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

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) – kin routes, a consistency check, not an over-determination. We derive a coding theorem of value, $\mathrm{\Delta }G\le I(X;Y)$; realized value decomposes as $G=D(q\mathrm{\parallel }r)-D(q\mathrm{\parallel }p)$. For populations, value is frame-relative while price is frame-independent; a fleet that pools its resource and fuses its perception inherits the ceiling $G_{\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{e}\mathrm{t}}\le I(X;Y_{1:m})\le H(X)$ (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. We test the single-frame laws on live language models, pre-registered: perception mutual information tracks realized capability (Spearman $\rho =0.977$ over 30 model$\times$domain points); out-of-sample $\mathrm{\Delta }G$ tracks $I(X;Y)$, shape-invariant across four task shapes ($n=42$, slope $0.953$); over-confidence is measurable dissipation. The stated continuation gate has since been run (pre-registered, frontier-model population): the coupled capacity-region prediction – growth-gap law, coalition submodularity with an XOR synergy control, joint ceiling, Kelly selection – is confirmed within its frozen bands on real agents; the mean-field residual law $\mathrm{\parallel }Vg\mathrm{\parallel }\mathrm{/}\gamma$ found no domain (populations hold no goal dispersion) and is retired to its mathematical scope. The contribution is the unification and the governance mapping that follows.