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

Abstract: We propose that value -- the quantity goal-directed agents create, destroy, and exchange -- is a lawful structural quantity in the same category as information. Following Shannon's method, we make one ruthless abstraction: value is the rate at which an agent converts a resource into goal-progress, relative to a frame fixed by its goal. A scale-invariance axiom forces a logarithmic measure, $V=\sum_i k_i \ln e_i$; compounding of a reinvested resource forces the same form via the ergodicity argument of Peters (2019). The two routes are kin rather than independent; their agreement is a consistency check, not an over-determination. We derive a coding theorem of value: $ΔG \le I(X;Y)$, achieved by Bayes-proportional allocation; realized value decomposes as $G=D(q|r)-D(q|p)$, identifying misalignment with measurable waste. For populations, value is frame-relative while price is frame-independent; a fleet that pools its resource and fuses its perception inherits the ceiling $G_{\mathrm{fleet}} \le I(X;Y_{1:m}) \le H(X)$ (a corollary; an earlier sum-form claim was wrong and is corrected in v5). A dynamical layer yields an is/ought asymmetry from which alignment emerges as a control-stability condition with a closed-form residual. We test the single-frame laws on live LLMs in a pre-registered scale-up: perception mutual information tracks realized capability rather than parameter count (Spearman $ρ= 0.977$ pooled over 30 model$\times$domain points), out-of-sample $ΔG$ tracks $I(X;Y)$, and over-confidence is measurable dissipation; a further pre-registered test shows the bridge is shape-invariant across four task shapes ($n=42$, slope 0.953). None of the mechanisms is individually new -- generalized Kelly, Armstrong & Mindermann (2018), classical control; the contribution is their unification and the governance mapping (incentive design over oversight) that follows.