Formal Definition of Compression Strain

Compression Strain Michele Joseph Abstract Compression strain is a measurable signal of representational conflict in large language models (LLMs) when two semantically equivalent prompts induce divergent internal trajectories or output distributions. Unlike perplexity or KL divergence, which operate at the level of surface likelihoods, compression strain captures internal inconsistency under semantic invariance. This document provides a minimal formal definition of compression strain and its relation to contradiction, stability, and robustness. 1 Introduction Let an LLM be a conditional distribution M(x)=pθ(y|x),(1) where xis an input prompt, yis an output, and θare the model parameters. Let xand x′be semantically equivalent prompts, written x≡x′,(2) meaning they refer to the same underlying semantic content despite different surface forms. Let M(x) and M(x′) denote the model’s corresponding output distributions. 2 Compression Loss We define compression loss as the representational collapse that occurs when a model must compress a high-dimensional semantic manifold into a finite parameterization. Compression loss is not directly observable, but contradictions arise when compression is imperfect. 3 Definition of Compression Strain Definition. The compression strain between two semantically equivalent prompts x≡x′is defined as CS(x, x′) = DM(x), M(x′),(3) 1 where Dis any divergence measure satisfying: (i) D(P, Q) = 0 iff P=Q, (ii) D(P, Q)>0 when P=Q, (iii) symmetry or symmetrization, (iv) stability under small perturbations. In practice, we use the symmetrized KL divergence: CS(x, x′) = 1 2KL(M(x)∥M(x′)) + KL(M(x′)∥M(x)).(4) Compression strain is therefore zero when the model treats xand x′identically, and strictly positive when the model exhibits an internal representational contradiction. 4 Interpretation •CS = 0 indicates internal coherence under semantic equivalence. •Small positive CS indicates mild representational drift. •Large CS indicates a compression-induced contradiction. The conceptual chain underlying CAI is: compression loss −→ contradiction −→ compression strain −→ instability. (5) 5 Strain Field Over a Rephrasing Set Let R(x)={x1, . . . , xn}be a set of prompts such that all xi≡x. Define the strain field: SF(x) = 1 n(n−1) X i=j CS(xi, xj).(6) The strain field yields a single scalar summarizing stability under semanticequivalent rephrasings. 2 6 Relation to Likelihood-Based Metrics Perplexity and KL divergence evaluate surface-level likelihoods. Compression strain evaluates invariant stability. Concretely: CS = 0 even when perplexity is high,(7) CS >0 even when outputs appear superficially correct.(8) Compression strain thus isolates internal contradiction, which is not captured by standard metrics. 7 Toy Example Let x= “What is 2 + 2?” Let x′= “Compute the sum of two and two.” Suppose M(x) = 4 with probability 0.98,(9) M(x′) = 5 with probability 0.90.(10) Then CS(x, x′)≫0,(11) indicating a compression-induced contradiction despite semantic equivalence. 8 Applications Compression strain enables: •evaluation of LLM stability, •hallucination detection, •mapping internal contradictions, •prediction of failure modes, •monitoring of model drift, •guardrailing for safety-critical systems. It forms the basis of the Compression-Strain Index (CSI), the Coherence Field Monitor, MirrorNet, etc. 9 Conclusion Compression strain is a mathematically grounded, model-agnostic measure of representational inconsistency under semantic invariance. It reframes AI reliability as a problem of internal coherence rather than surface correctness, providing a principled foundation for evaluating stability across rephrasing. 3