A Mathematical Theory of Value: a synthesis on goal-directed agency under resource constraints

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 = Σᵢ kᵢ ln eᵢ, via a Cauchy functional equation; the compounding dynamics of a reinvested resource force the same form independently via the ergodicity argument of Peters (2019). The two routes are kin rather than independent — the scale-invariance axiom is the static shadow of multiplicative compounding — so their agreement is a consistency check on the form, not an independent over-determination. From the compounding dynamics we also 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 a fleet which pools its resource and fuses its perception is one agent and so inherits the capacity ceiling, G_fleet ≤ I(X;Y_1:m) ≤ H(X) (a corollary; an earlier sum-form claim was wrong and is corrected in v5), 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 ΔG tracks I(X;Y) (slope CI excludes 0), and over-confidence is measurable dissipation in every domain — the two single-frame laws generalize. A further pre-registered test shows 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 (n=42) — the slope is 0.953, statistically indistinguishable from the classification-only value, promoting the bridge from a demonstration toward a law. 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 paper's stated continuation gate has since been run (pre-registered, on a frontier-model population): the coupled capacity-region prediction — the growth-gap law, coalition submodularity with a designed XOR synergy control, the joint ceiling, and Kelly selection — is confirmed within its frozen bands on real agents, the first real-agent confirmation of a prediction no component theory makes separately; the mean-field residual-scaling law ‖Vg‖/γ, by contrast, found no domain — capable populations hold no goal dispersion (V → 0) — and is retired to its mathematical scope. 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 single-agent core is generalized Kelly, the is/ought asymmetry's value-side is Armstrong & Mindermann (2018), and the alignment-stability algebra is classical control; the contribution is their unification under one substrate-grounded quantity and the governance mapping (incentive design over oversight) that follows.