CRISP: grade-separated chaos modeling for random RIS architectures via Clifford Wiener-Itô expansion

Abstract This paper presents CRISP, a grade-separated chaos modeling framework for random reconfigurable intelligent surface (RIS) architectures. Conventional RIS models typically represent the surface response by scalar phase shifts and treat the resulting channel as deterministic, Gaussian, or scalar stochastic. Such abstractions do not capture two structural features of realistic programmable environments: random spatial deployment and multigrade electromagnetic field structure. CRISP models the RIS-induced received field as a Clifford-valued functional of a Poisson point process (PPP) and expands it through a Clifford Wiener–Itô chaos representation. The resulting decomposition separates stochastic interaction order, indexed by Poisson chaos order, from electromagnetic structure, indexed by Clifford grade. This enables grade-wise projection beamforming, residual decomposition, finite-order approximation, and semantic-grade performance metrics. We establish the Clifford–Poisson chaos decomposition, Parseval identity, grade-wise projection optimality, PPP-density residual decay, chaos-truncation bounds, and semantic distortion control. Reproducible experiments show that scalar RIS suppresses mainly the grade- $$\\varvec{0}$$ 0 component, while CRISP achieves more balanced multigrade residual suppression, higher-grade fidelity, and lower semantic distortion across random deployments.