value

A mathematical theory of value — built the way Shannon built information.

This is the research repository behind the preprint A Mathematical Theory of Value (arXiv:2606.12502 · Zenodo (latest version) · concept DOI 10.5281/zenodo.20487041, always resolves to the latest version · byline Cheng Qian). It develops one bet to its consequences:

Value is a lawful, structural quantity — in the same category as information — currently left undefined only because, like information before 1948, it is confused with its semantic clothing (morality, price, preference). Strip the clothing and a structural quantity remains, with a measure, a capacity limit, and an exact accounting of value created versus dissipated. The end goal is concrete: a quantity precise enough to govern populations of separated AI individuals.

The motivating premise: in the near future, all AI agents are driven by values. If so, a theory of value is not philosophy — it is the control theory for agent populations.

What is and isn't new (up front). None of the underlying mechanisms is individually new: the single-agent core is generalized Kelly (J. L. Kelly, 1956) — the capacity theorem ΔG ≤ I(X;Y) and the fleet's Kelly-portfolio operating point are Kelly/Cover results re-read for an arbitrary scarce resource. The contribution is their unification under one substrate-grounded quantity, the cross-frame/price layer, and the is/ought asymmetry — beliefs have a target the world supplies, goals do not — from which alignment falls out as a control-stability condition. Claims are scoped to match; the honesty is deliberate.

The method (copied from Shannon, deliberately)

Shannon's 1948 paper worked because of one ruthless abstraction: he threw away meaning and defined information as nothing but reduction of uncertainty. From that he got a measure (−log p), a unit (the bit), a system model, and hard limits (channel capacity, the coding theorems).

Our analogous abstraction: throw away morality, price, and human psychology. What remains is value as a frame-relative but lawful quantity — relative to an agent's goal the way information is relative to a prior and energy is relative to a reference frame. Relativity does not disqualify a quantity from being fundamental; it forces the theory to make the frame explicit.

A theory of value is to agency what relativity is to motion. Value is frame-dependent on the agent's goal; the laws relating value across frames are universal.

The results so far

Doc	Contribution	Headline result
00	Framing	Value = exergy (created by gradients, dissipated by misalignment), not conserved energy; frame-relative; substrate = free energy
01	The measure	Logarithmic Value Law V = Σ kᵢ ln eᵢ (forced by one Cauchy axiom); optimal allocation eᵢ* ∝ kᵢ; focus penalty V* = K ln E − K·H(k̂)
02	The capacity theorem	ΔG ≤ I(X;Y) — value-throughput is bounded by information-throughput; Second Law G = D(q‖r) − D(q‖p) exact
03	Cross-frame / multi-agent	Value is frame-relative; price is frame-independent. Exchange via shadow price λ=K/E; invariant = Fisher–Rao metric (Čencov); alignment cos θ sets positive/negative-sum
04	Fleet governance	Pooled-fleet value-throughput ≤ H(X) — a fleet that fuses its perception is one agent, so doc 02's ceiling applies (a corollary, not a new theorem; the earlier sum-form claim was wrong and is corrected in §2's erratum); region shaped by alignment; operating point set by price
05	The equations of motion	Learning = value-recovery, dissipation = regret; natural-gradient (Fisher) belief flow, tâtonnement price flow, control+selection goal flow; alignment = a stability condition; a moving world forbids zero dissipation
06	Real-agent test (v1, n=4)	Four live LLMs obey the laws on their own outputs: I(X;Y) tracks realized capability not size (a cross-family 8B, bigger-but-weaker, lands at lower I); ΔG∝I(X;Y); over-confidence dissipates value (weakest → negative growth); diversity > redundancy; pricing beats pooling and wins under a compute budget — but no demon on correlated agents (honest negative). Generalized in 09/14.
07	Alignment theorem	Coupled control+selection flow solved: residual misalignment = ‖Vg‖/γ; stable iff γ > λ_max(∂(Vg)/∂k̄); incentive design (g→0) beats brute-force control (↑γ)
08	Field theory (speculative)	Continuum limit: value as a field; demand-shocks are waves (telegrapher's eq, dispersion crossover); collective goals are a phase transition (active matter / Toner–Tu); is/ought = beliefs massive, goals Goldstone. A research program, not a result — §9 is blunt about it.
09	v2 empirics (pre-registered)	Scale-up to 3 domains × 10 models with 95% CIs: I(X;Y) tracks capability at ρ=0.977 (R1 generalizes); over-confidence dissipates in every domain (R2 generalizes); value-price routing beats naive + cost-blind hand-tuned under a compute budget (ties under tokens, loses to cost-aware) — pricing = cost-awareness derived, scoped honestly

Supporting: docs/related-work.md — paper-ready prior-art section (utility theory, Kelly/info theory, RL, thermodynamics/FEP, general equilibrium, alignment) with an honest statement of contribution. The preprint built from these docs is in paper/ (paper/main.pdf). New here? docs/FAQ.md answers the common first objections (isn't value just a price? can it price the unpriced? is value stored like money? isn't this just Kelly?) honestly and in brief.

The three load-bearing equations

V   = Σ kᵢ ln eᵢ                  the measure            (doc 01)
ΔG  = I(X;Y)                      the capacity            (doc 02)
G   = D(q‖r) − D(q‖p)             available − dissipated  (doc 02)

Two structural facts give some confidence the abstraction is real, not decorative:

1. The logarithm arrives twice — from a scale-invariance axiom (doc 01) and from multiplicative compounding dynamics (doc 02). The two routes are kin, not independent (scale-invariance is compounding's static shadow), so their agreement is a consistency check on the form — not an independent over-determination, and not a proof of fundamentality.
2. Shannon's quantities recur — entropy H as a value-dissipation penalty (doc 01), KL divergence D as available-minus-dissipated value (doc 02), mutual information I as the value ceiling (doc 02), and the Fisher metric as the cross-frame invariant (doc 03). The information bridge is load-bearing, not decorative.

Why this serves "governing separated AI individuals"

• Per agent: value-generation rate is capped by its perception's mutual information I(X;Y_a). A macro raises I per joule — the formal reason a cheap reflex can match expensive deliberation on a narrow task.
• Across agents: their values are not cardinally comparable (Arrow / interpersonal-utility impossibility — and we do not pretend otherwise). Coordinate them through an emergent price on shared resources, exactly as economies coordinate incomparable utilities — never through a god's-eye utility sum.
• Fleet design: cultivate goal-diversity (trades to positive-sum) and perception-diversity (covers more of H(X)); suppress belief-divergence (zero/negative-sum speculation) and goal-conflict (negative cos θ). Headcount past perception-saturation adds nothing to what the fleet jointly knows (the pooled-fleet reading of doc 04 §2).

Evidence — the theory predicts its own instantiation

sim/ is a minimal numerical test: take each closed-form prediction and check a Monte-Carlo world produces those numbers. 20/20 checks pass (python3 sim/experiments.py):

• E1 ΔG = I(X;Y) to ~0.001 nats — value-throughput is information-throughput.
• E2 G = D(q‖r) − D(q‖p) exactly, including value going negative under confident error.
• E3 fleet ceiling ΔG_fleet ≤ H(X); diversity lifts it, redundancy adds exactly 0.
• E4 the headline: a priced/rebalanced fleet beats ad-hoc — Shannon's demon, a fleet that grows at +0.021/round from agents that individually don't grow at all.
• E5 dissipation = regret; a drifting world floors dissipation at ~0.18 nats while a stationary one → 0 (the Dynamical Second Law).

These confirm the math is self-consistent and correctly derived. The sim is necessary but circular (the worlds are built from the distributions the formulas assume), so the next step is real agents.

sim/real/ instantiates the theory with four live local LLMs (1.5B / 3B / 7B + a cross-family 8B) on a frozen 100-item decision task — X = the correct action, Y_a = the model's chosen action — and measures the value quantities from their actual outputs (python3 sim/real/experiments_real.py; full write-up in docs/06):

• R1 I(X;Y) tracks realized capability, not parameter count (here r ≈ 0.99 vs accuracy on just four models — small-n, generalized below to Spearman ρ = 0.977 over 30 model×domain points): the cross-family 8B is bigger yet weaker than the 7B and its I lands lower accordingly. Out-of-sample ΔG tracks I — value-throughput is information-throughput, on real behavior.
• R2 over-confidence is dissipation in nats, shrinking with capability (4.17 → 0.69 over the qwen ladder); the least-capable models' confident error drives realized growth negative.
• R3 the cheapest model delivers the most I(X;Y) per second of compute (0.74 vs 0.30 nats/s).
• R4 a diverse pair covers more of H(X) than either alone (+0.09 nats); an identical re-run adds 0.
• R5 (honest) a price/Kelly fleet beats equal-weight pooling but not the single best model — no demon on correlated agents — yet under a compute budget an I/cost price wins (0.33 vs 0.24 nats/s). Pricing pays where agents are priced and diverse, not merely pooled.

So the laws hold on real agents where the single-frame theory is testable; the one fleet-level claim with a genuine precondition (perception diversity) is reported with its boundary, not oversold.

v2 scale-up — generalization across domains and a 10-model ladder

sim/real/v2/ answers the v1 reviewer's binding critique ("one task, four models") by scaling to three domains (intent routing K=6, MMLU-style MCQA K=4, AG-News topic K=4; 240 items each) and a ten-model cross-family ladder (Qwen / Llama / Gemma / Phi / Mistral, 0.5B→8B), pre-registered before any model ran (PREREGISTRATION.md) and reported with 95% CIs (full write-up docs/09):

• R1-v2 (PASS): pooled over 30 model×domain points, I(X;Y) tracks realized capability at Spearman ρ = 0.977 (CI [0.916, 0.996]); out-of-sample ΔG tracks I with slope 0.935 (CI excludes 0). The capacity bridge generalizes.
• R2-v2 (PASS): over-confidence dissipates value in every domain; the weakest models go sharply negative (−11 nats on MMLU, −14 on topic).
• Fleet-R5 (headline, honest): a value-price router (∝ I_a/cost_a) beats round-robin and equal-weight on both cost metrics (governance claim non-empty), and beats the strong cost-blind hand-tuned router under a compute/latency budget (Δ = +6.4, CI [3.9, 9.2]) — while tying it under a token budget (token cost is ~uniform, so pricing reduces to quality-first) and losing to a cost-aware hand-tuned router. Pricing's edge is cost-awareness derived from first principles, bounded exactly.
• Ceiling (PASS): all pairwise joint I ≤ H(X); diverse specialists beat redundancy in every domain.

v2 moves the evidence from demonstration to generalization — the laws hold across task shapes and a real model ladder, with the fleet result scoped to where pricing actually pays.

Honest status

• Proven (within stated axioms): the log law (01), the capacity theorem with achievability + converse (02), the Second-Law decomposition (02), the cross-frame exchange rate + Fisher invariant (03), and the fleet outer bounds (04).
• Conjecture: the Second Law of Value as a global tendency (00 §4); equilibrium-price existence beyond the convex regime (03 §8).
• Open: the exact multi-agent capacity region (04) — likely as hard as open problems in network information theory; the cross-frame cardinal comparison (03, deliberately unsolved); the full dynamics of how k, p, π evolve (Thread C — unwritten).

Each doc ends with an explicit "where this breaks" section, in Shannon's spirit of naming the model's assumptions.

Reading order

00 (framing) → 01 (the measure) → 02 (the limit) → 03 (many agents) → 04 (the fleet) → 05 (dynamics) → 06 (the real-agent test) → 07 (the alignment theorem) → 08 (the field-theory frontier). Each builds on the last; each is self-contained enough to skim from its headline box. Docs 01–04 are the statics (the measure, the capacity bound, equilibria); 05 and 07 are the dynamics (equations of motion + the alignment-stability theorem); 06 is the evidence (live LLMs measured against the laws); 08 is the speculative frontier (a continuum/field-theory extension — read its §9 caveats; a research program, not a result). Together they form a thermodynamics of value, with a real-agent test. docs/related-work.md situates it against prior art.

Provenance & non-goals

• Author byline for any external write-up: Cheng Qian. Repo git identity: jamesavechives.
• This is a theory repo — definitions, derivations, and honest limits. It is not a product, and it does not aim to resolve interpersonal value comparison (provably non-canonical) or to produce a single fleet-wide "total value" number (there isn't one — that's the point of price).

License

Copyright © 2026 Cheng Qian. Dual-licensed by material type (see LICENSE): the written work (docs, paper, prose, figures) under CC BY 4.0 — reuse is welcome with attribution to Cheng Qian; the code (sim/, tools/, the website) under Apache 2.0. Cite as: Qian, Cheng. A Mathematical Theory of Value. arXiv:2606.12502; Zenodo, concept DOI 10.5281/zenodo.20487041.