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 scarce resource into goal-progress, relative to a frame fixed by the agent's goal. From a single scale-invariance axiom we derive a logarithmic measure of value, V = Σ kᵢ ln eᵢ. From the compounding dynamics of a reinvested resource 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 then 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 (0.5B–8B): perception mutual information tracks realized capability rather than parameter count (Spearman ρ = 0.977 pooled over 30 model×domain points), out-of-sample value-growth tracks I(X;Y) (slope CI excludes 0), and over-confidence is measurable dissipation in every domain — the two single-frame laws generalize. A fleet-pricing experiment is reported scoped and primary-metric-first: value-pricing recovers cost-aware routing from first principles — it ties good hand-tuned routing under a token budget, beats a cost-blind router under a compute budget, and does not outperform a cost-aware engineer; its contribution is principled measurement, not outperformance. The laws hold in the smooth, concave (diminishing-returns) regime; threshold, satiation, and risk-seeking goals lie outside it. 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.