Rigorous 512-Bit Interval Certification of Semilocal Weil Positivity up to L = 3.00 and the Von Koch Spectral Equivalence

We establish an unconditional, computer-assisted certification of the semilocal Weil quadratic form associated with the Riemann zeta function on bounded spatial domains[−L,L]. Using 512-bit and 256-bit rigorous ball arithmetic via the Arb library, we resolve thelongstanding numerical and structural bottlenecks associated with finite-dimensional Galerkindiscretizations. Specifically, we prove: (1) Kato-Lax-Milgram-Nelson lower semiboundednessof the semilocal Weil operator HW ≥ −8.51I on the Sobolev form domain Hlog /20 ([−1, 1]); (2)the Exact Orbit-Block Spectral Theorem, proving that the compressed prime shift operatorsatisfies ∥P[−1,1]V ′LP[−1,1]∥L2 ≤ 3.129253 at L = 1, accompanied by a dynamic frequencycutoff escalation schedule T(L) ∈ {400, 500, 1000, 2000, 3000} guaranteeing exterior complementcoercivity across [L7,L17] and live 256-bit multi-anchor finite Galerkin certificationH8(L) ≻ 0 across all discrete prime-power thresholds L ∈ {L7, 1,L8,L9,L11,L13,L16,L17}with macroscopic Markov continuum protection radii ρ8(L) ∈ [1.20 × 10−8, 6.81 × 10−7]; (3)an unconditional positive definite certificate for the full 650-dimensional Galerkin blockP650 at L = 1, establishing via a four-factor metric pullback that the physical ambientfloor satisfies aphys650 ≥ 2.6674020235268543 × 10−31 > 0; (4) an explicit proof thatspatial-spectral commutator mismatch ∥[P[−1,1], χ(D)]∥ ≥ 0.25 defeats standard two-blockscalar Schur complements by > 1016×, and the rigorous construction of a three-block(12 + 638 + Q650) directional Schur complement proving operator positivity of the lowerhybridoperator Hlower ≻ 0 on L2([−1, 1]); (5) a higher-order Sobolev integration-by-partshierarchy overcoming the critical s = 1/2 divergence and m = 1 Poincaré barrier, provingthat for m = 3 the Stieltjes tail enclosure at T0 = 3 × 1012 satisfies ϵ3(T0) ≤ 1.66822 × 10−62,which guarantees that the truncated Platt-Trudgian zero-Gram quadratic form is strictlypositive definite, Q≤T0 [f] ≥ 8.0976 × 10−32∥f∥2 > 0, across all degrees 6 ≤ N ≤ 649 onthe boundary-vanishing polynomial subspace VN,3 ⊂ P650; (6) the first certified zero-gapcontinuous continuum covering of the entire macroscopic prime-power frontier [L7,L41] =[12log 7, 12log 41] ≈ [0.972955, 1.856786] using Neumaier’s Centered Mean-Value Form, uncutPlancherel-monotone dimensionless frequency rescaling, and exact rational eigenbasis congruencepreconditioning, establishing strict positive definiteness across 6468 overlapping intervalboxes with uniform physical floor λmin(H8(L)) ≥ 7.138792 × 10−10 > 0 and overcoming the2 × 107× monomial interval dependency barrier; and (7) the infinite-dimensional KLMNform closure and multi-degree Galerkin hierarchy across N ∈ {8, 12, 16, 24} and representativeanchors {L7,L1,L9,L13,L17,L25,L31,L41}: using 220-bit bracket Gauss-Legendre quadrature (reducing matrix entry radii to < 10−55) and 64-bit dyadic rational eigenbasiscongruence preconditioning RN ∈ QN×N (σmin(RN) ≥ 0.9999999999999999), we certifystrict positive definiteness across all 32 out of 32 anchor-degree pairs via Ostrowski-Sylvester pullback (λmin(HN(Lk)) ≥ σmin(RN)2dmin(eHN(Lk)) > 0). Rayleigh-Ritz variationalanalysis demonstrates rapid ground-state concentration: > 98.36% in shell [0..7],> 99.979% in k ≤ 11, > 99.9999% in k ≤ 15, with spectator modes k ∈ [16..23] carryingonly 3.604348 × 10−7 of total eigenvector energy. Furthermore, combining Watson’s superexponentialspherical Bessel bound η650(400) ≤ 1.675199 × 10−109 with the shell couplingbound ∥HQN,PN ∥ ≤ 1.481088 yields the explicit Block-Schur complement remainder bound∥HQN,PNH−1PNHPN,QN ∥ ≤ 2.825896 × 10−87 ≪ βref = 4.30, establishing infinite-dimensionalform closure with certified exterior complement floor SN ≥ 4.2999999999I > 0, while restrictionto the conforming Dirichlet subspace ϕk = Pk − Pk+2 ∈ Hlog /20 ([−L,L]) ensuresthat boundary energy vanishes and all eigenvalues remain strictly positive up to N = 32;and (8) Method C Pole-Prime Stieltjes Principal-Value Renormalization and Method BMontgomery-Vaughan mean-value form bounds: canceling the 2eL leading-order divergencebetween the discrete prime distribution and the smooth pole kernel collapses the effectivebandlimit threshold from Tunrenormmin (2.50) ≥ 42, 650 down to Trenormmin (2.50) ≤ 15.334 (a2, 782× reduction factor), enabling live 256-bit Arb certification of positive definiteness acrosseight new escalation anchors up to L = 2.50 (n ≤ 139 prime powers) with minimum pivot≥ 4.2245×10−10 > 0 and unconditional Theorem 5A zero-form positivity; and (9) Method ASlepian-Pollak-Landau Prolate Spheroidal Wave Function (PSWF) concentration and live256-bit Arb certified frontier extension to L = 3.00 (X = e6 ≈ 403.43, encompassing 87 activeprime powers up to p ≤ 397): constructing the exact rational symmetric tridiagonal matrixrepresentation of the Slepian Sturm-Liouville differential operator Lc := − ddξ ((1−ξ2) ddξ )+c2ξ2(c = LT) in the normalized Legendre basis, we prove that low-lying PSWF modes achieve> 1, 919× boundary trace suppression for Mode 0 (|Φ0,15(1)| = 3.6833 × 10−4 vs. 0.7071 forψ0(1)), 255.7× for Mode 1, and 108.4× for Mode 2, eliminating boundary Gibbs leakagewithout artificially restricting the form domain. At L = 3.00, discrete prime shifts andpole singularities cancel under Method C Stieltjes pairing from ∥Mprime∥2 + ∥Ppole∥2 =48.526 down to ∥Vrenorm(3.00)∥2 = 4.114 ≤ 4.30 (11.80× cancellation gain), collapsingthe Montgomery-Vaughan bandlimit cutoff from Tunrenormmin (3.00) = 1.416 × 107 down toTrenormmin (3.00) = 16.176 ≤ 16.691 (8.752 × 105× complexity reduction factor). Live 256-bitArb certification rigorously proves positive definiteness of both H8(L) and HPSWF8 (L) acrossall six escalation anchors {L169,L199,L243,L289,L343,L403 = 3.00} (with minimum pivots≥ 5.4502 × 10−11 > 0 and PSWF pivots ≥ 1.6029 × 10−8 > 0), the conforming Dirichlethierarchy is certified for N ∈ {8, 12, 16, 24}, and the rescaled Platt-Trudgian Stieltjes tailbound ϵ3(3 × 1012; 3.00) ≤ 1.4221 × 10−61 yields an unconditional Theorem 5A zero-formnet floor Q≤T0,3.00[f] ≥ 5.7906 × 10−11 > 0; (10) the unbroken, zero-gap continuous continuumcovering of the entire macroscopic parameter interval [L7, 3.00] = [12log 7, 3.000000] ≈[0.972955, 3.000000]: proving the C2-Smooth Threshold Activation Theorem on H10 ([−L,L]),where homogeneous Dirichlet boundary vanishing ϕk(±L) = 0 forces the convolution overlapintegral Ijk(L; s) = O((2L − s)3) with vanishing first and second derivatives at every primethreshold, eliminating boundary jump impulses and allowing centered mean-value intervalboxes to straddle all 73 prime powers between L41 and L = 3.00; paired with Method C precongruencedirectional derivative cancellation (|(R⊤H′R)00| ≤ 2.1802 × 10−7 ≤ 5.0 × 10−6),Segment A ([L41, 2.50], 644 boxes) and Segment B ([2.50, 3.00], 500 boxes) establish positivedefiniteness across 1,144 boxes (min pivot ≥ 7.5502 × 10−11 > 0), unifying withprior stages into an unbroken chain of 7,612 overlapping rational interval boxes with consecutiveoverlap ≥ 10−5 everywhere and unconditional Theorem 5A net positivity floorQ≤T0,L[f] ≥ 7.927 × 10−11∥f∥2 > 0 across all 87 active prime powers up to n = 397; and(11) the Renormalized Chebyshev-Stieltjes Remainder and Von Koch Equivalence Theorem:on any conforming space H10 ([−L,L]), integration by parts yields the exact quadratic form identity ⟨f, Vrenorm(L)f⟩ = −2Cf (0) + 2R 2L0 KV K(s)(C′f (s) − 12Cf (s))ds + Rlow(f) whereKV K(s) = (ψ(es) − es)/es/2, matching direct operator assembly in 256-bit Arb across{L7,L41, 2.50, 3.00} to < 1.30 × 10−45; empirical evaluation across all 198 jump points upto X = 403.43 verifies |KV K(s)| ≤ 1.0982 ≤ 1.25 on [2L7, 6.0] and ≤ 1.4142 ≤ 1.42 on[0, 6.0] (explaining the 11.8× operator norm cancellation over unrenormalized sum 48.526);and proving bijectively that while local positivity at L ≤ 3.00 is unconditional and safebecause 3.00 < Lbreak(δ, γ0) = 12δ log( |ρ0|λarch2 ) for all δ ≤ 0.25, uniform form-boundedness∥V Dirrenorm(L)∥2 = O(L2) as L → ∞ is mathematically equivalent to Von Koch’s (1901) theoremψ(x) − x = O(x1/2 log2 x), establishing that extrapolating empirical Stieltjes remainderboundedness to ∞ without proving RH is mathematically circular. Finally, we providea forensic audit of subtle failure modes in automated mathematical reasoning pipelines,detailing the design of a deterministic, zero-fixture referee gate. We emphasize that theseresults establish local continuum stability and finite-block positivity; they do not prove theRiemann Hypothesis, as global infinite-dimensional coercivity across all L ≥ 1 remains anopen obligation strictly equivalent to the hypothesis itself.