Deterministic Johnson--Lindenstrauss Projections from Pisot $\beta$-Transformations for Zero-Knowledge Private Routing

Abstract

Zero-knowledge (ZK) proofs certify that a message belongs to an allowed semantic class without revealing the message, but the certificate compares a high-dimensional embedding against class centroids, so its cost grows with the embedding dimension $d$. A Johnson--Lindenstrauss (JL) projection lowers $d$ to $m\ll d$ while preserving pairwise distances, yet a random JL matrix must be committed and its sampling proved inside the circuit, which is costly and a leakage risk. We construct a public deterministic projection from the standardized orbit of a Pisot $\beta$-transformation, analyzed through the spectral gap of the $\beta$-map, the geometric decay of its correlations, rather than equidistribution. We prove that the induced squared-norm estimator is unbiased up to a term decaying geometrically with a sampling gap, and that its variance is $V_0/m$ with a constant $V_0$ that is dimension-free in experiment and, under one stated concentration hypothesis, in theory. A single public seed preserving all pairwise centroid distances therefore exists and is found by search. Against six standard projections, including the chaotic-sequence matrix of Yu \emph{et al.}, the construction matches statistical quality to within measurement noise, and it is the only one simultaneously free of in-circuit randomness and exactly reproducible in a fixed finite field at a per-step cost $\log_2\beta$ rather than $2^{k}$.