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

arXiv: https://arxiv.org/abs/2606.12502 · HTML: https://mlsanity.com/paper/2606.12502/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 33 references that are in the map.
- First posted on arXiv 10 Jun 2026.

## Builds on (most cited first, 6 of 6)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Program Synthesis with Large Language Models](https://mlsanity.com/paper/2108.07732/) (2021)
- [Character-level Convolutional Networks for Text Classification](https://mlsanity.com/paper/1509.01626/) (2015)
- [An Evaluation Dataset for Intent Classification and Out-of-Scope Prediction](https://mlsanity.com/paper/1909.02027/) (2019)
- [Occam's razor is insufficient to infer the preferences of irrational agents](https://mlsanity.com/paper/1712.05812/) (2017)

## Built on by (most cited first, 1 of 1)

- [Information Limits and Attractor Dynamics in Economies of Frontier LLM Agents: A Pre-Registered Test](https://mlsanity.com/paper/2607.06001/) (2026)

## Abstract (arXiv)

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=sumi kiln ei; 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 ≤ I(X;Y); realized value decomposes as G=D(q|r)-D(q|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 Grm fleet≤ I(X;Y1:m)≤ 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 ρ=0.977 over 30 model×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 |Vg|/γ 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.