The Perceptual Game-Theoretic Transform (PGTT): Axiomatizing Signal Compression via the Fourier–Shapley Isomorphism

The mathematical representation of discrete signals is classically governed by linear basis transforms, such as the Discrete Fourier Transform (DFT), which treat spectral projection as a rigid, deterministic geometric operation. Under this paradigm, signal compression and thresholding rely on heuristic error metrics like the Minimum Mean Squared Error (MMSE). To mathematically axiomatize these fundamental operations, this paper reformulates computational basis transforms through the lens of Cooperative Game Theory. By defining discrete signal reconstruction as a Grand Coalition of orthogonal frequency players, we establish a strict mathematical isomorphism between functional analysis and cooperative game theory. We prove that the spectral energy assigned to each frequency coefficient is exactly its Shapley Value and that classical MMSE minimization is mathematically equivalent to maximizing the retained Shapley payout. Furthermore, we extend this framework to physical hardware and linear shift-invariant (LSI) systems, modeling 8-bit quantization and the Modulation Transfer Function (MTF) as sub-additive “economic taxes” on the coalition. Finally, by intentionally violating the Shapley Symmetry Axiom to mimic the Contrast Sensitivity Function (CSF) of the human visual system, we propose a fundamentally new mathematical basis: the Perceptual Game-Theoretic Transform (PGTT). Unlike classical methods that rely on post hoc quantization for signal compression, the PGTT acts as an inherently efficient transform that structurally guarantees sub-Nyquist computational complexity and dynamic range reallocation prior to physical hardware saturation.