Covariance-Dominated Delta Encoding for Coordinate-wise Product QuantizationA Rate–Distortion Theorem for the Mnemosyne Project (Part III)

This paper presents a self-contained rate–distortion theorem for a simple but practically relevant class of product quantizers under a covariance domination condition that arises naturally when delta encoding is applied to temporally correlated embedding sequences. We consider coordinate-wise uniform scalar quantization in an eigen-basis of the input covariance and prove that the mean squared error (MSE) is upper-bounded by a constant times the covariance trace. When the covariance of a delta representation is dominated by that of the original representation in the Loewner order, this immediately implies a worst-case MSE advantage for delta encoding under a fixed quantization configuration. Although the core result is fully abstract, the theorem is motivated by, and designed for, the Mnemosyne Project, an edge-oriented large language model (LLM) infrastructure. Part III of Mnemosyne focuses on compressing activation vectors via Soft-ZCA whitening and product quantization (PQ). In this context, our theorem supplies the mathematical backbone for Chapter 3.2.1 (Theorem 5.1‑R), which claims that inserting a delta-encoding stage before whitening+PQ can strictly improve MSE-type distortion on a large subset of tokens, while respecting updated security requirements that no longer treat delta encoding as a cryptographic entropy source. The present version is purely theoretical: we provide complete proofs of the scalar and vector quantization lemmas, the Loewner-order trace monotonicity, and the coordinate-wise PQ trace bound. We state LLM-specific claims as conditional on an empirically testable covariance domination assumption. We also discuss how recent empirical work on residual-based KV compression, angle-based KV quantization, and semantic/context compression for LLMs makes this assumption highly plausible, suggesting that subsequent experiments on Mnemosyne’s embedding streams are likely to match the theoretical predictions.