A Mathematical Theory of Value

We propose that value — the quantity that goal-directed agents create, destroy, and exchange — is a lawful structural quantity, in the same category as information, once stripped of its semantic clothing (morality, price, psychology). Following the method of Shannon (1948), we make one ruthless abstraction: value is the rate at which an agent converts a physical resource into goal-progress, relative to a frame fixed by the agent's goal. A scale-invariance axiom forces a logarithmic measure of value, V = Σ kᵢ ln eᵢ, via a Cauchy functional equation; the compounding dynamics of a reinvested resource force the same form independently via an ergodicity argument — two routes with no shared premises. From the compounding dynamics we derive a coding theorem of value: the rate at which an agent can create value through a perception channel Y of the world X is bounded by the mutual information, ΔG ≤ I(X;Y), achieved by Bayes-proportional allocation; and realized value decomposes exactly as available potential minus dissipation, G = D(q‖r) − D(q‖p), identifying misalignment with measurable waste. For populations we show value is frame-relative while price is frame-independent, that the collective value throughput of a fleet is capped by the world's entropy, Σ Gₐ ≤ H(X), and that the fleet's operating point is a Kelly portfolio over agents selected by an emergent price. A dynamical layer gives the equations of motion and an is/ought asymmetry — beliefs have a target the world supplies, goals do not — from which alignment emerges as a control-stability condition with a closed-form residual misalignment. We test the single-frame laws on live language models in a pre-registered scale-up across three task domains and a ten-model, five-family ladder, and further show the out-of-sample bridge ΔG ~ I(X;Y) is shape-invariant: pooled across four qualitatively different task shapes — classification, reasoning (GSM8K), sequential decision, and code (MBPP), all with discrete gold so I is computed, not estimated — the slope (0.953) is statistically indistinguishable from the classification-only value, promoting the bridge from a demonstration toward a law. Over-confidence is measurable dissipation in every domain; the fleet-pricing claim is reported with its honest boundary. The laws hold in the smooth, concave regime. None of the underlying mechanisms is individually new; the contribution is their unification under one substrate-grounded quantity, and the is/ought asymmetry that follows.