# A Mathematical Theory of Interpretation: Rational Entropy, Spectral Readout, and Confusability as a Resource

arXiv: https://arxiv.org/abs/2608.23892 · HTML: https://mlsanity.com/paper/2608.23892/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 24 references that are in the map.
- First posted on arXiv 24 Aug 2026.

## Builds on (most cited first, 5 of 5)

- [Probing Classifiers: Promises, Shortcomings, and Advances](https://mlsanity.com/paper/2102.12452/) (2021)
- [Towards Automated Circuit Discovery for Mechanistic Interpretability](https://mlsanity.com/paper/2304.14997/) (2023)
- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Causality is Key for Interpretability Claims to Generalise](https://mlsanity.com/paper/2602.16698/) (2026)
- [Explainable AI needs formalization](https://mlsanity.com/paper/2409.14590/) (2024)

## Abstract (arXiv)

This article presents the abridged core of A Mathematical Theory of Interpretation (MTI), which treats interpretation as observer-relative spectral measurement under an access structure. MTI makes interpretation a method-design problem: access, query, utility, and medium determine what an observer can select, identify, communicate, or refuse. On a learning-invariant Hilbert realization, Rational Entropy measures residual uncertainty across knowledge, utility, and medium. In the finite-effective regime, we classify its zero set. Pairwise confusability is equivalent to uniform atomic collapse, while a unique utility maximum can select one atom even when other zero-cost states remain non-atomic. This reverses the usual zero-error role of confusability: agreement in at least one observer direction excludes unresolved multi-atom readings, while the joint label preserves identification. The corresponding free-design capacity is the product of all but the smallest direction budget. A four-condition certificate characterizes sharp, decodable, medium-faithful, and order-independent readout on a finite commuting code sector and returns typed obstructions when those guarantees fail. Together, these results establish MTI as a theoretical basis for constructing interpretation methods with explicit access assumptions, guarantees, and failure modes.