Singularity-Scale Modulation (SSM)

This work introduces Singularity-Scale Modulation (SSM), a proposed physical-layer framework that encodes information jointly in two independent domains of a wavefield: the topological geometry of intensity singularities (optical or phononic caustics), and the recursive scaling of phase velocities across separately tunable spatial and temporal axes.SSM is presented as a formal unification of Multi-Scale Spatio-Temporal Fractal Modulation (MS-STFM) and Caustic Data Modulation (CDM), whose mathematical domains are argued to be orthogonal. The manuscript develops a multi-fractional wave equation for the temporal-scale domain via a continuum limit of discrete recursive phase accumulation, with Caputo derivatives and a Fox H-function kernel for the general incommensurate case; an eikonal and catastrophe-theoretic formulation of the spatial-topological domain, classifying symbols by fold, cusp, swallowtail, and umbilic structure; and a coupled propagator in which a fractal-preconditioned drive signal modulates a caustic-generating phase mask.Three claims are advanced: a multiplicative capacity scaling in which each of K = N𝔾 × N𝕊 sub-channels hosts an alphabet of M caustic topologies; a product-rule bit error rate under assumed statistical independence between temporal and spatial noise; and a cryptographic key space formed by the Cartesian product of continuous speed vectors and high-dimensional phase masks. A mode-switch condition based on a unified Fresnel number is proposed.Physical realizability is discussed in terms of programmable digital signal processing, programmable phase modulation, and passive fractal or dispersive media. Extensions proposed for future work include adaptive closed-loop optimization, caustic braiding, non-Euclidean media, super-oscillatory masks, and spacetime-lattice encoding.The status of the results: the catastrophe-theoretic caustic classification and eikonal ray mapping are established mathematics applied here to information encoding. The capacity and error-rate expressions follow from the stated model assumptions but depend on conditions—channel orthogonality, an invertible matched kernel, and cross-domain noise independence—that are asserted rather than demonstrated. The security claim rests on key-space cardinality and lacks a formal adversary-model proof. No experimental validation is reported. The work is offered as a conceptual.