Reconstruction from Compressed Signals: Markets and Neural Networks as a Single Information-Theoretic Problem

Derived from the forthcoming Prolegomena to the Unification of Economics, Computer Science, Mathematics, and Physics, Volume I, The Price Libertarian: Why Free Markets Need Free Prices, Edition 1. 0 (Beier, forthcoming 2026) Abstract This paper identifies a structural parallel between two systems that appear unrelated: financial markets coordinating through prices, and neural networks learning from training data. Both perform lossy reconstruction from compressed signals, though the resemblance has gone unnoticed because the fields that study them do not share a literature. A price compresses human judgment into a scalar. A training example compresses the reasoning that produced it into a data point. The downstream system - whether a buyer or a neural network - must then reconstruct enough of the lost structure to act effectively, and when it cannot, the consequences differ in form but not in kind: capital goes to the wrong place, or the model fails on data it has not seen before. This paper defines a quantity, Beier Extropy (EB = H (P, D) - H (P) ), that measures the information restored to a compressed signal when structured metadata (Disclosure) is attached. EB is not a new quantity - it is the standard conditional entropy H (D|P) - but the identification that this single quantity governs performance on two surfaces simultaneously, through an identical mechanism, is new. The central claim is that EB operates on market coordination and AI training at once, and that this dual operation is not a metaphor but a consequence of both systems processing the same kind of compressed signal. The paper develops the formal argument, addresses objections, and identifies testable predictions on both surfaces.