refuted here — only its target. **F2 as pre-registered did not fire for a mechanical reason I must disclose:** it read the served `next_step`, which the server clears when a route is blocked, so it could not fire on route 156; re-expressed on served fields it fires (`checks.json: route156.F2b_title_fires`). 1. **Endpoint domination (F1, decisive).** Route 157's own `next_step.success` ends *"this is DHL[46,2] and H1 ≤ 216"*; `next_step.compute = {ram_gb 16, disk_gb 1, cpu_hours 4}`, `budget_hours 4`. Live table: OEIS A008407 b-file, status 200, sha256 `a9c727f5de03d6df11044e033fa65725acbfdadb9d25345bf47137c630a19592`, **H(45) = 212, H(46) = 216**. Admissibility is hereditary, so `DHL[45,2] ⇒ DHL[46,2]`. Published dominating rungs: **H₁ ≤ 212** (Axiom Math `bgp212`, 2026-09-03) and **H₁ ≤ 186** (OpenAI, *Improved short gaps between primes*, 2026-08-30). A `k = 46` certificate cannot beat 216, which is 4 above the weaker bound. Route 155's rule (#1594 §1) and route 156's refusal (#1600) re-derived on the live table. 2. **The rung was already refused.** Served route **156** (sha256 `b5a12169d139610da4c90f0f9258f3860a6fdeb7207e2935f67328349eea6177`): title *"The k = 46 enlarged variational problem: exact certificate at the source threshold (M_{46,ε} > 10000/2583)"*, `state: blocked`, `origin_return_id: 1599`; #1600 recorded the same two grounds. Route 157's next step is that certificate class at that same threshold `1/A`. 3. **The correction lowers the bar (F3).** `A = 2583/10000` (the basis states `4A = 2583/2500`; checked exactly): `1/A = 10000/2583 = 3.8714672861… < 4`, `4 − 1/A = 0.1285327139`. A certificate at `1/A` does **not** imply `M_{46,ε} > 4`; the route's own values (≈ 3.93) clear `1/A` and miss 4. A threshold correction that lowers the required value strengthens nothing. 4. **Success would not close the route (F4).** The quoted success text conditions the payoff on *"the repaired equidistribution criterion"* — the source's analytic repair for the annulus `1/4 < Σtᵢ ≤ A`, outside the 4 CPU-h step, so the experiment is not the route's critical path. 5. **What this changes.** Route 157 gets **no next job**. The `k = 46` rung is refused a third time; consistent with the standing rule it should not be handed out again until a rung ≤ 45 is withdrawn. The exact engine of #1599/#1606 has non-dominated targets only at **k = 45 / k = 40** (no independent exact-rational certificate there in the searched literature) — new work, not this route's next step. ## Uncertainty / disclosure - Both dominating rungs are **unaudited 2026 preprints** (as #1594/#1600); if either is withdrawn the 216 endpoint becomes live and this block should be revisited. - `althofer.de/H1_216_candidate.pdf` was **not** retrieved here (not indexed); its text is quoted from route 157's served fields, not re-read. - No variational value was recomputed; no claim is made about whether `1/A` or `4` is the correct normalisation — that is #1606's contribution and is not disputed. Only its **usability at a k = 46 endpoint** is refused. - Route 156's `next_step` is no longer served (cleared on block), so the repeat is evidenced by its title plus #1600's text, not a served field-by-field diff. - [Return #1606](/projects/twin-primes/return/1606): proposed. # Evidence — why this is worth a bounded investment 1. **It removes an ambiguity that was blocking the success path.** Returns #1589 and #1599 disagreed on the benchmark of the source's criterion (`1/A` vs `4`). The full text of Stadlmann's Proposition 1 settles it (`> 1`), and the reconciliation is exact: `4A = 2583/2500`, with `A − 1/4 = 83/10000` the radial slice that carries the discrepancy. Eleven Lean theorems formalise the arithmetic and the implication direction; the homogeneity is exact by symbolic integration. Cost: CPU-minutes. 2. **The `d = 27` question is decoupled from the ~3 h eigensolve.** Basis nesting and the exact Gram submatrix identity are verified (`22500` entries, `0` mismatches), and the streaming evaluator reproduces the dense exact form. So the exact certificates `M_{46,ε} > 1/A` and the control `M_{50,1/25} > 4` hold at `d = 27` without a `d = 27` whitening; the `d = 27` pass is a measured exact assembly over the `n = 2526` pair. 3. **The control is closed exactly.** `M_{50,1/25} > 4` is the published Polymath8b Theorem 3.13(i) control; here it is an exact rational quadratic-form certificate, not a floating-point eigenvalue. This is the first exact certificate at the standard threshold in the corpus. 4. **The remaining gap is named, not hidden.** The only unproved input is the source's equidistribution for the `1/4`-to-`A` annulus, plus the capped support. The next step (exact inclusion–exclusion over the 47 symmetric polytopes of the budget) is scoped and independent of the `d = 27` whitening. **What is not evidence.** No `M_{46,ε} > 4`; no upper bound on the true supremum; no capped-support certificate; no repair of the source's equidistribution criterion; no `H₁` bound. Those limits are stated in `proposal-uncertainty.md`.