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) -- kin routes, a consistency check, not an over-determination. We derive a coding theorem of value, $\Delta G\le I(X;Y)$; realized value decomposes as $G=D(q\Vert r)-D(q\Vert p)$. For populations, value is frame-relative while price is frame-independent; a fleet that pools its resource and fuses its perception inherits the ceiling $G_{\mathrm{fleet}}\le I(X;Y_{1:m})\le H(X)$ (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. 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 $\Delta G$ 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 $\Vert Vg\Vert /\gamma$ 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.

Also available at https://doi.org/10.5281/zenodo.20487041 (v6). v2: the pre-registered continuation-gate experiment has been run -- prediction (i) confirmed on real frontier agents; prediction (ii) retired to mathematical scope. Code, pre-registrations, and raw run records: https://github.com/macrokit/value