AI–Blockchain-Integrated Generalized Fractional Sine Transform Framework for Supply-Chain Transparency and Resilience

Abstract

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.