Mathematical Framework Proposes Value as Measurable Structural Quantity in Goal-Directed Agents
A new preprint on arXiv proposes a formal mathematical framework treating 'value' — what goal-directed agents create and exchange — as a lawful structural quantity analogous to information in Shannon's theory. The authors derive a logarithmic value measure from scale-invariance and ergodicity arguments, then validate it empirically on language models, finding that perceptual mutual information tracks model capability better than parameter count (Spearman ρ = 0.977 across 30 model-domain points). The work matters because it attempts to unify several existing frameworks — Kelly criterion, alignment theory, and classical control — into a single governance-relevant theory with implications for AI incentive design and oversight.
The paper, submitted to arXiv in June 2026, proposes that value can be defined rigorously as the rate at which an agent converts a resource into goal-progress relative to a fixed goal frame. Two independent derivations — one from a scale-invariance axiom and one from an ergodicity argument following Peters (2019) — both yield the same logarithmic form V = Σ kᵢ ln eᵢ, which the authors treat as a consistency check. A central result is a 'coding theorem of value' (ΔG ≤ I(X;Y)), showing that goal-progress is bounded by the mutual information between environment and agent perception, achieved by Bayes-proportional resource allocation. For multi-agent fleets, the theory shows that pooling resources and fusing perception raises this ceiling up to the entropy of the environment H(X). The framework is tested in pre-registered experiments on live language models, where perceptual mutual information outperforms parameter count as a predictor of realized capability, and a shape-invariance test across four task types (n=42) yields a slope of 0.953. The authors also derive an is/ought asymmetry from the dynamical layer, framing AI alignment as a control-stability condition with a computable residual. The contribution is explicitly positioned as a synthesis of existing mechanisms — generalized Kelly, Armstrong & Mindermann (2018), and classical control — rather than the introduction of entirely new components.
What's missing
As a preprint, this work has not yet undergone peer review, and the empirical validation relies on a relatively small sample (30 model-domain points; n=42 for the shape-invariance test), leaving open questions about generalization to broader model families and task distributions. It is also unclear how the proposed governance and incentive-design applications would translate into practical AI oversight mechanisms.
What different sources said
- arXiv cs.AICenter
A Mathematical Theory of Value: a synthesis on goal-directed agency under resource constraints
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