The generalized two-dimensional fractional sine transform (2D-FrST) is a powerful analytic instrument for the spectral analysis of bivariate signals that arise in multi-scale supply-chain dynamics, hybrid-vehicle systems and resilient logistics networks. In this paper we state and rigorously prove Modulation Theorem-2, which expresses the 2D-FrST of a function modulated by a product of cosines (or by mixed sine–cosine factors) as a linear combination of four fractional sine or cosine transforms evaluated at frequency points shifted by the modulation frequencies. We further establish the Shifting Property, showing that a spatial translation of the original function maps, under the 2D-FrST, into a linear combination of fractional sine and cosine transforms of a phase-modulated version of the function. Both results are derived by means of elementary product-to-sum and angle-addition identities together with a careful accounting of the quadratic-phase factors that appear in the fractional kernel. The theoretical developments are illustrated by concrete applications to artificial-intelligence pattern mining and blockchain-based immutable logging, thereby enhancing transparency, traceability and resilience of digital supply-chain ecosystems. All results are placed in the broader context of the FKF-transform framework and related spectral methods recently introduced for multi-echelon lead-time analysis and secure logistics networks. Keywords— generalized two-dimensional fractional sine transform; modulation theorem; shifting property; fractional cosine transform; AI–blockchain integration; supply-chain resilience; FKF transform; spectral analysis; transparency; traceability.
Let $\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s),\qquad$ $F(x)=\frac{\xi'}{\xi}\!\left(\frac1{1-x}\right)=\sum_{m\ge0}f_mx^m,$ and define the symmetric Toeplitz--Hankel coefficients $g_{ij}=f_{|i-j|}-f_{i+j+1}+\delta_{ij}f_0.$ The adjacent matrices $M_n=\begin{pmatrix}g_{nn}&g_{n,n+1}\\g_{n,n+1}&g_{n+1,n+1}\end{pmatrix}$ form the local family in a previously established criterion equivalent to the Riemann hypothesis. We prove an unconditional finite-range positivity theorem for this family. The zero-pair moment representation of $g_{ij}$ expresses $M_n$ as a sum of rank-one polynomial atoms. A finite verification of the Riemann hypothesis up to height $H$ then splits this sum into a positive-semidefinite verified part and an unrestricted high-zero tail. Two rigorously isolated low zeros provide a positive core; every zero above $H$ is controlled in operator norm by an explicit zero-counting estimate. For the polynomial recurrence $B_0(t)=1,\quad B_1(t)=3-t,\quad B_{m+1}(t)=(2-t)B_m(t)-B_{m-1}(t),$ we derive the exact oscillatory form $B_m(4\sin^2\alpha)=\frac{\sin((2m+1)\alpha)}{\sin\alpha}$ and a closed formula for its two-point Christoffel--Darboux kernel. Using the Platt--Trudgian verification height $H=3{,}000{,}175{,}332{,}800$, the first two LMFDB/Platt zero intervals, and directed-rounding MPFR arithmetic, we certify the kernel away from zero at every integer level $0\le n\le99{,}999$. The resulting positive-core lower bound exceeds the adversarial high-zero tail bound by more than ten orders of magnitude. Consequently $M_n\succ0\qquad(0\le n\le99{,}999),$ or equivalently the first $100{,}000$ adjacent Toeplitz--Hankel determinants are strictly positive. No assumption is made about zeros above the verified height. To the best of our knowledge, this is the first finite-height transfer theorem for this adjacent Toeplitz--Hankel family and the first rigorous certification of its initial $100{,}000$ strict inequalities. This is not a proof of the Riemann hypothesis.
Let $\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s),\qquad$ $\frac{\xi'}{\xi}\!\left(\frac1{1-x}\right)=\sum_{m\ge0}f_mx^m,$ and define $g_{ij}=f_{|i-j|}-f_{i+j+1}+\delta_{ij}f_0,\qquad$ $M_n=\begin{pmatrix}g_{nn}&g_{n,n+1}\\g_{n,n+1}&g_{n+1,n+1}\end{pmatrix}.$ The condition $M_n\succeq0$ for every $n\ge0$ is a previously established criterion equivalent to the Riemann hypothesis. We prove an unconditional finite-range extension for this family without scanning individual zero ordinates or individual Christoffel--Darboux values. Writing $q=n+1$, a critical-line zero $\frac12+i\gamma$ contributes a rank-one atom generated by a two-dimensional polynomial vector. In diagonal and anti-diagonal coordinates its exact phase is controlled by $x=2q\arctan\frac1{2\gamma}.$ We use two level-dependent ordinate windows $(q,5q/4],\qquad (2q/5,q/2],$ whose phase slopes have opposite signs. Every cross-window pair has wedge at least $c_*/q^3$, while an explicit zero-counting estimate supplies at least $q\log q/100$ zeros in each window. The resulting moving Gram core satisfies $\lambda_{\min}(A_{q-1}^{\rm mov}) \ge \frac{c_*^2}{10400}\frac{\log q}{q^3}.$ Combining this with the Platt--Trudgian verification height $H=3{,}000{,}175{,}332{,}800$ and an unrestricted high-zero tail estimate gives $M_n\succ0\qquad(0\le n\le2{,}030{,}956).$ Thus the first $2{,}030{,}957$ local inequalities are proved unconditionally. We also show that every fixed finite zero core has smallest eigenvalue with liminf zero, explaining why level adaptation is structurally necessary for this method. To the best of our knowledge, the moving-window frame transfer and this finite-range theorem are new. The result is not a proof of the Riemann hypothesis.
This paper establishes, inside the Lean 4 proof assistant, a three-level formal identification. The levels are: (i) Belnap multilattice axioms for Weyl–Heisenberg covariant SIC-POVMs at $d=2^n$; (ii) the Zauner conjecture; and (iii) the mixed-signature Stark conjecture for the ray class field $K_d=\mathbb{Q}(\sqrt{(d-3)(d+1)})$, a real-quadratic case of Hilbert's Twelfth Problem. Fiducials are unit-normalized and satisfy $(d+1)|\langle\psi,D_{a,b}\psi\rangle|^2=1$. The equivalence hilbert_embedding_equiv_zauner is proved by rfl: the Belnap embedding into $\mathbb{C}^{2^n}$ and the Zauner conjecture at $d=2^n$ are definitionally the same proposition. The Belnap skeleton (orbit size $4^n$, Frobenius closure $\mu\circ\delta=\mathrm{id}$, join-equiangularity, Born rule) contains zero sorries. Open arithmetic content is marked by named gap axioms for Stark units on WH frames; a proof of Stark would close all three levels at once. For dimension $d=12$ we prove SICPOVM_Exists 12 outright. We construct an exact fiducial in a finitely presented $\mathbb{Q}$-algebra, verify 143 overlap identities with native_decide, and transfer everything to $\mathbb{C}^{12}$ along a ring homomorphism. The theorem crystal_forces_d12_sic depends on no axiom beyond Lean 4's standard foundations and compiler trust. This is, to our knowledge, the first machine-checked SIC-POVM existence in any dimension. For the frontier dimension $d=2048=2^{11}$ the transport apparatus is formalized and sorry-free. It includes a forward map $\varphi\colon B^{\oplus 11}\to\mathbb{C}^{2048}$, a reduction $\psi$ with $\psi\circ\varphi=\mathrm{id}$, a conditional reduction to Stark, and a non-real character obstruction that blocks the false branch. Unconditional existence remains open; the machinery that surrounds it is closed.
1. A. Weil [3] constructed a universal distribution t on the Weil group.The values of I at various test functions give the contributions from the zeros of L-functions which appear in the .explicitformulas.In this note, we shall construct a universal distribution zi on GL(n) and prove the explicit formula for automorphic L-functions using z/ when n-2.For n 2, to derive such a result, we must assume certain property of characters of infinite dimensional representations of GL(n) over a local field.This property, formulated as Conjecture, seems to lie slightly beyond our present knowledge of harmonic analysis.The distributions A have striking formal resemblance to Weil's one.Furthermore they are related to each other so that zl is the "direct image" of z/ for m n.This is a pleasant fact since we think that a discovery of new functorial properties related to zeros of zeta functions would be crucial for the proof of the Riemann hypothesis.