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

Cheng Qian

REVIEW 2 major objections 2 minor 1 cited by

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Value is the rate at which an agent converts a resource into goal-progress relative to a goal-fixed frame, taking the logarithmic form V = sum k_i ln e_i.

2026-06-27 07:34 UTC pith:ASRIMKOR

load-bearing objection The paper unifies existing pieces around a rate-based value definition but adds a scale-invariance axiom to get the log form and reports only summary correlations without visible controls. the 2 major comments →

arxiv 2606.12502 v2 pith:ASRIMKOR submitted 2026-06-10 physics.soc-ph cs.AI

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

Cheng Qian This is my paper

classification physics.soc-ph cs.AI

keywords value theorygoal-directed agencyresource constraintsmutual informationcoding theoremscale invarianceBayes allocationmisalignment

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines value as the rate at which goal-directed agents convert resources into progress within a frame set by the goal itself. Scale invariance applied to this rate produces a logarithmic measure, and the same form arises from compounding under ergodicity. The resulting structure yields a coding bound on progress change by mutual information, realized through Bayes-proportional resource allocation. Realized value is then expressed as the difference of two divergences, converting misalignment directly into a measurable quantity of waste. Empirical checks on language models show mutual information tracking capability gains across scales and task shapes.

Core claim

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=∑_i k_i ln e_i. The same form follows from reinvestment compounding. This produces a coding theorem ΔG ≤ I(X;Y) achieved by Bayes-proportional allocation, with realized value decomposing as G = D(q||r) − D(q||p). For populations, value remains frame-relative while price is frame-independent, and a pooled fleet is bounded by the joint mutual information ceiling.

What carries the argument

The scale-invariance axiom applied to the definition of value as a resource-to-goal-progress conversion rate, which produces the logarithmic measure and enables the coding theorem ΔG ≤ I(X;Y) with Bayes-proportional allocation.

Load-bearing premise

The scale-invariance axiom that forces the logarithmic form of the value measure from the definition of value as a conversion rate.

What would settle it

A controlled setting in which an agent's measured goal-progress increase exceeds the mutual information between its inputs and outputs, despite using Bayes-proportional allocation, would disprove the coding theorem.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Realized value decomposes as G = D(q||r) − D(q||p), identifying misalignment with measurable waste.
Value is frame-relative while price is frame-independent for populations.
A fleet that pools its resource and fuses its perception inherits the ceiling G_fleet ≤ I(X;Y_{1:m}) ≤ H(X).
Alignment emerges as a control-stability condition with a closed-form residual from the dynamical layer.
Perception mutual information tracks realized capability rather than parameter count, and the relation is shape-invariant across task shapes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The unification would allow incentive design to target the mutual information term directly instead of output-level oversight.
The frame-relativity of value versus price independence suggests modeling multi-agent markets by separating individual goal frames from shared pricing.
If the bound holds across domains, the same coding relation could apply to biological or economic agents without new postulates.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

The paper unifies existing pieces around a rate-based value definition but adds a scale-invariance axiom to get the log form and reports only summary correlations without visible controls.

read the letter

The main thing here is an attempt to treat value as a conversion rate from resource to goal progress in a fixed frame, then derive a coding bound and a misalignment decomposition from it. The unification of generalized Kelly, the Armstrong-Mindermann point, and control ideas is explicit, and the fleet corollary correction in v5 shows the authors caught an earlier error. That is the actual contribution on offer.

The log form V = sum k_i ln e_i does not follow from the rate definition alone; the paper adds a scale-invariance axiom to force it, and the ergodicity route is labeled a consistency check. Without the axiom the functional form is under-determined, which is the central soft spot. The LLM tests give high Spearman numbers but are reported only as pooled statistics with no pre-specified controls or exclusion rules visible in the abstract, and k_i remain free parameters. Full derivations, code, and raw data are not inspectable from what is provided.

The work is aimed at readers who already work at the intersection of information theory, ergodicity economics, and alignment. A serious referee could check whether the coding theorem holds once the axiom is granted and whether the empirical claims survive proper controls. The paper is coherent on its own terms and engages the cited literature directly, so it clears the bar for peer review even though the current evidence is thin and the axiom is doing heavy lifting.

Referee Report

2 major / 2 minor

Summary. The paper claims that value is the rate at which an agent converts a resource into goal-progress relative to a goal-fixed frame. A scale-invariance axiom leads to the logarithmic value measure V = ∑ k_i ln e_i, which is consistent with an ergodicity argument. It derives a coding theorem ΔG ≤ I(X;Y) achieved by Bayes-proportional allocation, with realized value G = D(q||r) - D(q||p). The theory is extended to populations and a dynamical layer for alignment, and tested on language models with pre-registered experiments showing high correlations between mutual information and capability (Spearman ρ = 0.977) and shape-invariance across task shapes (slope 0.953).

Significance. If the results hold, this provides a unified mathematical framework for value in goal-directed systems, bridging information theory, resource allocation under constraints, and control theory. The pre-registered empirical tests on LLMs offer concrete, falsifiable support for the single-frame laws, and the correction of an earlier claim in v5 demonstrates scientific integrity. The governance mapping for incentive design is a notable implication.

major comments (2)

[Value measure derivation (abstract)] Value measure derivation (abstract): The scale-invariance axiom is invoked to obtain V=∑k_i ln e_i from the conversion-rate definition. The definition itself supplies no functional equation, and the paper explicitly labels the ergodicity route a consistency check rather than an independent derivation. This axiom is load-bearing for the logarithmic form, the coding theorem ΔG ≤ I(X;Y), and all downstream results; additional discussion of why alternative functional forms are ruled out by the rate definition alone would strengthen the foundation.
[Empirical validation (abstract)] Empirical validation (abstract): The reported Spearman ρ = 0.977 (30 model×domain points) and shape-invariance test (n=42, slope 0.953) support the claim that perception mutual information tracks realized capability. However, the abstract provides no indication that the k_i coefficients or domain-specific frames were fixed before seeing the data; this is load-bearing for the out-of-sample ΔG tracking I(X;Y) claim and the assertion of pre-registration robustness.

minor comments (2)

[Notation (abstract)] Notation (abstract): The symbols e_i and the precise meaning of the goal-fixed frame should be defined explicitly at first use to aid readability.
[Population extension (abstract)] Population extension (abstract): The corrected fleet ceiling G_fleet ≤ I(X;Y_{1:m}) ≤ H(X) is noted as a corollary; a brief comparison to the prior (incorrect) sum-form claim would help readers track the change.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive overall assessment and for identifying two points where the manuscript's foundation and empirical claims can be clarified. We address each major comment below and will make the indicated revisions.

read point-by-point responses

Referee: Value measure derivation (abstract): The scale-invariance axiom is invoked to obtain V=∑k_i ln e_i from the conversion-rate definition. The definition itself supplies no functional equation, and the paper explicitly labels the ergodicity route a consistency check rather than an independent derivation. This axiom is load-bearing for the logarithmic form, the coding theorem ΔG ≤ I(X;Y), and all downstream results; additional discussion of why alternative functional forms are ruled out by the rate definition alone would strengthen the foundation.

Authors: The conversion-rate definition supplies the conceptual object (a rate of resource-to-goal-progress conversion relative to a fixed goal frame) but does not by itself select a functional form; the scale-invariance axiom is therefore introduced as the minimal additional structural requirement that selects the logarithmic measure. We agree that an explicit paragraph contrasting why other candidate forms (linear, power-law, etc.) violate scale-invariance under this rate definition would strengthen the presentation. We will insert such a paragraph immediately after the axiom statement in Section 2, while retaining the explicit statement that the ergodicity argument is only a consistency check. revision: yes
Referee: Empirical validation (abstract): The reported Spearman ρ = 0.977 (30 model×domain points) and shape-invariance test (n=42, slope 0.953) support the claim that perception mutual information tracks realized capability. However, the abstract provides no indication that the k_i coefficients or domain-specific frames were fixed before seeing the data; this is load-bearing for the out-of-sample ΔG tracking I(X;Y) claim and the assertion of pre-registration robustness.

Authors: The pre-registration document (linked in the supplementary materials) fixed both the domain-specific frames and the functional form of the k_i coefficients on theoretical grounds before any data were collected or inspected. The abstract's phrase 'pre-registered scale-up' is intended to encompass this protocol, but we acknowledge that the abstract itself does not explicitly state the a-priori status of the frames and coefficients. We will revise the abstract and the opening paragraph of the empirical section to make this timing explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit axiom added to rate definition; consistency check with external reference; pre-registered tests independent of derivation.

full rationale

The paper defines value as the conversion rate relative to a goal-fixed frame, then states that a scale-invariance axiom is required to obtain the logarithmic form V=∑_i k_i ln e_i. It does not claim the log form follows from the rate definition alone and explicitly labels the Peters (2019) route as a consistency check rather than independent derivation. The coding theorem ΔG ≤ I(X;Y) and decomposition G=D(q||r)−D(q||p) are derived from the resulting measure. No equation reduces a claimed result to its inputs by construction, and no load-bearing step relies on self-citation chains. Pre-registered empirical tests (Spearman ρ=0.977, shape-invariance test n=42) supply external content.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on a scale-invariance axiom introduced to force the log form and on the ergodicity argument imported from Peters 2019; no new entities are postulated and the k_i coefficients appear as goal-frame parameters whose values are not derived from first principles.

free parameters (1)

k_i
Goal-frame coefficients in V=∑k_i ln e_i; their numerical values are not fixed by the axioms and must be supplied per application.

axioms (2)

ad hoc to paper Scale-invariance axiom forces logarithmic measure
Invoked in abstract to obtain V=∑k_i ln e_i from the rate definition.
domain assumption Ergodicity argument for compounding resources (Peters 2019)
Second route to the same log form; treated as external input.

pith-pipeline@v0.9.1-grok · 5905 in / 1446 out tokens · 31801 ms · 2026-06-27T07:34:33.576510+00:00 · methodology

0 comments

read the original 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 $G=D(q\|r)-D(q\|p)$. For populations, value is frame-relative while price is frame-independent; a fleet that pools its resource and fuses its perception inherits the ceiling $G_{\rm fleet}\le I(X;Y_{1:m})\le H(X)$ (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 $\|Vg\|/\gamma$ 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.

discussion (0)

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