WINNEX UB Width: Adaptive Dimensionality Folding for Provable High-Dimensional Retrieval (The Fold Limit Theorem and a Manifold-Aligned Bound Engine)

This document is the pre-patent technical specification for WINNEX UB Width, an adaptive, deterministic, training-free operator that restores the pruning power of a Cauchy-Schwarz bound engine at high embedding dimension (d = 1536). It formalizes the mathematics of dimensionality folding along the corpus' principal directions, proves the Fold Limit theorem (the exact condition under which folding improves or collapses retrieval), specifies a variance-profile router that selects between the folding path and the exact-scan path, and states the patent claims with an honest viability analysis. Contribution: not a new similarity index, but a provable operator layer that composes with an existing bound engine (winnex-madhava) and manifold projector (winnex-xfactor). The projection basis is aligned to the corpus' principal directions, the Cauchy-Schwarz residual shrinks from ~sqrt(1-s/d) to the manifold residual, and the bound remains valid in the original space with 0 bound violations by construction. Measured on real arXiv embeddings (d = 1536, N = 50,000, 200 queries, top-10): random basis gives residual 0.940 and 0% pruning; the manifold-aligned basis gives residual 0.244, 92.7% pruning, recall 1.000, and 0 bound violations. Native C++ latency at N = 100,000: 1.99 ms/query vs 11.08 ms for exact scan (5.6x faster). The paper documents the float32/float64 equivalence, the fragility of a hand-rolled Jacobi eigensolver at d=1536, and the correction of the router from a binary spectral-gap threshold to the variance-profile criterion.