Pisot-Orbit Deterministic JL Layer

(1) Integration point: insert the fixed projection immediately after a high-dimensional embedding or pooled transformer representation and before cosine retrieval, nearest-centroid classification, or an MoE router. For a 768-dimensional embedding and m=128, use a 128x768 deterministic matrix. During training, freeze this projection initially; optionally add a learned diagonal scale after projection so the representation can compensate for mild anisotropy without learning a full dense matrix. (2) Pseudocode: choose beta, seed u0, offset s, and gap g; iterate u=T_beta(u) and collect at least m+d orbit values; compute mu and sigma on a long burn-in orbit; set P[r,i]=z[s+(r-1)*g+i]/sqrt(m); compute a scalar alpha from calibration data using alpha^2=(sum_a ||c_a||_2^2)/(sum_a ||P*c_a||_2^2); output y=alpha*P*x. Enumerate several hundred seeds and gaps, evaluate maximum relative pairwise centroid distortion, and freeze the best matrix. (3) Compute from the mathematics: the beta orbit, centered coefficients, projection entries, and the proposed geometric dependence on g. Estimate empirically: mu, sigma, alpha, the best seed, pairwise distortion, and V0 from held-out vectors. Measure correlation decay by estimating |corr(z_t,z_{t+k})| and fitting an exponential envelope A*rho^k. For a proof-oriented prototype, implement the orbit with fixed-point or finite-field arithmetic and compare projection hashes across machines. (4) First cheap experiment: use CIFAR-10 class embeddings or a small language-model retrieval benchmark; compare against Gaussian JL, sparse JL, and a learned linear projection at equal m. Measure retrieval recall, centroid accuracy, maximum pairwise distortion, projection-generation time, stored bytes, and downstream training loss. Success means matching Gaussian-JL accuracy within measurement noise while being exactly reproducible, reducing generation or commitment cost, observing approximately geometric bias reduction with larger g, and observing variance proportional to 1/m.