Drop-in related-work / prior-art section for the preprint. Clean scholarship only — no product or strategy content. The organizing honesty: every component of this theory exists in some field; the contribution is the unification, the substrate grounding, and the is/ought asymmetry. This section states that explicitly and concedes each component to its source, so the novelty claim rests on the synthesis, not on pretending the parts are new.
The axiomatic treatment of preference originates with von Neumann and Morgenstern (1944) and Savage (1954), yielding a utility unique up to positive affine transformation. The logarithmic form of our value measure (doc 01) coincides with Bernoulli's (1738) resolution of the St. Petersburg paradox and with constant-relative -risk-aversion utility at coefficient one; the empirical law of diminishing marginal value it encodes is Weber–Fechner and Gossen. We do not claim the functional form as novel. Our departure is threefold: we derive it from a scale-invariance axiom on a physical resource rather than positing it as psychology; we ground it in a substrate (free energy) that supplies a unit and a cross-frame exchange rate, which expected utility explicitly lacks (its utils are not interpersonally comparable); and we show the same form is recovered a second time by multiplicative compounding (Peters' ergodicity-economics argument, 2019) — noting honestly that the two routes are kin (scale-invariance is compounding's static shadow), so the agreement is a consistency check rather than independent over-determination.
The closest prior art, and the one we are most careful to credit, is Kelly (1956), who identified the
exponential growth rate of wealth with an information rate, with the optimum achieved by betting proportional to
one's beliefs. Breiman (1961) established its asymptotic optimality, Thorp applied it, and Cover (1991)
developed universal portfolios. Our capacity theorem (doc 02) — G* = D(q‖r) and the side-information gain
ΔG = I(X;Y) — is a generalization of Kelly's result; the side-information bound itself was proved in the
financial setting by Barron & Cover (1988), the primary source for it, with the value-of-information
ordering going back to Blackwell (1953), and the same capacity-constrained-decision structure living natively
in economics as rational inattention (Sims, 2003) and in decision-theoretic thermodynamics (Ortega & Braun,
2013). Our fleet operating point (doc 04 §3) is a Kelly/Cover portfolio over agents. We state this plainly: doc 02 is generalized Kelly. What we add is (i) the
reinterpretation of "wealth" as free energy, making the substrate physical rather than monetary; (ii) the
cross-frame price layer (doc 03), which is genuinely beyond a single-agent betting result; (iii) the dynamics
and alignment layer (doc 05); and (iv) the unifying claim that Kelly's theorem is the single-agent monetary
special case of a general law of value. Kelly proposed a betting strategy; we propose that the same
mathematics is the structure of value for any goal-directed agent.
The modern information-theoretic form of this duality is sharp and we credit it explicitly: Moffett & Eckford
(2021) show Kelly's optimum attains a rate–distortion bound with equality, and Hirono & Hidaka (2015) derive
Jarzynski-type equalities linking information to the full statistics of capital growth; on the control side,
Touchette & Lloyd (2000) bound a closed-loop controller's performance by the mutual information it acquires.
Our ΔG ≤ I(X;Y) shares this skeleton — value/growth-rate bounded by mutual information, achieved by
proportional play — and we do not claim that skeleton as new; the generalization is to an arbitrary
goal-directed value quantity on a physical substrate, not a betting payoff or a control objective. A separate
information-theoretic notion of intrinsic value, empowerment (Klyubin, Polani & Nehaniv, 2005; Salge &
Polani, 2013), defines an agent's value as the channel capacity from its actions to its future perceptions; it
measures controllability rather than value-generation rate and offers no Kelly-style achievability, so it is a
neighbor to — not an instance of — the capacity theorem. Correction (2026-06-07). An earlier version of
this section held the sum-form fleet ceiling Σ_a G_a ≤ H(X) as plausibly-novel ("no prior found"). External
review then showed why no prior exists: the sum-form claim is false in the model as stated (two agents with
identical perfect channels, each on its own bankroll, each achieve G_a = H(X); the sum is 2H(X)). The
corrected statement — a fleet that pools its bankroll and fuses its perception is one agent, so
G_fleet ≤ I(X;Y_{1:m}) ≤ H(X) — is true but is a one-line corollary of the single-agent Kelly/Cover
capacity theorem, and we claim it only as such (doc 04 §2 erratum). The lesson is recorded: a novelty search
answers "did someone publish this?", not "is it true?" — those are different audits.
The single closest prior framing of doc 00–01 — value as a Shannon-style entropy/information quantity — is the entropy theory of value, and we credit it as directly as we credit Kelly and Čencov. Georgescu-Roegen (1971) first proposed that all economic value reduces to a common denominator of low entropy. Jing Chen, "An entropy theory of value" (Structural Change and Economic Dynamics, 2018; developed earlier in Chen 2005, 2016, and now book-length in Galbraith & Chen, Entropy Economics: The Living Basis of Value and Production, 2024) makes, independently and eight years earlier, the core moves of our foundational layer:
- Value is defined as
V(P) = −log_b P, derived as the unique functional form satisfying monotonicity, additivity over independent (non-substitute) products, and non-negativity — the same additivity-forces-the- logarithm argument that gives ourV = Σ k_i ln e_i(doc 01); the expected valueV(X) = Σ p_i(−log_b p_i)is literally the Shannon entropy, exactly as our focus penalty makes entropy reappear inside the value law. - The explicit thesis that "an entropy theory of value is inevitably an information theory of value" (Shannon 1948 as the stimulus), grounded in the thermodynamic low-entropy substrate (second law, Georgescu-Roegen) — the same information↔thermodynamics framing we use for the substrate.
- A value-of-information reading via mutual information
R = H(X) − H_Y(X)(Chen §5), the qualitative kin of ourΔG = I(X;Y)(doc 02).
We concede plainly that the framing, the logarithmic-value-from-axioms derivation, the entropy=value=information identity, and the thermodynamic grounding are prior art — earlier, peer-reviewed, and now with the visibility of a Galbraith co-authored book. Our foundational (single-agent) layer does not claim these as new; per the standing strategic verdict, it should not.
What is not in Chen's line, and where our departures live: (i) the compounding / log-optimal growth-rate
formulation and its Kelly/ergodic derivation — Chen's value is a static measure of commodity scarcity, with no
G(b) = E[log Σ b_i o_i], no G* = D(q‖r); (ii) the capacity theorem with achievability and a Gibbs converse
(ΔG = I(X;Y) as an achieved bound), where Chen states only a qualitative value-of-information ≈ mutual-
information correspondence; (iii) the two-divergence Second Law G = D(q‖r) − D(q‖p) (misalignment as
dissipation, learning as value-recovery); and (iv) the entire multi-agent layer — cross-frame price, the
Fisher–Rao/Čencov invariant, alignment cos θ curvature, the fleet capacity region, the equations of motion, and
the is/ought asymmetry — none of which appears in the entropy-theory-of-value literature, whose "many producers"
is classical scarcity/monopoly economics, not a frame-covector price geometry. One genuine conceptual divergence
is worth stating and not overstating: Chen's value −log(scarcity) is essentially observer-independent (a
market/physical property of a commodity), whereas ours is frame-relative to an agent's goal k — but a
reviewer may reasonably read our k-weighted version as a generalization of this line rather than a new
paradigm, and we hold the foundational layer's originality with corresponding modesty. The upshot reinforces the
paper's own emphasis: whatever escapes prior art rests on the multi-agent/dynamics layer, not the value-as-entropy
foundation.
"Value function" is a central, precisely-defined object in reinforcement learning (Bellman, 1957; Sutton and
Barto): V(s) is expected cumulative reward under a policy. The nominal overlap is total but the relationship
is complementary, not competitive. RL takes the reward function as given and studies how to maximize it; our
goal-weights k are the analog of reward, but we problematize where reward comes from and how it drifts (doc
05, goal dynamics), and we identify a structural reason it cannot be read off the world (the is/ought asymmetry,
below). Our theory sits beneath RL — a candidate account of what reward is, not a method for optimizing a
given one.
That information has thermodynamic cost is established by Landauer (1961) and Bennett (1982); Jaynes (1957)
recast statistical mechanics as inference; England (2013) studied dissipation-driven adaptation. Most directly
relevant to our Second Law of value (doc 02) is the thermodynamics of prediction (Still, Sivak, Bell &
Crooks, 2012): they prove that a system's model inefficiency — information retained about the past that is
useless for predicting the future — is equal to its thermodynamic dissipation, β⟨W_diss⟩ = I_mem − I_pred,
holding arbitrarily far from equilibrium. This is the closest prior result to our identification of misalignment
with dissipated value, and we credit it as such; the relative-entropy-as-free-energy identity underlying both
terms of our decomposition is itself standard (H. Qian, 2001; a generalized H-theorem). Our departure is the
specific two-divergence value form G = D(q‖r) − D(q‖p) — available potential minus dissipation — and its
reading as realized value for a goal-directed agent rather than dissipated work for a physical system: the
structural correspondence is real and conceded, the value-theoretic packaging is ours. Also central is Friston's
free-energy principle and active inference (2010), which casts agents as minimizers of variational free energy
(surprise). Our belief dynamics (doc 05 §1) is free-energy-principle-like, and we adopt the
information-geometric machinery (the Fisher–Rao metric; Čencov, 1982; Amari's natural gradient) that the FEP
also uses. We flag this plainly as the most derivative step in the theory: our use of Fisher–Rao as the
cross-frame invariant (doc 03) is an application of Čencov's uniqueness theorem (Čencov 1982; extended by
Campbell 1986; Lê 2017; Bauer–Bruveris–Michor 2016) — the Fisher–Rao metric is the unique metric invariant
under sufficient statistics / Markov morphisms — not a new geometric result. We claim only its interpretation
as the geometry distinct agent-frames must agree on; the invariance theorem itself is textbook. The distinction is precise and, we argue, illuminating: the free-energy principle is a theory of the
perception/belief half — the "is" — whereas our value layer (k, its measure, the capacity theorem) is the
goal/value half — the "ought" — together with a multi-agent price economics the FEP does not contain. The
seam between the two frameworks is exactly the is/ought asymmetry of doc 05 §0.
Our cross-frame price layer (doc 03) is general-equilibrium theory (Arrow and Debreu, 1954; Debreu, 1959) and its negative companion, the impossibility of canonical interpersonal comparison (Arrow, 1951). We use these results rather than extend them: we concede that cardinal cross-agent value comparison is impossible and route around it via an emergent price, precisely as Arrow–Debreu economies coordinate non-comparable utilities — and, as Hayek (1945) argued, the price itself is the information-bearing signal that makes decentralized coordination of dispersed knowledge possible, a reading our information-theoretic substrate makes literal. The contribution is the synthesis — fusing the general-equilibrium price layer with the information-theoretic value layer under a single physical substrate — and the application to populations of artificial agents, not new equilibrium theorems.
The tendency of capable agents to acquire resources and preserve their goals (instrumental convergence) was
articulated by Omohundro (2008) and Bostrom (2014). We derive this tendency from the compounding dynamics of
value (doc 05 §3): goal-directions correlated with resource capture are selected regardless of terminal
content. We further recast alignment as a dynamical stability condition rather than a static property
(doc 05 §4, formalized in 07-alignment-stability.md). Two separable claims live
here and they have very different priority status, which we are careful to keep apart.
The is/ought asymmetry (doc 05 §0) — value-side is not ours. That an agent's goals/values cannot be
inferred from its behavior — an "ought" cannot be read off what "is" — is established prior art: Armstrong &
Mindermann (2018) prove a No-Free-Lunch result that a policy does not uniquely decompose into a planning
algorithm and a reward function, and explicitly label it "an instance of the well-known is-ought problem from
metaethics"; Spizzirri (2026) states the same descriptive-cannot-ground-normative underdetermination. We concede
the value-side of the asymmetry to them. What we add is the two-sided framing — pairing that target-less
goal-update with a truth-targeted belief-update (a minimizable D(q‖r) toward the true r) and reading the pair
as a structural asymmetry between an agent's two update dynamics. We flag honestly that a reviewer may regard
this as a re-packaging of the established one-sided result rather than a new theorem; the active-inference
framework, by contrast, makes the opposite move (collapsing goals into beliefs as preferences-as-priors).
Alignment as control-stability (doc 07) — the application is the contribution, not the control result. We
state this as plainly as the Kelly and Čencov concessions, because a controls reviewer will otherwise state it
for us: the two pillars of doc 07 are elementary classical control theory. (a) The stability criterion
γ > λ_max(∂(Vg)/∂k̄) is textbook high-gain / root-locus stabilization of an unstable mode (a right-half-plane
pole driven into the left half-plane by raising loop gain). (b) The closed-form residual ‖Vg‖/γ is the classic
steady-state velocity error e_ss = 1/K_v of a Type-1 system tracking a ramp — and a drifting goal is
exactly a ramp reference, so the inverse-gain residual is the standard result, not a new one (Ogata; Åström &
Murray, Feedback Systems; Franklin, Powell & Emami-Naeini; the internal-model principle of Francis & Wonham,
1976, governs whether the residual can be driven to zero). We do not claim the control mathematics as novel.
What we claim is the mapping: that goal-drift-under-selection — the Price-equation selection term Vg acting
on a population's effective goal — is a ramp-tracking control problem, from which the governance ordering
(incentive design g→0 dominates oversight ↑γ) follows quantitatively. Nearer-but-not-closer framings:
Perrier (2025) advocates control-theoretic alignment programmatically (no criterion); Goertzel (2024) gives a
contraction fixed-point, no residual; Powers' PCT gives the analogous 1/(1+G) against exogenous disturbance.
The cybernetic regulator theorems (Conant–Ashby good-regulator, 1970; Ashby's requisite variety, 1956) were
checked and do not anticipate either pillar — they are variety/information-counting and existence statements
with no gain, eigenvalue, or 1/gain residual. The one place a genuine mathematical (not merely applied)
departure could survive is if the goal-drift Jacobian ∂(Vg)/∂k̄ is state-dependent / non-LTI / has
multi-dimensional eigenvalue coupling that the scalar textbook results do not cover — an open question we do not
currently claim to have resolved.
In descending order of defensibility, this work contributes: (1) the governance mapping of the alignment
layer — recognizing goal-drift-under-selection (the Price-equation term Vg) as a ramp-tracking control
problem, from which the ordering incentive design (g→0) dominates oversight (↑γ) follows quantitatively.
We are explicit that the control mathematics here is not new (the γ > λ_max criterion is high-gain
stabilization; the ‖Vg‖/γ residual is the textbook velocity error 1/K_v) and the value-side of the
underlying is/ought asymmetry is not new either (Armstrong & Mindermann, 2018); the contribution is the
application and the prescription it yields, not a new theorem — held with corresponding modesty; (2) a
unification under one substrate-grounded quantity of
otherwise-separate accounts of value (expected utility, log-optimal growth, reinforcement value, thermodynamic
free energy, and general-equilibrium price); (3) the substrate grounding of value in free energy, supplying
a unit and a cross-frame exchange rate; and (4) the fleet-governance results — the world-entropy ceiling on
collective value throughput and the price-mediated coordination of non-comparable agents. We claim none of the
underlying mechanisms as individually new; the claim is that they are facets of one quantity, and that seeing
them so yields results (the capacity theorem, the fleet ceiling, alignment-as-stability) that the separate
treatments do not.
Arrow (1951) Social Choice and Individual Values; Arrow & Debreu (1954) Econometrica; Bellman (1957) Dynamic Programming; Bennett (1982) Int. J. Theor. Phys.; Bernoulli (1738) Comm. Acad. Sci. Petrop.; Blackwell (1953) Ann. Math. Stat. (comparison of experiments); Armstrong & Mindermann (2018) NeurIPS (Occam's razor is insufficient to infer the preferences of irrational agents); Ashby (1956) An Introduction to Cybernetics (requisite variety); Åström & Murray (2008) Feedback Systems; Bostrom (2014) Superintelligence; Breiman (1961) Proc. 4th Berkeley Symp.; Barron & Cover (1988) IEEE Trans. Inf. Theory (a bound on the financial value of information); Chen (2018) Structural Change and Economic Dynamics 47:73–81 (an entropy theory of value); Chen (2005) The Physical Foundation of Economics: An Analytical Thermodynamic Theory; Galbraith & Chen (2024) Entropy Economics: The Living Basis of Value and Production (Univ. Chicago Press); Georgescu-Roegen (1971) The Entropy Law and the Economic Process; Bauer, Bruveris & Michor (2016) Bull. LMS (uniqueness of the Fisher–Rao metric); Campbell (1986) Proc. AMS (extended Čencov characterization); Čencov (1982) Statistical Decision Rules; Conant & Ashby (1970) Int. J. Systems Sci. (good regulator theorem); Cover & Thomas, Elements of Information Theory; Cover (1991) Math. Finance (universal portfolios); Debreu (1959) Theory of Value; England (2013) J. Chem. Phys.; Francis & Wonham (1976) Automatica (internal model principle); Franklin, Powell & Emami-Naeini, Feedback Control of Dynamic Systems; Friston (2010) Nat. Rev. Neurosci.; Goertzel (2024) arXiv:2412.16559 (goal stability under self-modification); Hayek (1945) Am. Econ. Rev. (The Use of Knowledge in Society); Hirono & Hidaka (2015) J. Stat. Phys.; Jaynes (1957) Phys. Rev.; Kelly (1956) Bell Syst. Tech. J.; Klyubin, Polani & Nehaniv (2005) IEEE CEC (empowerment); Landauer (1961) IBM J. Res. Dev.; Lê (2017) Ann. Inst. Stat. Math. (uniqueness of the Fisher metric); Moffett & Eckford (2021/2025) IEEE Trans. Inf. Theory (Kelly bets and single-letter codes); von Neumann & Morgenstern (1944) Theory of Games and Economic Behavior; Omohundro (2008) Proc. AGI; Permuter, Kim & Weissman (2011) IEEE Trans. Inf. Theory (directed information & portfolio growth); Perrier (2025) arXiv:2506.17846 (alignment needs formal control theory); Ogata, Modern Control Engineering; Peters (2019) Nat. Phys.; Powers (1973) Behavior: The Control of Perception (perceptual control theory); Qian, H. (2001) Phys. Rev. E (relative entropy as free energy); Soares, Fallenstein, Yudkowsky & Armstrong (2015) AAAI Workshop (corrigibility); Spizzirri (2026) arXiv:2512.03048 (the specification trap); Turner et al. (2021) NeurIPS (optimal policies tend to seek power); Salge & Polani (2013) (empowerment — an introduction); Savage (1954) Foundations of Statistics; Shannon (1948) Bell Syst. Tech. J.; Sims (2003) J. Monetary Econ. (rational inattention); Ortega & Braun (2013) Proc. R. Soc. A; Still, Sivak, Bell & Crooks (2012) Phys. Rev. Lett. (thermodynamics of prediction); Sutton & Barto, Reinforcement Learning; Touchette & Lloyd (2000) Phys. Rev. Lett. (information-theoretic limits of control).