A Mathematical Theory of Value: A Synthesis on Goal-Directed Agency Under Resource Constraints

Jul 16, 2026

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Proposes value as a lawful structural quantity created, destroyed, and exchanged by goal-directed agents, in the same formal category as information once stripped of moral, price, and psychological semantics. Using Shannon’s method, defines value as conversion rate from physical resources to goal-progress relative to an agent-fixed goal frame, yielding a logarithmic law via scale invariance. Establishes a coding bound ∆G ≤ I(X;Y) and an exact decomposition into potential minus dissipation to quantify misalignment and waste.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages20

Reading Time50 min

Introduction

Statics: the measure, the limit, the Second Law

The measure: a logarithmic law of value

The capacity theorem: value is bounded by information

The Second Law of Value

Multi-agent: price and the fleet

Cross-frame value and the frame-independence of price

The fleet capacity region

Dynamics: motion and alignment

The equations of motion and the is/ought asymmetry

Alignment as a stability condition

Evidence

Synthetic validation

Real agents

Generalization: a pre-registered scale-up

arXiv :2606 . 12502v2 [physics .soc-ph] 3 Jul 2026 A Mathematical Theory of Value a synthesis on goal-directed agency under resource constraints Cheng Qian 2026 Abstract We propose that value—the quantity that goal-directed agents create, destroy, and exchange— is a lawful structural quantity, in the same category as information, once stripped of its semantic clothing (morality, price, psychology) . Following the method of Shannon (1948), we make one ruthless abstraction: value is the rate at which an agent converts a physical resource into goal-progress, relative to a frame fixed by the agent’s goal. A scale-invariance axiom forces a logarithmic measure of value, V = Pi ki ln ei , via a Cauchy functional equation; the compounding dynamics of a reinvested resource force the same form independently via the ergodicity argument of Peters (2019) . The two routes are kin rather than independent—the scale-invariance axiom is the static shadow of multiplicative compounding—so their agreement is a consistency check on the form, not an independent over-determination. From the compounding dynamics we also derive a coding theorem of value: the rate at which an agent can create value through a perception channel Y of the world X is bounded by the mutual information, ∆G ≤ I (X;Y ), achieved by Bayes-proportional allocation; and realized value decomposes exactly as available potential minus dissipation, G = D (q ∥r) − D(q ∥p), identifying misalignment with measurable waste. For populations we show value is frame-relative while price is frame-independent, that a fleet which pools its resource and fuses its perception is one agent and so inherits the capacity ceiling, Gfleet ≤ I (X;Y1:m) ≤ H (X) (a corollary; an earlier sum-form claim was wrong and is corrected in v5), and that the fleet’s operating point is a Kelly portfolio over agents selected by an emergent price. A dynamical layer gives the equations of motion and an is/ought asymmetry—beliefs have a target the world supplies, goals do not—from which alignment emerges as a control-stability condition with a closed-form residual misalignment. We then test the single-frame laws on live language models in a pre-registered scale-up across three task domains and a ten-model, five-family ladder (0 .5B–8B): perception mutual information tracks realized capability rather than parameter count (Spearman ρ = 0 .977 pooled over 30 model×domain points), out-of-sample ∆G tracks I (X;Y ) (slope CI excludes 0), and over-confidence is measurable dissipation in every domain—the two single-frame laws generalize. A further pre-registered test shows the out-of-sample bridge ∆G ∼ I (X;Y ) is shape-invariant: pooled across four qualitatively different task shapes—classification, reasoning (GSM8K), sequential decision, and code (MBPP), all with discrete gold so I is computed, not estimated (n = 42)—the slope is 0 .953, statistically indistinguishable from the classification-only value, promoting the bridge from a demonstration toward a law. A fleet-pricing experiment is reported scoped and primary-metric-first: valuepricing recovers cost-aware routing from first principles—it ties good hand-tuned routing under a token budget, beats a cost-blind router under a compute budget, and does not outperform a cost-aware engineer; its contribution is principled measurement, not outperformance. The paper’s stated continuation gate has since been run (pre-registered, on a frontier-model population): the coupled capacity-region prediction—the growth-gap law, coalition submodularity with a designed XOR synergy control, the joint ceiling, and Kelly selection—is confirmed within its frozen bands on real agents, the first real-agent confirmation of a prediction no component theory makes separately; the mean-field residual-scaling law ∥Vg∥/γ, by contrast, found no domain — capable populations hold no goal dispersion (V → 0)—and is retired to its mathematical scope. The laws hold in the smooth, concave (diminishin

FAQs

How does the paper define value for goal-directed agents under resource constraints?

Value is defined as the rate at which an agent converts a physical resource into goal-progress relative to a frame fixed by its goal. This structural definition is treated as analogous to information once semantic dressing is removed.

What are the main theoretical results connecting value and information?

A coding theorem bounds realized value increase by mutual information, with ∆G ≤ I(X;Y) achieved by Bayes-proportional allocation. Value also decomposes exactly into potential minus dissipation, identifying misalignment with measurable waste.

How does the paper test and validate the proposed laws?

It reports pre-registered experiments on live language models, including a scale-up across multiple domains and model sizes, and an out-of-sample bridge test across qualitatively different task shapes. Results show that mutual information tracks realized capability and that dissipation corresponds to over-confidence, supporting generalization.

Jul 16, 2026

Upload by Jiven