Thermodynamic Measure of Intelligence Ishanu Chattopadhyay1, 2, ∗
arXiv:2606.20231v1 [cs.AI] 18 Jun 2026
1
Institute for Biomedical Informatics, University of Kentucky, Lexington, Kentucky, USA 2 Department of Computer Science, University of Kentucky, Lexington, Kentucky, USA (Dated: June 19, 2026)
Can intelligence be measured? We propose that intelligence can be defined as the lawful amplification of rare but valid futures: a system increases the probability of outcomes that would be unlikely under passive dynamics but remain admissible under the constraints of the domain. We start with the premise that an intelligent system must model the world and its own place within it. Because the system is part of the world it models, this leads naturally to recursive self-simulation: the system represents futures in which its own actions are part of the trajectory. Our central results give a necessity statement and a conditional near-sufficiency statement connecting this architecture to a precise thermodynamic measure of lawful amplification of rare-valid futures: high rare-valid lift is impossible unless the internal simulation identifies rare-valid futures with high fidelity; conversely, when rare-valid fidelity is high and the simulation contains an effective policy, the achievable lift approaches the actuation-limited optimum. Thus recursive self-simulation is not merely a plausible feature of intelligence but, under the stated assumptions, is necessary and nearly sufficient for high thermodynamic intelligence. The resulting framework makes intelligence measurable on a universal scale, from passive matter and feedback controllers, large language models, and humans as text generators to Maxwell-demon-like information engines.
I.
INTRODUCTION
Can intelligence be framed as a measurable physical quantity? We start from the observation that any system perceived to be intelligent models the world with itself inside it, simulates possible futures conditioned on its own actions, and uses that internal model to make some futures more likely than they would be under passive dynamics. This suggests that a relevant measurable quantity is rare-valid lift: the increase in probability assigned to futures that are unlikely under a passive baseline but remain valid under the constraints of the domain. Our central result is that high rare-valid lift cannot be obtained merely from randomness or strong actuation. Under bounded amplification, it requires high-fidelity selfsimulation: the system’s internal model must identify rare-valid futures accurately enough to target them. Standard definitions of intelligence emphasize behavior: imitation or conversational indistinguishability [1], learning, reasoning, planning, generalization, compression, reward maximization, or task success [2–4]. These criteria are useful, but they do not by themselves identify a substrate-independent operation common to brains, large language models, microbial communities, immune repertoires, controllers, and idealized information engines such as Maxwell’s demons. All of these systems transform information into action. We therefore ask a different question: what does an intelligent system do to the likelihood of possible futures? This question is related to, but distinct from, existing task-facing accounts of intelligence. Legg–Hutter intelligence defines an agent’s intelligence by its expected
∗ ishanu [email protected]
reward over a universal distribution of computable environments [2]. Chollet’s ARC framework instead emphasizes skill-acquisition efficiency: the ability to infer abstract rules and generalize from sparse experience under human-like priors [4]. These accounts evaluate performance across environments, benchmarks, or problem classes. Our framework is path-facing. We ask what physical or probabilistic operation underlies such performance once a level of description, baseline law, validity criterion, and observational resolution are fixed. Reward maximization, benchmark generalization, theorem proving, symbolic problem solving, and biological adaptation can then be treated as special cases. We begin with recursive self-simulation. A system acts intelligently, in the present sense, when it carries a model of the world that includes itself as a causal object. Such a model can represent possible futures, including the system’s own interventions, the observations those interventions may produce, and the way its own information state may later change. Minsky anticipated this point in his account of internal models: advanced problem solving requires a system to represent its own goals, resources, and problem-solving activity, and self-understanding can involve models of models of oneself [5–7]. Related ideas appear in work on self-reference and strange loops [8–10]. To measure the capabilities enabled by recursive selfsimulation, we use path laws. Passive dynamics induce a baseline distribution P0 over trajectories. A controlled or agent-like system induces another law P . A self-simulation becomes observable through the way it changes these probabilities. Intelligence, viewed this way, is lawful trajectory reweighting: some futures become more likely, others less so, and the change must respect the thermodynamic accounting required by measurement, memory, computation, control, and erasure. Not all trajectory manipulations are the same. The
2
a. Recursive self-simulation
b. Rare-valid lift
base
reality
𝑥(𝑡 ′)
𝑥(𝑡 ′)
fidelity
𝑥(𝑡)
𝑥(𝑡)
b Φ
𝑥𝑟𝑣 (𝑡 ′) Identify rare-valid futures
Lift rare-valid futures
𝑥𝑟𝑣 (𝑡 ′)
∫ I = 1𝛿 𝑉 ( 𝑑𝑃 − 𝑑𝑃0 ) 𝛿
c. Examples on the compressed thermodynamic-intelligence scale
Λ = log10 (log10 (I + 1) + 1)
15
10
5
0
Passive baseline (rock)
Fixedfeedback controller
Repeated control
Sparse velocity demon
GPT-5 symbolic
Human symbolic
Maxwell demon
Velocity demon (1 mm3 air)
FIG. 1. Conceptual summary. (a) Recursive self-simulation: a system represents the world at one level together with models of b measures how accurately the simulation its own future states and actions at higher simulated levels. The rare-valid fidelity (Φ) identifies targetable rare-valid futures. (b) Thermodynamic intelligence: relative to a passive trajectory law R (P0 ), an induced law (P ) shifts probability mass toward rare-valid trajectories (Vδ ), producing rare-valid lift (Iδ = δ −1 V (dP − dP0 )). (c) δ Representative systems on the compressed scale (Λ = log10 (log10 (I + 1) + 1)). The plotted examples are finite-resolution calibrations of probability lift, with symbolic and demon entries interpreted under the assumptions stated in the text.
most informative changes occur in the tail of the trajectory distribution. Moving probability among futures already common under P0 may reflect stabilization or regulation, but it does not strongly test whether the system can reach beyond passive dynamics. Rare futures probe that ability because they would otherwise remain effectively unrealized. Rarity alone, however, is not a measure of intelligence; random noise also produces improbable events. The futures must remain valid; and hence we focus on rare-valid futures: trajectories that have low probability under the passive law but remain admissible under the constraints of the domain. Rarity
supplies counterfactual difficulty; validity, interpreted as physical realizability, biological viability, semantic coherence, executable correctness, or functional success, prevents the measure from rewarding noise. To make this notion precise and computable, we consider the thermodynamics of system trajectories, and we define thermodynamic intelligence as rare-valid probability lift: the fractional increase, under the induced law P , in the probability of an exceedingly rare but valid set. The definition turns the quantification of intelligence into a question about path measures. The thermodynamic machinery needed for this analysis is
3 well developed [11–15], including non-equilibrium fluctuation theorems, which quantify the relative likelihood of entropy-producing and entropy-reducing trajectories under passive dynamics [16–19]. The symbolic case uses the complementary information-theoretic language of entropy rate and coding [20, 21]. We illustrate the framework through examples spanning passive systems with zero lift, simple controllers with modest amplification, symbolic generators, including GPT-5 and human text, to Maxwell-demon-like information engines. Maxwell’s demon provides the canonical historical case: a hypothetical microscopic observer using information about particle states to sort thermal fluctuations and create an apparent local entropy reduction. Here the demon serves as an idealized high-lift limit, where near-perfect microstate simulation, rare-valid identification, and actuation produce extreme rare-valid trajectory amplification before implementation costs are paid. Conversely, the symbolic examples show how the same formalism can be applied at a finite linguistic resolution, once the baseline ensemble, validity criterion, sequence length, and generatorinduced probability shift are specified.
II.
RECURSIVE SELF-SIMULATION
To act with intelligence, a system needs a deep model of its world, with itself in it. Minsky anticipated this point in his account of internal models: advanced problem solving requires a system to represent not only the external situation, but also its own goals, resources, and problem-solving activity, and self-understanding can involve models of models of oneself [5–7]. We formalize this self-in-world requirement as an embedded representation hierarchy. Let B denote an agent-like system embedded in an environment E, and let U = B ∪ E. At a minimal level, the system maintains an internal representation of its observable world, (0)
(0)
rB = rB (U ). (1) Because B ∈ U , a sufficiently general model of the local universe must also represent the system itself: its state, memory, uncertainty, actions, and possible future updates, inducing a recursive hierarchy: rB = rB (rB ),
(1)
(1)
(0)
(2)
(2)
(2)
(1)
(3)
(k)
(k)
(k−1)
rB = rB (rB ), .. .
rB = rB (rB ). (4) The hierarchy encodes predictions about the world, predictions about the system’s own future actions, and predictions about how its information state may change; recursive self-simulation is therefore a finite self-referential loop. If the environment contains other agents Bj , the same idea extends to nested representations: (k) (ℓ) rB rBj , j ̸= B, k, ℓ ≥ 0. (5)
Social and biological environments can therefore generate interacting recursive models. This is the thermodynamic analogue of theory-of-mind style modeling: an agent’s future depends partly on what it predicts other agents will perceive, infer, and do [22, 23]. Recursive self-simulation becomes operational when a system evaluates consequences of its own future actions. Let ht denote the history available at time t, let At denote the available action set, and let Γt:T denote a future trajectory segment. Suppose an agent evaluates a trajectory functional G under its k-level internal model and selects a⋆t ∈ arg max Er(k) [G(Γt:T ) | ht , a] . (6) a∈At
B
Equivalently, the agent may implement a policy πt (· | ht ) concentrated near such maximizing actions. To evaluate this expectation, the model must represent the future environment, the agent’s possible actions, and the agent’s own future information state. Our construction is distinct from predictive-processing and active-inference views of perception and action [24]; our measured quantity is rare-valid probability lift relative to P0 , not free energy itself. The hierarchy of recursive simulation determines which futures the system can identify, evaluate, and target. To connect this architecture to a measurable quantity, we now introduce the level-relative rare-valid lift. Let L0 denote the base reality, or physical level, under consideration. Throughout the paper, I denotes a fractional probability lift of a rare-valid set relative to a passive baseline. At level k, this lift compares the probability assigned to a rare-valid event under an induced or simulated law with its probability under the corresponding passive law. Realized intelligence is the lift actually induced in the level-k path law. Intelligence potential is the largest such lift available inside a level-(k + 1) simulation of level k. Section III gives the corresponding trajectory-space definition and identifies it as thermodynamic intelligence. For k ≥ 0, let Ωk be the trajectory space at level k, let Fk be its σ-algebra, let P0,k be the passive or baseline law, and let ηk be the observational resolution. Let Vδ,k ⊆ Ωk be a measurable rare-valid event at level k with baseline mass δk ≜ P0,k (Vδ,k ) > 0. (7) When the target mass is fixed or clear from context, we write δ rather than δk . Define Lk ≜ (Ωk , Fk , P0,k , Vδ,k , ηk ). (8) For any level-k path law Qk satisfying Vδ,k ∈ Fk , define the level-k rare-valid lift Qk (Vδ,k ) − P0,k (Vδ,k ) Iδ,k (Qk ; P0,k , Vδ,k ) ≜ P0,k (Vδ,k ) (9) Qk (Vδ,k ) − δk = . δk The hierarchy is recursive in the upward direction: Lk+1 is a simulation or model of Lk . We write hatted quanti-
4 Next we have our necessity result: under bounded amplities for the representation of level k inside level k + 1: b b b b b Lk+1→k = (Ωk+1→k , Fk+1→k , P0,k+1→k , Vδ,k+1→k , ηbk+1→k ). fication, high level-relative intelligence potential cannot be obtained from low rare-valid simulation fidelity. Here Vbδ,k+1→k is the level-(k + 1) representation of the Theorem 1 (Rare-valid self-simulation fidelity is neceslevel-k rare-valid set, with simulated baseline mass sary). Work inside the level-(k + 1) simulation of level b b b δk+1→k ≜ P0,k+1→k (Vδ,k+1→k ) > 0. (10) k, and abbreviate Pb0 = Pb0,k+1→k , Pbπ = Pbπ,k+1→k , When no ambiguity is possible, write Pb0 , Pbπ , Vbδ , and δb b = A bk+1→k , δb = Pb0 (Vbδ ) > 0, and Vbδ = Vbδ,k+1→k , A for the corresponding level-(k + 1 → k) quantities. The b let Φ denote the fidelity in Eq. (14). Assume Pbπ ≪ Pb0 . realized rare-valid lift of B at level k, when the system Suppose there exists αmax ≥ 1 such that induces the actual level-k path law PB,k , is dPbπ real b ∩ Vbδ , Iδ,k (B) ≜ Iδ,k (PB,k ; P0,k , Vδ,k ) (ω) ≤ αmax Pb0 -a.e. on A (16) b0 d P (11) PB,k (Vδ,k ) − P0,k (Vδ,k ) = . and P0,k (Vδ,k ) dPbπ real b (ω) ≤ 1 Pb0 -a.e. on Vbδ \ A. (17) If PB,k = P0,k , then Iδ,k (B) = 0 relative to that baseb0 d P line. This does not say the system lacks intelligence; it Then says that intelligence is not realized as a path-law change (k+1→k) b at that level. Ibδ (π) ≤ (αmax − 1)Φ. (18) The intelligence potential for level k is computed inside (k+1→k) b And, if αmax > 1 and for some I0 > 0, Iδ (π) ≥ b k+1→k be the level-(k + 1) simulation of level k. Let Π I0 , then b k+1→k , define the the simulated policy class. For π ∈ Π I0 b≥ Φ . (19) simulated rare-valid lift αmax − 1 b b b P ( V ) − δ π,k+1→k δ (k+1→k) Thus high intelligence potential requires high rare-valid . (12) Ibδ (π) ≜ δb simulation fidelity relative to the available amplification The intelligence potential of B for level k, as represented budget. In particular, if I0 > αmax − 1, no policy satisfyat level k+1, is the supremal simulated rare-valid lift over ing the amplification bound can attain I0 . the policy class available in that representation: (k+1→k) Proof. See Appendix A. pot Iδ,k (π). (13) (B) ≜ sup Ibδ The near-converse requires an implementation assumpb k+1→k π∈Π tion: the simulation must contain a policy that amplifies Thus actuation is not assumed at level k when potential the correctly identified rare-valid region. is computed. The simulated action variables live in Lk+1 ; realization at level k is the separate question of whether a Theorem 2 (Near-sufficiency under effective simulated simulated policy can indeed be implemented as an actual actuation). Use the notation of Theorem 1. Suppose path-law change. there exists a simulated policy π and constants αmin > 1, a. Rare-valid simulation fidelity The relevant fi0 ≤ βmin ≤ αmin such that delity is not generic prediction accuracy. A model may dPbπ predict common trajectories well while missing the rareb ∩ Vbδ , (ω) ≥ αmin Pb0 -a.e. on A (20) dPb0 valid set, or be coarse in irrelevant coordinates while accurate on the rare-valid futures and actions that matter. and We therefore define fidelity directly on the target set. dPbπ b (ω) ≥ βmin Pb0 -a.e. on Vbδ \ A. (21) bk+1→k ⊆ Ω b k+1→k denote the set of trajectories Let A dPb0 identified by the level-(k + 1) simulation as targetable Then rare-valid futures for level k. We define the level-specific (k+1→k) b + βmin (1 − Φ) b − 1. Ibδ (π) ≥ αmin Φ (22) rare-valid self-simulation fidelity b b b b In particular, if 0 ≤ ε ≤ 1 and Φ ≥ 1 − ε, then b k+1→k ≜ P0,k+1→k (Ak+1→k ∩ Vδ,k+1→k ) . (14) Φ (k+1→k) b δk+1→k Ibδ (π) ≥ (αmin − 1) − (αmin − βmin )ε. (23) b b → 1, effective simulated actuation When the represented level is fixed, we write Φ for Therefore, as Φ b k+1→k . The set Vbδ is “true” only relative to the speciΦ drives the intelligence potential toward the actuationfied level-k description and its represented validity critelimited value αmin − 1. b = 1 means that, at the simulated baseline rion. Thus Φ Proof. See Appendix A. resolution, the targetable set covers the represented rareTheorems 1 and 2 are the formal bridge between reb = 0 means that the simulation misses it. The valid set; Φ cursive self-simulation and thermodynamic intelligence. corresponding rare-valid self-simulation error is Low rare-valid fidelity caps the achievable lift; high rareb εbRV (15) k+1→k ≜ 1 − Φ. valid fidelity, together with a policy that amplifies the
5 correctly identified region, yields high lift. These statements concern intelligence potential for level k as computed in the level-(k+1) simulation. Realized intelligence at level k additionally requires implementation as an actual level-k path-law change. III.
THERMODYNAMIC INTELLIGENCE
The previous section described recursive selfsimulation as the internal architecture. We now define the observable: probability lift over rare-valid regions of trajectory space. Because trajectory spaces may be continuous or high-dimensional, rarity is defined at finite observational resolution. Definition 1 (Rare-valid set at finite resolution). Let V ⊂ Ω denote the set of valid trajectories. Validity is domain-dependent. In a physical system, validity means physical admissibility. In a biological system, it may mean viability or functional organization. In a symbolic system, it may mean grammaticality, semantic coherence, factual consistency, and task relevance. Let Πη be a finite measurable partition of Ω at observational resolution η. For a cell C ∈ Πη , P0 (C) is the passive probability of observing a trajectory in that cell. A rare-valid set Vδ,η is a union of valid cells with small passive probability and target passive mass δ. When exact normalization is possible, we choose P0 (Vδ,η ) = δ.
Definition 2 (Thermodynamic intelligence at resolution δ). Let P be the trajectory distribution induced by a system, and let Vδ satisfy P0 (Vδ ) = δ. Define the δ-scale thermodynamic intelligence of P relative to P0 and Vδ as P (Vδ ) − δ P (Vδ ) − P0 (Vδ ) = . δ δ
Equivalently, in density notation, Z 1 Iδ (P ; P0 , Vδ ) = ( dP − dP0 ) . δ Vδ
(25)
(26)
When the limit exists, define I(P ; P0 , V ) = lim+ Iδ (P ; P0 , Vδ ).
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δ→0
If P = P0 , then Iδ = 0. If a system deterministically realizes a rare-valid cell in Vδ , so that P (Vδ ) = 1, then Iδ =
1 1−δ ≈ . δ δ
Lemma 1 (Rare-valid amplification implies path-measure divergence). Let P0 (Vδ ) = δ, let p = P (Vδ ), and suppose 0 < δ < 1. Define Iδ =
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p−δ . δ
(29)
Then DKL (P ∥ P0 ) ≥ d(p ∥ δ),
(30)
where d(p ∥ δ) = p log
1−p p + (1 − p) log δ 1−δ
(31)
is the binary KL divergence. Equivalently, when p = δ(1 + Iδ ) ≤ 1, DKL (P ∥ P0 ) ≥ d(δ(1 + Iδ ) ∥ δ) .
(24)
For a finite partition, exact equality need not hold for every δ. In that case one may either choose an attainable value of δ, use P0 (Vδ,η ) ≤ δ, or obtain exact normalization by randomized inclusion of a boundary cell. Equivalently, Vδ,η may be taken as the lowest-baseline-probability valid region of total passive mass δ, up to this boundary convention. When the resolution η is fixed, we write Vδ for Vδ,η .
Iδ (P ; P0 , Vδ ) =
Thus the measure ranges naturally from zero for passive systems to very large values for ideal information engines that select extremely rare valid trajectories. The definition credits only probability mass moved into futures that are both low-probability under P0 and valid under the domain constraints. In flexible settings, recursive self-simulation should improve both the identification of such futures and the actions that make them more likely. Lemma 1 records the information-theoretic consequence: amplifying a rare-valid set requires path-measure divergence from the passive baseline.
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Proof. See Appendix A. A.
Trajectory-Space Thermodynamics
Rare-valid lift is a change in trajectory probabilities, so its natural thermodynamic setting is path space. Let (Ω, F) denote a measurable space of trajectories ω over [0, τ ]. Let P0 be the passive path measure and PB the path measure induced by an agent B acting through feedback. Let S(ω) denote physical entropy production along ω, and define the dimensionless entropy production S(ω) σ(ω) = . (33) kB − For matched entropy-production bins A+ s and As , where + − As collects trajectories with σ(ω) ≈ +s and As collects the corresponding time-reversed or otherwise matched trajectories with σ(ω) ≈ −s, the passive fluctuationtheorem relation is written at event level as P0 (A+ s ) log ≃ s. (34) P0 (A− s ) Equivalently, in dimensional units, if the matched bins correspond to entropy productions +∆S and −∆S, then the right-hand side is ∆S/kB . The symbol ≃ marks event-level coarse-graining: exact equality requires bins and matching conventions that preserve the underlying trajectory-level fluctuation relation.
6 Now, feedback replaces the passive law P0 by the controlled law PB . Microscopic feedback fluctuation relations generally contain trajectory-dependent measurement and information terms, so a universal scalar correction need not exist after coarse-graining. For the fixed bins used here, we therefore define the event-level information correction directly: PB (A+ s ) Js (B) ≜ s − log . (35) PB (A− s ) Equivalently, PB (A+ s ) log (36) − = s − Js (B). PB (As ) Thus Js (B) records how feedback changes the entropyproduction log-ratio on the chosen bins relative to the passive fluctuation-theorem scale. It is a coarse-grained diagnostic, not a complete thermodynamic balance; measurement, memory, computation, control, and erasure remain part of the full physical accounting. a. Coarse-grained path-deviation stability We next record a stability bound for the entropy-bin signatures just defined. Let P and Q be path measures on (Ω, F). − For matched entropy-production events A+ s and As , define Q(A+ P (A+ s ) s ) ∆s (P, Q) ≜ log . (37) − − log P (As ) Q(A− s ) For P = PB and Q = P0 , write ∆s (B) = ∆s (PB , P0 ). If the passive fluctuation relation holds in the dimensionless convention, then PB (A+ s ) ∆s (B) = log − s, (38) PB (A− s ) with s replaced by ∆S/kB in dimensional units. Assumption 1 (Nondegenerate entropy bins). For the two path measures being compared, there exists ms > 0 such that − + − P (A+ (39) s ), P (As ), Q(As ), Q(As ) ≥ ms . Theorem 3 (Coarse-grained path-deviation bound). Under Assumption 1, √ 2p DKL (P ∥ Q). (40) |∆s (P, Q)| ≤ ms In particular, √ 2p |∆s (B)| ≤ DKL (PB ∥ P0 ). (41) ms Proof. See Appendix A. Theorem 3 gives the thermodynamic role of pathmeasure divergence. Lemma 1 shows that rare-valid amplification requires divergence from the passive law. Theorem 3 shows that such divergence also controls how much coarse-grained entropy-production log-ratios can change on fixed nondegenerate bins. Thus the rarevalid lift is not an isolated score: when a controller reweights trajectory probabilities, the induced change is constrained in the same path-measure geometry that governs coarse-grained thermodynamic signatures. The
bound is intentionally finite-bin and moderate-event; rare-event amplification itself is handled by Lemma 1. b. Auxiliary model-to-control continuity The fidelity theorems above are rare-set results. A separate continuity statement compares coarse-grained thermodynamic signatures induced by nearby path laws. It does not prove high thermodynamic intelligence from high fidelity; it only says that finite-depth controlled laws inherit the entropy-bin signatures of an ideal controlled law when the induced path laws are close. (k) Let PB denote the controlled path law induced by a policy computed from the k-level recursive internal model (k) rB . Let PB⋆ denote the ideal controlled path law induced by the limiting or perfectly faithful recursive model for the same objective and admissible control class. Let εk ≥ 0 denote intervention-relevant prediction error. Assumption 2 (Model-to-control stability). There exist a constant C > 0 and a modulus ρ, with ρ(ε) → 0 as ε → 0, such that (k) DKL PB ∥ PB⋆ ≤ Cρ(εk ). (42) Proposition 1 (Recursive fidelity controls convergence to the ideal thermodynamic signature). Assume modelto-control stability. Suppose the entropy bins are nonde(k) generate under PB and PB⋆ , with lower bound ms > 0. Then √ 2C p (k) ⋆ ∆s (PB , P0 ) − ∆s (PB , P0 ) ≤ ρ(εk ). (43) ms Consequently, if εk → 0, then the finite-depth thermodynamic signature converges to the ideal controlled thermodynamic signature at the rate determined by ρ. Proof. See Appendix A.
B.
Imperfect Rare-Set Identification
The definition above assumes access to the true rarevalid set Vδ . Real agents infer an estimated set Vδ′ , so amplification can be spent on false-positive trajectories that are rare but not valid. We model this protocol-level bookkeeping penalty. a. Perfect identification Assume that P is absolutely continuous with respect to P0 on the true rarevalid set and amplifies that set by a constant likelihood factor α: dP (ω) = α, dP0
ω ∈ Vδ ,
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with P0 (Vδ ) = δ and αδ ≤ 1. Outside Vδ , P is renormalized so that it remains a probability distribution. Substituting into Definition 2 gives Iδ =
αδ − δ P (Vδ ) − P0 (Vδ ) = = α − 1. δ δ
(45)
7 Thus, at fixed resolution δ, α = Iδ + 1. If the limit defining I exists and the amplification factor has a corresponding limiting value, then α = I + 1 in that limit. The local log-likelihood entropy bookkeeping associated with an amplified rare-valid trajectory is dP loc = −kB log α = −kB log(Iδ + 1). Sideal = −kB log dP0 This is a local likelihood-accounting term, not a complete entropy balance without a specified physical protocol. b. Imperfect identification Let the agent identify an approximate rare-valid set Vδ′ instead of Vδ . For the main theorem we analyze the conservative-identification case in which the estimated set contains the true rare-valid set, Vδ ⊆ Vδ′ . (46) This isolates false-positive cost and excludes false negatives. Define Eδ = Vδ′ \ Vδ , perr = P0 (Eδ ). (47) ′ Assume that the agent amplifies Vδ by the same likelihood factor α: dP (ω) = α, dP0
ω ∈ Vδ′ .
(48)
Require the normalization condition α(δ + perr ) ≤ 1, (49) so that the amplified mass assigned to Vδ′ remains compatible with a probability law. Under (46), P (Vδ ) = αδ, hence α = Iδ + 1, and P (Eδ ) = αperr . False-positive correction depends on the physical protocol used to store, test, correct, or erase erroneous assignments. We therefore keep the cost explicit and report both the expected per-trial term and the version normalized per amplified true rare-valid trajectory. Assumption 3 (Error-resolution protocol). For a falsepositive region Eδ with baseline mass perr = P0 (Eδ ), the protocol used to resolve, correct, or erase amplified erroneous assignments has entropy cost kB c(perr ) per unit amplified false-positive mass, where c(p) ≥ 0. The baseline-surprisal Landauer bookkeeping protocol corresponds to c(p) = log(1/p). This convention charges erroneous assignments according to their rarity under the passive baseline P0 , not according to their mass after amplification. Other physical implementations may induce different cost functions. Theorem 4 (Conditional imperfect rare-set identification accounting). Let Vδ be the true rare-valid set with P0 (Vδ ) = δ, and let Vδ′ be the agent’s estimated rare-valid set. Suppose Vδ ⊆ Vδ′ , let perr = P0 (Vδ′ \ Vδ ), assume dP/ dP0 = α on Vδ′ , and assume α(δ + perr ) ≤ 1. Under Assumption 3, the expected false-positive overhead per trial is ∆S err ≜ αkB perr c(perr ). (50)
The corresponding expected protocol-adjusted bookkeeping per trial is S imperfect = −αδkB log α + ∆S err . (51) Equivalently, normalizing by the amplified true rare-valid mass P (Vδ ) = αδ, the protocol-adjusted local bookkeeping per amplified true rare-valid trajectory is perr loc Simperfect = −kB log α + kB c(perr ). (52) δ For the baseline-surprisal Landauer bookkeeping protocol c(p) = log(1/p), 1 ∆S err = αkB perr log , (53) perr and perr 1 loc Simperfect = −kB log α + kB log . (54) δ perr Using α = Iδ + 1, this becomes perr 1 loc Simperfect = −kB log(Iδ + 1) + kB log . δ perr The expected version is S imperfect = −(Iδ + 1)δkB log(Iδ + 1) (55) 1 + (Iδ + 1)kB perr log . perr Proof. See Appendix A. C.
Link to recursive self-modeling resolution
The dependence of perr on recursive model resolution is a statistical learning question, not a thermodynamic law. We therefore state rare-set stability as a hypothesis. In the conservative-identification case, the inferred set covers the true rare-valid set and only the excess falsepositive mass shrinks with model fidelity. (k) Let Vδ denote the rare-valid set inferred from the (k) k-level recursive model rB . Define (k) p(k) \ Vδ . (56) err = P0 Vδ Let εk be the intervention-relevant prediction error from Assumption 2. Hypothesis 1 (Rare-set stability under model convergence). There exists a modulus of continuity ρδ , with ρδ (ε) → 0 as ε → 0, such that p(k) err ≤ ρδ (εk ).
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This hypothesis is nontrivial: small KL model error need not imply small false-positive mass on a rare set unless the rare-valid boundary is stable. A concrete finiteresolution sufficient condition is given in Appendix A 9; it shows that Hypothesis 1 follows from uniform validityscore convergence together with a margin bound on the rare-valid boundary. Corollary 1 (Conditional recursive self-modeling entropy efficiency). Assume the baseline-surprisal Landauer bookkeeping protocol and Hypothesis 1. Suppose
8 (k)
(k)
(k)
Vδ ⊆ Vδ , (Iδ + 1)(δ + perr ) ≤ 1, perr < e−1 , and ρδ (εk ) < e−1 . Define the expected excess bookkeeping penalty (k)
∆S err ≜ S k + (Iδ + 1)δkB log(Iδ + 1).
(58)
Then (k)
0 ≤ ∆S err = (Iδ + 1)kB p(k) err log
1 (k) perr
(59) 1 . ρδ (εk ) Equivalently, normalized per amplified true rare-valid trajectory, ≤ (Iδ + 1)kB ρδ (εk ) log
loc 0 ≤ ∆s(k) err ≜ Sk + kB log(Iδ + 1) kB (k) kB 1 1 = perr log (k) ≤ ρδ (εk ) log . δ δ ρδ (εk ) perr
(60)
Consequently, (k)
εk → 0 =⇒ p(k) ∆S err → 0, ∆s(k) err → 0 =⇒ err → 0, and therefore Skloc → −kB log(Iδ + 1) (61) from above. Proof. See Appendix A. Together, model-to-control stability and rare-set stability give the intended chain: recursive fidelity improves the controlled law, improves rare-set identification, and reduces false-positive bookkeeping waste. Amplification power and rare-set resolution remain distinct: a powerful but poorly resolved controller can amplify wrong futures, while a high-resolution but weakly actuating model can identify rare-valid futures without making them likely. IV. A.
INTELLIGENCE CALCULATIONS Intelligence of Maxwell’s Demons
Maxwell’s demon is the canonical limiting case of realized thermodynamic intelligence. In level-relative terms, the demon carries an effectively perfect simulation of the relevant gas microstate and the consequences of its gate action. It observes, stores, and acts to separate particles by velocity. Under passive dynamics, a temperature gradient is exponentially unlikely; under demon-assisted dynamics, the trajectory can become likely or deterministic. The demon is not a free entropy machine, but an idealized upper bound: near-perfect rare-valid fidelity, target identification, and actuation at the measured level. Let ∆S > 0 denote the magnitude of a local entropy reduction in the gas. For a matched entropy-producing trajectory, the fluctuation-theorem scale gives ∆S P0 (+∆S) ∼ exp . (62) P0 (−∆S) kB Equivalently, ∆S P0 (−∆S) ∼ P0 (+∆S) exp − . (63) kB
If the positive-entropy counterpart has order-one probability at the chosen coarse-graining, then the entropyreducing event has passive probability on the scale exp(−∆S/kB ). An ideal demon that realizes it with probability near one therefore has amplification ∆S ∆S/kB I + 1 ∼ exp , log10 (I + 1) = . (64) kB log 10 a. Illustrative entropy-reduction calculation Suppose ∆S = 10−19 J/K. Since kB = 1.380649×10−23 J/K, we obtain 10−19 ∆S ≈ 7243. (65) ≈ kB 1.380649 × 10−23 Therefore, at an O(1) positive-counterpart coarsegraining, P0 (−∆S) ∼ e−7243 ≈ 10−3146 , I + 1 ∼ e7243 ≈ 103146 . Thus an ideal demon that deterministically realizes this trajectory has a thermodynamic-intelligence scale on the order of I ∼ 103146 . This should be read as a fluctuationtheorem scale calculation, not as a complete microscopic probability model for a particular gas protocol. b. Velocity-selection demon Consider a classical ideal gas of N particles at temperature T . Let the demon select particles whose kinetic energy exceeds a threshold mv 2 ϵ = 2kBcT . For a three-dimensional Maxwell–Boltzmann gas, the dimensionless kinetic energy x = mv 2 /(2kB T ) follows a Gamma distribution with density 2 f (x) = √ x1/2 e−x . (66) π The fraction of particles above threshold is √ Γ(3/2, ϵ) 2 √ −ϵ q(ϵ) = P(x > ϵ) = = erfc( ϵ) + √ ϵe . Γ(3/2) π The conditional mean dimensionless energy above threshold is Γ(5/2, ϵ) E[x | x > ϵ] = . (67) Γ(3/2, ϵ) The excess dimensionless kinetic energy above the thermal mean 3/2 is Γ(5/2, ϵ) 3 − . (68) ϕ(ϵ) = Γ(3/2, ϵ) 2 The following is a local Q/T -scale estimate, not a full entropy accounting. Interpreting the selected particles’ excess kinetic energy as the heat scale sorted at ambient temperature T , the dimensionless local entropy-reduction proxy is |∆S| N ≈ q(ϵ)ϕ(ϵ). (69) kB 2 Here the factor 1/2 corresponds to the idealized oneshot protocol in which the demon acts on one half of the chamber. Therefore N log10 (I + 1) ≈ q(ϵ)ϕ(ϵ). (70) 2 log 10 c. Concrete numerical instance (ambient nitrogen) To make the velocity-selection estimate explicit, con-
9 TABLE I. Concrete velocity-selection demon estimates for nitrogen-like air at T = 300 K, using the corrected threedimensional Maxwell–Boltzmann tail. Nsel is the selectedparticle count in the one-shot half-chamber protocol, Qexcess is the local sorted excess kinetic-energy scale for 1 mm3 of air, and L = log10 (I + 1). Scaled columns report the indicated common multipliers. Values are local amplification-scale estimates before measurement, memory, control, and erasure costs. ϵ
vc (m/s)
2 3 4 5
597 731 844 944
3
q
ϕ
3
3
1 mm mm Q1excess L1 mm 100 L100 Nsel14 Nsel (10 ) (10−6 J) (1014 )
0.261 1.652 13.1 9.38 0.112 2.615 5.58 6.34 0.0460 3.593 2.30 3.59 0.0186 4.578 0.928 1.85
31.9 13.6 5.61 2.27
21.8 14.8 8.35 4.30
22.9 15.5 8.76 4.50
sider nitrogen-like air at T = 300 K, with molecular mass mN2 ≈ 4.65 × 10−26 kg. The cutoff speed corresponding to the dimensionless kinetic-energy threshold ϵ is s 2kB T ϵ vc (ϵ) = . (71) mN2
information-theoretic language of entropy rate, coding, and sequence likelihood [20, 21]. Let X1:n = (X1 , . . . , Xn ) be a symbolic sequence over a finite alphabet Σ. For a prompt or task context y, let P0 (· | y) denote a baseline symbolic distribution on Σn , and let PG (· | y) denote the distribution induced by generator G. The baseline may be an n-gram model, a low-order Markov model, a prompt-independent language model, a lower-capacity reference generator, or a fixed decoding policy. Let Vn (y) ⊆ Σn be the valid outputs for context y. Validity is task-dependent—syntactic well-formedness, semantic coherence, factual consistency, entailment, executable correctness, biological admissibility, or task relevance—and must be measurable under both P0 (· | y) and PG (· | y).
1.
Symbolic rare-valid amplification
A rare-valid symbolic set must be defined by baseline mass, not merely by a pointwise likelihood cutoff. Fix 0 < δ < 1. Let Rδ,n (y) ⊆ Vn (y) be a valid set whose baseline probability is δ: X P0 (Rδ,n (y) | y) = P0 (x | y) = δ. (72)
At T = 300 K, this gives cutoff speeds of approximately 597, 731, 844, and 944 m/s for ϵ = 2, 3, 4, and 5, respectively. The excess kinetic energy sorted by the demon in the one-shot half-chamber protocol is x∈Rδ,n (y) N When exact equality is not attainable because Σn is Qexcess ≈ q(ϵ)ϕ(ϵ)kB T 2 discrete, one may use the nearest attainable mass, use Qexcess Qexcess P (R (y) | y) ≤ δ, or randomize inclusion of a boundary N ⇒ ≈ kB q(ϵ)ϕ(ϵ), and log10 (I + 1) ≈ . 0 δ,n T 2 kB T log 10 element to obtain exact mass. A natural construction At one atmosphere and 300 K, an ideal gas has density is to choose Rδ,n (y) from the lowest-baseline-probability 2.44 × 1025 m−3 , so 1 mm3 contains N = 2.44 × 1016 valid outputs until total baseline mass δ is reached. particles. Table I reports the selected-particle count Nsel , The symbolic thermodynamic intelligence of generator sorted excess-energy scale Qexcess , and L = log10 (I + 1). G, at length n, context y, and rare-valid mass δ, is The scale is extreme because the same selection rule PG (Rδ,n (y) | y) − P0 (Rδ,n (y) | y) acts on many microscopic degrees of freedom. For N = Iδ,n (G | y) = 3 P0 (Rδ,n (y) | y) 100, log10 (I + 1) ≈ 1.85–9.38. In 1 mm of air, it acts (73) PG (Rδ,n (y) | y) on 1014 –1015 eligible particles; the sorted excess energy is = − 1. only O(10−5 ) J, but corresponds to roughly 1015 thermalδ scale selections. Equivalently, Scaling from 1 mm3 to 1 cm3 multiplies Nsel , Qexcess , PG (Rδ,n (y) | y) = δ (1 + Iδ,n (G | y)) . (74) and L by 103 , leaving vc , q(ϵ), and ϕ(ϵ) unchanged. The Thus I = 0 when the generator matches the baseline δ,n local entropy reduction must still be compensated by on the rare-valid set; positive values indicate amplificameasurement, memory, control, and erasure costs; the tion and negative values suppression. For a distribution table gives idealized local amplification capacity, not a µ over prompts or tasks, second-law violation. µ Iδ,n (G) = Ey∼µ [Iδ,n (G | y)] . (75) The symbolic definition inherits the binary coarseB. Symbolic Generation: Human and LLM Text graining bound from the path-space theory. Let pG (y) = PG (Rδ,n (y) | y) = δ (1 + Iδ,n (G | y)) . (76) Many intelligent systems emit symbolic sequences, inThen data processing for KL divergence under the binary cluding human speech or writing and LLM token streams. partition {Rδ,n (y), Rδ,n (y)c } gives Even when direct thermodynamic work on the environDKL (PG (· | y) ∥ P0 (· | y)) ≥ d(pG (y) ∥ δ), (77) ment is unobserved, symbolic output provides an observp 1 − p able trajectory. The symbolic case therefore gives a finite where d(p ∥ δ) = p log + (1 − p) log . δ 1−δ empirical version of the rare-valid framework, using the
10 Equivalently, DKL (PG (· | y) ∥ P0 (· | y)) (78) ≥ d (δ(1 + Iδ,n (G | y)) ∥ δ) . Thus symbolic rare-valid amplification requires measurable divergence from the baseline symbolic law.
2.
mass into invalid or incoherent strings. Therefore val Iδ,n (GT | y) ̸≡ Hn (GT | y). (86) The empirical prediction is an intermediate optimum: low temperature is valid but generic, high temperature is rare but often invalid, and the validity-weighted score peaks when outputs are both rare under P0 and valid.
Validity-weighted operational estimator 4.
Empirical validity is often graded. Let v(x, y) ∈ [0, 1] be a calibrated validity score and define wδ,n (x, y) = v(x, y) 1{0<P0 (x|y)≤qδ (y)} , (79) where qδ (y) is chosen so that the baseline weighted mass is X wδ,n (x, y)P0 (x | y) > 0. (80) Z0,δ (y) = x∈Σn
The validity-weighted symbolic intelligence is then P n wδ,n (x, y) (PG (x | y) − P0 (x | y)) val . Iδ,n (G | y) = x∈Σ Z0,δ (y) (81) This reduces to the hard rare-valid definition when v is an indicator of Rδ,n (y), and it penalizes entropy inflation because rare invalid outputs receive little or no weight. For a concrete task family, let D+ contain valid prompt–output pairs and D− invalid, corrupted, or adversarial outputs. Train a calibrated classifier cθ (x, y) ∈ [0, 1] estimating validity. For threshold τ , define Vn,τ (y) = {x ∈ Σn : cθ (x, y) ≥ τ }. (82) Given P0 (· | y), choose a rare-tail threshold qδ (y) so that Rδ,n,τ (y) = {x ∈ Σn : cθ (x, y) ≥ τ, 0 < P0 (x | y) ≤ qδ (y)}, with P0 (Rδ,n,τ (y) | y) = δ, (83) up to the finite-alphabet boundary convention. The corresponding hard-threshold symbolic intelligence is PG (Rδ,n,τ (y) | y) − 1. (84) Iδ,n,τ (G | y) = δ Here cθ may combine grammaticality, entailment, and task relevance. For executable reasoning, validity can instead be defined by a parser, unit tests, proof checker, or verifier.
3.
Temperature as a negative control
Let GT denote an LLM decoded at temperature T . The entropy of the generated symbolic distribution is X Hn (GT | y) = − PGT (x | y) log PGT (x | y). (85) x∈Σn
The validity-weighted rare-tail score is P n wδ,n (x, y) (PGT (x | y) − P0 (x | y)) val Iδ,n (GT | y) = x∈Σ . Z0,δ (y) Entropy and rare-valid lift need not be monotone. Increasing temperature may raise Hn (GT | y) while shifting
Set-level lift and symbolic scale calculations
For the hard rare-valid set Rδ,n (y), define the set-level log-lift PG (Rδ,n (y) | y) λδ,n (G | y) = log . (87) P0 (Rδ,n (y) | y) Since P0 (Rδ,n (y) | y) = δ, this gives the exact identity λδ,n (G | y) = log (1 + Iδ,n (G | y)) . (88) Equivalently, Iδ,n (G | y) = exp (λδ,n (G | y)) − 1. (89) This set-level identity replaces an informal pointwise average log-lift. The latter, PG (x | y) Ex∼PG (·|Rδ,n (y),y) log (90) P0 (x | y) is generally unequal to λδ,n (G | y), except when PG (x | y)/P0 (x | y) is nearly constant over Rδ,n (y). For sequence length n, define the per-symbol structured log-lift in bits by λδ,n (G | y) . (91) ∆ℓδ,n (G | y) = n log 2 Then log10 (Iδ,n (G | y) + 1) = n ∆ℓδ,n (G | y) log10 2. (92) This is a structured log-lift decomposition, not an ordinary entropy-rate statement. The latter becomes relevant only after an additional finite-resolution AEP approximation has been specified.
5.
Sentence-scale human and AI estimates
We now instantiate the symbolic calculation at the sentence scale. Louwerse’s simplified combinatorial estimate [26] gives a finite-resolution ensemble of interpretable English sentences of roughly 3–20 words on the order of NV ≜ |Vn | ≈ 5 × 1021 . (93) This is not a universal linguistic constant; it is a usable baseline cardinality for the present finite-resolution calculation. To define a comparable rare-valid target, let G⋆n ⊂ Vn denote sentence-scale strings that satisfy an additional high-quality human predicate: literary force, explanatory compression, originality, memorability, or comparable semantic/aesthetic force. We estimate its cardinality by taking a broad human canon of B⋆ ∼ 2 × 103 high-quality long-form works and S⋆ ∼ 5 × 103 sentence-
11 scale units per work, NG ≜ |G⋆n | ≈ B⋆ S⋆ ∼ (2 × 103 )(5 × 103 ) = 107 . (94) The scale of this assumption is conservative relative to large public-domain corpora; the Standardized Project Gutenberg Corpus, for example, contains more than 5 × 104 books and more than 3 × 109 word tokens [27]. We treat NG = 106 –108 as a sensitivity range. Taking the passive symbolic baseline to be approximately uniform over Vn , the baseline mass of the exemplary human-quality set is NG 107 = 2 × 10−15 . δ⋆ = ≈ (95) NV 5 × 1021 If qG = PG (G⋆n ) is the probability that generator G lands in this target set under the specified task condition, then qG NV IG + 1 = = qG . (96) δ⋆ NG For an expert human process conditioned on producing exemplary sentence-scale text, qH ≈ 1, giving 5 × 1021 = 5 × 1014 . (97) IH + 1 ≈ 107 Equivalently, LH = log10 (IH + 1) = 14.699 (98) ΛH = log10 (LH + 1) = 1.196. (99) For the LLM estimate we use only one machine generator and one human reference corpus: Gutenberg prose and GPT-5 long-form prose. The entropy-rate estimates are generated by the self-contained procedure in Appendix B. Both corpora are mapped to a common 27symbol alphabet, consisting of the letters a–z and space, with punctuation, digits, and non-ASCII characters removed. The resulting mean entropy rates are HH = 0.77 bits/character (for Gutenberg prose), (100) HGPT5 = 0.74 bits/character. (101) Since Louwerse’s valid-sentence estimate is a sentencescale count, the entropy-rate correction must use a character-scale length. We take a 20-word sentence-scale unit to have n⋆ = 100 characters after the same coarsegraining. The correction below is an AEP-style supportsize approximation, not an exact finite-length theorem. More precisely, we write qGPT5 log2 = n⋆ (HGPT5 − HH ) + ρn⋆ , (102) qH where ρn⋆ collects finite-length typical-set error, boundary effects of the rare-valid set, and the fact that entropyrate support size is only a proxy for overlap with the exemplary-output subset. The numerical value below sets ρ100 = 0, and should be read as a central orderof-magnitude estimate. With n⋆ = 100, this gives qGPT5 ≈ 2100(0.74−0.77) = 2−3 = 0.125. (103) qH Combining (97) and (103) gives the central estimate IGPT5 + 1 ≈ (5 × 1014 )2100(0.74−0.77) = 6.25 × 1013 . (104)
Thus LGPT5 = log10 (6.25 × 1013 ) = 13.796 (105) ΛGPT5 = log10 (LGPT5 + 1) = 1.170. (106) Since HGPT5 < HH , the sentence-scale choice n⋆ = 100 is conservative relative to longer independently composable symbolic units: increasing n⋆ would decrease the GPT-5 estimate, provided the entropy-rate gap persists and the rare-valid construction is consistently extended. 6.
Algorithmic-complexity interpretation
The symbolic construction has an algorithmicstatistics interpretation, but not a direct estimator. Kolmogorov complexity is uncomputable and machinedependent up to additive constants [28]; the useful point is structural: algorithmic statistics separates the description of a model class from the index of an object inside it [29]. Let K(x) denote the prefix Kolmogorov complexity of a finite string x. A two-part code describes x through a finite set or model class S ∋ x: K(x) ≲ K(S) + log |S|. (107) Here K(S) describes the regularity class and log |S| indexes x within it. Thus high-complexity strings need not be noise; they may lie in rich valid classes. The target is not raw unpredictability, but probability mass assigned to rare strings that remain valid under task constraints. Levin’s coding theorem relates universal a priori probability m(x) to Kolmogorov complexity [30]: K(x) = − log m(x) + O(1). (108) This gives an algorithmic analogue of symbolic rarity. If M is a universal semimeasure or computable approximation, normalize it over Σn by M (x) Mn (x) = P . (109) z∈Σn M (z) For an algorithmic rare-valid set Aδ,n (y) ⊆ Vn (y) satisfying Mn (Aδ,n (y)) = δ, define PG (Aδ,n (y) | y) K Iδ,n (G | y) = − 1. (110) δ Then the same binary KL bridge gives K DKL (PG (· | y) ∥ Mn ) ≥ d δ(1 + Iδ,n (G | y)) ∥ δ . (111) For a specified task distribution and baseline, larger symbolic thermodynamic intelligence means more probability mass on valid strings that are rare under that baseline. Empirical comparison therefore requires finite baselines, validity functions, and prompt distributions. V.
DISCUSSION
The central insight in this work is that perceived intelligence is measurable by what a system does to the probability distribution over possible futures. The architectural claim is that intelligence requires recursive
12 TABLE II. Numerical thermodynamic-intelligence scale sorted from small to large. The table reports the approximate rarevalid lift I and the stabilized double-log scale Λ = log10 (log10 (I + 1) + 1). Rows are ordered by the midpoint of the underlying L = log10 (I + 1) range when a range is reported. Symbolic entries are central finite-resolution estimates; the GPT-5 entry sets the finite-length AEP correction ρ100 = 0. Demon entries are local amplification scales before measurement, memory, control, and erasure costs. I
Example / regime
Calculation basis
Passive matter / passive gas
Baseline dynamics, P = P0 .
0
0
Narrow fixed-feedback controller
Constant rare-valid lift, P (Vδ ) = αδ, with α = 2–102 .
1–99
0.114–0.477
Repeated dynamic controller
Sequential binary lift over 7–10 controlled stages, I + 1 ≈ 27 –210 . Maxwell–Boltzmann velocity selection with N = 100 particles and ϵ = 2–5. Central sentence-scale rare-valid lift with NV = 5 × 1021 , NG = 107 , n⋆ = 100, HH = 0.77, HGPT5 = 0.74 bits/character, and ρ100 = 0. Same sentence-scale calculation with qH ≈ 1: IH + 1 = NV /NG . Fluctuation-theorem scale with ∆S/kB = 7242.97.
1.27 × 102 –1.02 × 103
0.493–0.603
7.0 × 101 –2.4 × 109
0.455–1.016
6.25 × 1013
1.170
5.0 × 1014
1.196
∼ 103146
3.498
Maxwell–Boltzmann velocity selection with N = 2.44 × 1016 and ϵ = 2–5.
104.50×10 – 15 102.29×10
Sparse velocity-selection demon GPT-5 symbolic generation
Expert human symbolic generation Maxwell demon with ∆S = 10−19 J/K Velocity-selection demon in 1 mm3 air
self-simulation: a system models a world in which it is itself an acting component, evaluates action-conditioned futures, and uses that model to select interventions. The operational claim is that this architecture becomes measurable as rare-valid probability lift: amplification of futures that were unlikely under a passive baseline but remain valid under the constraints of the domain. The main mathematical claim is that these two ideas are not merely associated. Under bounded amplification, high rare-valid lift requires high rare-valid fidelity in the system’s self-simulation, and high fidelity is nearly sufficient when an effective amplifying policy is available. This perspective is adjacent to, but distinct from, several existing formalisms. Legg–Hutter intelligence measures expected reward over a universal distribution of computable environments, whereas the present quantity measures probability lift of a specified rare-valid path set relative to a specified passive law [2]. Chollet’s ARC framework emphasizes skill-acquisition efficiency and abstraction from sparse examples, whereas the present framework asks what path-law operation such success corresponds to once the baseline, validity criterion, and resolution are fixed [4]. Free-energy and active-inference formulations describe perception and action through variational free-energy minimization; here the measured object is not free energy itself but the induced reweighting of rare-valid trajectories [24]. The closest thermodynamic relative is semantic information, in which information is meaningful when it is causally necessary for a system to maintain viability under counterfactual interventions [25]. In contrast, rare-valid lift measures how much an induced law amplifies valid low-baseline-probability futures, and Theorems 1 and 2 connect that amplification to rare-valid simulation fidelity. Thus the present contribution is not rarity, validity, self-modeling, or feedback
Λ
14
14.653– 15.360
thermodynamics in isolation [13, 14], but their combination into a level-relative path-measure definition with explicit fidelity bounds. This formulation makes intelligence a level-relative measurement. A claim of intelligence is not made relative to an inaccessible absolute reality, but relative to a specified level of description, baseline path law, validity criterion, and observational resolution. At level k, realized thermodynamic intelligence means that the system actually changes the level-k path law. Thermodynamic intelligence potential for level k is computed inside a level-(k + 1) simulation of level k, where counterfactual actions can be evaluated even if they are not implemented at level k. This distinction separates what a system can identify in simulation from what it actually makes more probable in the measured world. The numerical examples calibrate the measure rather than define a taxonomy. Table II reports the stabilized double-log scale Λ = log10 (log10 (I + 1) + 1), which allows passive systems, feedback controllers, symbolic generators, and idealized information engines to be placed on the same probability-lift scale. Passive matter has zero lift by construction. Simple feedback produces modest lift; repeated dynamic control compounds small gains; symbolic generators can amplify valid lowbaseline-probability sequences; and Maxwell-demon-like systems occupy the high end because microscopic information is used to select rare thermodynamic trajectories. The 1 mm3 velocity-selection demon is large because the same selection rule is applied across 1014 –1015 eligible particles. These are local amplification scales before full measurement, memory, control, and erasure costs are paid. The formal results separate the ingredients of the theory. Theorems 1 and 2 provide the central bridge from
13 recursive self-simulation to thermodynamic intelligence: rare-valid fidelity is necessary under bounded amplification and nearly sufficient with effective simulated actuation. Lemma 1 shows that rare-valid amplification entails path-measure divergence from the passive baseline. Theorem 3 and Proposition 1 relate path-law changes to coarse-grained thermodynamic signatures. Theorem 4 gives a protocol-dependent bookkeeping penalty for imperfect rare-set identification. Together, these results distinguish simulation fidelity, amplification power, thermodynamic accounting, and implementation. Empirical use of Iδ requires explicit choices of baseline, validity criterion, level of description, and trajectory resolution. These choices are not defects of the framework; they are the conditions under which the measurement is meaningful. Poor baselines can inflate or suppress measured lift, and continuous high-dimensional rare sets require statistical regularity conditions such as finite partitions, margin assumptions, or large-deviation structure. Natural testbeds include symbolic generation with executable or semantic checkers, closed-loop control with known passive dynamics, biological sequence evolution under viability constraints, and synthetic Maxwelldemon-like information engines with explicit measurement and erasure costs.
VI.
CONCLUSION
Intelligence can be treated as recursive self-simulation made observable through lawful amplification of rarevalid futures. A system is intelligent, in this sense, when it models a world containing itself, evaluates actionconditioned futures, and shifts probability mass toward futures that were rare under a specified passive baseline but remain valid. The central result is that high lift cannot be obtained from randomness or actuation alone: under bounded amplification it requires high rare-valid selfsimulation fidelity, and with effective simulated actuation that fidelity yields lift near the actuation-limited optimum. The framework is level-relative, thermodynamically accounted, and applicable across passive systems, feedback controllers, Maxwell-demon-like information engines, and symbolic generators once the level, baseline law, validity criterion, trajectory resolution, and induced probability shift are specified.
DATA AND CODE AVAILABILITY
The data, parameter files, and scripts used to generate the numerical calibration results reported in this manuscript are available through Harvard Dataverse at https://doi.org/10.7910/DVN/F5TGT3. The accompanying public GitHub repository, https: //github.com/zeroknowledgediscovery/tme, contains the reproducibility code used to generate the values in Fig. 1, Table I, Table II, and Appendix C.
The GPT–human entropy-rate values used as symbolic inputs are not re-estimated in the TME repository; they are imported as documented input constants from the workflow in https://github.com/ zeroknowledgediscovery/nero, with provenance specified in the repository metadata and with the estimation protocol described in Appendix B.
ACKNOWLEDGMENTS
This work was supported by the Defense Advanced Research Projects Agency (DARPA) under the MAGICS program, DARPA-EA-25-02-05-MAGICS-PA-025, Award No. HR0011-26-3-E016. The views, opinions, and conclusions expressed in this work are those of the author and do not necessarily represent the official position or policy of DARPA or the U.S. Government.
14 Appendix A: Proofs and Technical Qualifications 1.
Proof of Theorem 1
Since Pbπ ≪ Pb0 , the likelihood-ratio assumptions give b + Pb0 (Vbδ \ A). b Pbπ (Vbδ ) ≤ αmax Pb0 (Vbδ ∩ A) (A2) b − Φ). b = δ(1 b Pb0 (Vbδ \ A)
(A3)
Therefore h i b . Pbπ (Vbδ ) ≤ δb 1 + (αmax − 1)Φ
a a0 − log b b0 ≤ | log a − log a0 | + | log b − log b0 |.
|∆s (P, Q)| = log
b and Φ b δ, b for Proof. For readability, write Pb0 , Pbπ , Vbδ , A, the corresponding level-(k + 1 → k) quantities. Decompose the simulated rare-valid mass as b + Pbπ (Vbδ \ A). b Pbπ (Vbδ ) = Pbπ (Vbδ ∩ A) (A1)
By definition, b = δbΦ, b Pb0 (Vbδ ∩ A)
Then
Pbπ (Vbδ ) − δb (k+1→k) b ≤ (αmax − 1)Φ. (A5) Ibδ (π) = δb (k+1→k) If αmax > 1 and Ibδ (π) ≥ I0 > 0, then Eq. (18) implies b I0 ≤ (αmax − 1)Φ, (A6)
|∆s (P, Q)| ≤
1 (|a − a0 | + |b − b0 |) . ms
|a − a0 | + |b − b0 | ≤ 2 TV(P, Q).
(A15)
By Pinsker’s inequality, r
1 DKL (P ∥ Q). 2
(A16)
2p DKL (P ∥ Q). ms
(A17)
TV(P, Q) ≤ Therefore, √ |∆s (P, Q)| ≤
Proof of Theorem 2
Taking P = PB and Q = P0 gives (41).
Proof. Use the same abbreviations as in the previous proof. The lower likelihood-ratio assumptions give b + βmin Pb0 (Vbδ \ A). b Pbπ (Vbδ ) ≥ αmin Pb0 (Vbδ ∩ A) (A7) b − Φ), b = δbΦ b and Pb0 (Vbδ \ A) b = δ(1 b we Using Pb0 (Vbδ ∩ A) obtain h i b + βmin (1 − Φ) b . Pbπ (Vbδ ) ≥ δb αmin Φ (A8)
4.
Proof of Proposition 1
Proof. By definition, (k)
(k)
∆s (PB , P0 ) − ∆s (PB⋆ , P0 ) = log
Therefore
PB (A+ s ) (k)
PB (A− s )
− log
PB⋆ (A+ s ) − ⋆ PB (As )
(k)
Pbπ (Vbδ ) − δb (k+1→k) b + βmin (1 − Φ) b − 1. ≥ αmin Φ Ibδ (π) = δb (A9) b ≥ 1 − ε, then the right-hand side is minimized over If Φ b b = 1 − ε whenever αmin ≥ βmin . This Φ ∈ [1 − ε, 1] at Φ yields (k+1→k) Ib (π) ≥ αmin (1 − ε) + βmin ε − 1 (A10) δ
= (αmin − 1) − (αmin − βmin )ε.
(A11)
□
3.
= ∆s (PB , PB⋆ ). (k)
Applying Theorem 3 with P = PB and Q = PB⋆ gives √ q 2 (k) (k) ⋆ ∆s (PB , P0 ) − ∆s (PB , P0 ) ≤ DKL (PB ∥ PB⋆ ). ms Using (42) yields (43).
5.
Proof of Lemma 1
Proof. Apply the data-processing inequality for KL divergence to the binary coarse-graining {Vδ , Vδc }. The induced Bernoulli laws have success probabilities p = P (Vδ ) and δ = P0 (Vδ ). Therefore
Proof of Theorem 3
DKL (P ∥ P0 ) ≥ d(p ∥ δ).
Proof. Let a = P (A+ s ),
(A14)
The two event probabilities are components of the coarsegrained distributions induced by the partition containing − A+ s , As , and the complement. Hence
and rearrangement gives Eq. (19). □
2.
(A13)
Since all four probabilities are at least ms , the logarithm is 1/ms -Lipschitz on this interval, so
(A4)
(k+1→k) Substituting into the definition of Ibδ (π) gives
(A12)
b = P (A− s ),
a0 = Q(A+ s ),
b0 = Q(A− s ).
Substituting p = δ(1 + Iδ ) gives (32).
15 6.
Proof of Theorem 4
Proof. Because Vδ ⊆ Vδ′ and dP/ dP0 = α on Vδ′ , the true rare-valid mass is P (Vδ ) = αδ. Hence Iδ = (αδ − δ)/δ = α − 1. The false-positive region Eδ = Vδ′ \ Vδ has baseline mass perr , so its amplified controlled mass is P (Eδ ) = αperr . By Assumption 3, resolving this amplified erroneous mass costs kB c(perr ) per unit amplified false-positive mass. Therefore the expected overhead is ∆S err = αkB perr c(perr ). The expected ideal bookkeeping contribution from the amplified true rare-valid mass is −αδkB log α, giving (51). Dividing (51) by P (Vδ ) = αδ gives the local per-amplified-true-rare-valid expression (52). The Landauer forms follow by setting c(p) = log(1/p), and the substitution α = Iδ + 1 follows from the conservativeidentification identity above.
The protocol-dependent form in the main text keeps this choice explicit. Similarly, the path-deviation bound in Theorem 3 is a moderate-event stability bound because of the factor 1/ms ; rare-valid amplification is handled separately through Lemma 1.
9.
A sufficient condition for rare-set stability
Hypothesis 1 is a statistical regularity condition, not a thermodynamic law. Here we record a concrete finiteresolution setting in which it holds. Fix an observational partition Πη and a rare-tail region Rδ,η , chosen under the passive law P0 . Let s : Πη → R be a cell-level validity score and let τ be a validity threshold, so that the true rare-valid set at this resolution is [ Vδ,η = C. (A21) C∈Πη : C⊆Rδ,η , s(C)≥τ
7.
Proof of Corollary 1
Proof. For 0 < p < e−1 , the function g(p) = p log p1 is
Let sk : Πη → R be the validity score induced by the k-level recursive model, and define the inferred set [ (k) Vδ,η = C. (A22) C∈Πη : C⊆Rδ,η , sk (C)≥τ
(k)
increasing. Hypothesis 1 gives perr ≤ ρδ (εk ). Therefore, p(k) err log
1 (k) perr
≤ ρδ (εk ) log
1 . ρδ (εk )
Multiplying by (Iδ + 1)kB gives (59). Multiplying by kB /δ gives (60). Since ρδ (ε) → 0 and p log(1/p) → 0 as p → 0+ , both excess bookkeeping penalties vanish as εk → 0. Hence Skloc approaches the ideal local bookkeeping value from above.
Assume that the score error is uniformly controlled by the intervention-relevant model error: sup |sk (C) − s(C)| ≤ c εk (A23) C⊆Rδ,η
for some constant c > 0. Also assume a boundarymargin condition: there is a modulus mδ (t), with mδ (t) → 0 as t → 0, such that [ P0 C ≤ mδ (t). (A24) C⊆Rδ,η : 0≤τ −s(C)≤t
Then 8.
Additional protocol and rare-set qualifications
The main text uses the conservative-identification case Vδ ⊆ Vδ′ to isolate false-positive overhead and preserve the identity α = Iδ + 1. If false negatives are allowed, define t = P0 (Vδ ∩ Vδ′ ), f = P0 (Vδ′ \ Vδ ), m = δ − t. (A18) ′ ′ If dP/ dP0 = α on Vδ and dP/ dP0 = β outside Vδ , with 1 − α(t + f ) β= , P (Vδ ) = αt + β(δ − t), (A19) 1−t−f then αt + β(δ − t) − δ Iδ = . (A20) δ Thus α = Iδ + 1 is specific to the no-false-negative case. The baseline-surprisal Landauer form in Eq. (53) charges false-positive assignments according to their rarity under P0 . A controlled-distribution erasure protocol instead uses P (Eδ ) = αperr , giving, when αperr < 1, 1 1 αkB perr log = αkB perr log − log α . αperr perr
(k) P0 Vδ,η \ Vδ,η ≤ mδ (c εk ).
(A25)
Thus Hypothesis 1 holds with ρδ (ε) = mδ (cε). Indeed, if a cell C ⊆ Rδ,η is a false positive, then sk (C) ≥ τ but s(C) < τ . By Eq. (A23), τ − s(C) ≤ sk (C) − s(C) ≤ c εk . Hence every false-positive cell lies in the boundary band {C ⊆ Rδ,η : 0 ≤ τ − s(C) ≤ c εk }, and Eq. (A25) follows from the margin condition. A particularly simple case occurs when the validity boundary has a positive finite-resolution margin: if there exists γ > 0 such that no cell in Rδ,η satisfies 0 < |τ − s(C)| ≤ γ, then mδ (t) = 0 for t < γ. Consequently, whenever c εk < γ, the false-positive mass is zero: (k) P0 Vδ,η \ Vδ,η = 0. More generally, if the boundary band satisfies a polynomial margin bound mδ (t) ≤ Cδ ta for constants Cδ > 0
16 and a > 0, then
The cohort-level summary statistics used in the symbolic calculation are
a a p(k) err ≤ Cδ c εk .
Source
This gives an explicit modulus for Hypothesis 1.
b median H b s.d. H b count mean H
GPT-5 long-form prose Project Gutenberg prose
Appendix B: Entropy-rate Estimation for text
This appendix gives the self-contained entropy-rate protocol used in Section IV B 5. The goal is not to reproduce a full detector or model-comparison study, but only to obtain two character-level entropy-rate estimates on a common symbolic alphabet: a human prose reference and a GPT-5 long-form prose estimate. All texts are converted to a fixed 27-symbol alphabet, Σ27 = {a, b, . . . , z, space}. Text is lowercased; punctuation, digits, and non-ASCII characters are removed. The resulting character stream is treated as a finite sample path from an approximately stationary symbolic source. Entropy is reported in bits per character. For a preprocessed sequence s1:N , we use a nonparametric probabilistic-finite-state-automaton entropy-rate estimator [31, 32]. For a substring-frequency threshold m, substrings with fewer than m occurrences are excluded. Empirical next-symbol distributions are estimated from retained histories and histories inducing similar next-symbol laws are represented as states of an inferred finite-state source. If Qm is the resulting state set, π bm (q) is the empirical state frequency, and pbm (a | q) is the empirical next-symbol law, the threshold-specific estimate is X X b (m) = − H π bm (q) pbm (a | q) log2 pbm (a | q). q∈Qm
a∈Σ27
(B1) To reduce dependence on a single pruning threshold, the reported document-level estimate is the median across a fixed threshold grid, b = medianm∈{m ,...,m } H b (m) . H 1 M
0.74 0.77
0.74 0.78
0.08 0.12
197 4341
The main text uses the mean values HGPT5 = 0.74 and HH = 0.77 bits per character. The median values give the same qualitative ordering; using the Gutenberg median 0.78 instead would make the GPT-5 support correction smaller by an additional factor of 2 at n⋆ = 100.
Appendix C: Numerical Scale Calculations
This appendix supports the numerical values in Table II. The table does not report the raw lift I, because 15 the values span from 0 to powers such as 1010 . Instead it reports the compressed double-log scale L ≜ log10 (I + 1),
(C1)
Λ ≜ log10 (L + 1).
(C2)
The +1 terms keep the scale finite at the passive baseline. For all nonzero large examples, Λ behaves as an ordinary log log-scale. When a row is a range, the endpoints of the displayed Λ-range are obtained by applying Eq. (C2) to the endpoint values of L. The ordering in Table II uses the midpoint of the underlying L-range.
1.
Passive baseline
For a passive system, P = P0 , so I = 0. Therefore L = log10 (1) = 0, and Λ = log10 (1) = 0. (C3)
2.
Fixed-feedback amplification
(B2)
The estimator requires no language model, labels, or training corpus. The thresholding step removes poorly supported histories; the median aggregation stabilizes the estimate across admissible frequency cutoffs. The human reference corpus consists of English Project Gutenberg long-form prose. Legal headers and boilerplate are removed, and texts shorter than 150,000 postprocessed characters are excluded. This yields 4341 Gutenberg documents. The GPT-5 corpus consists of 197 long-form prose samples generated with a fixed narrativeprompt protocol using GPT-5 API access. Each sample is generated toward a target length of approximately 150,000 characters using repeated continuation calls under default sampling settings, with only maximum completion length controlled.
Suppose a controller amplifies a rare-valid set by a constant likelihood factor α. Then P (Vδ ) = αδ, αδ − δ Iδ = = α − 1, δ L = log10 (Iδ + 1) = log10 α.
(C4) (C5) (C6)
For the fixed-feedback row we use α = 2 to 102 . Thus α=2: 2
α = 10 :
L = 0.301,
Λ = 0.114,
(C7)
L = 2.000,
Λ = 0.477.
(C8)
This gives the table entry Λ = 0.114–0.477.
17 3.
Repeated dynamic control
Repeated control compounds set-level lift. If m stages have approximate lift factors α1 , . . . , αm , then I +1≈
L≈
m Y j=1 m X
αj ,
(C9)
The term ρn⋆ is not estimated here. It represents finitelength typical-set error, rare-valid boundary effects, and support-overlap error. This qualification matters because n⋆ = 100 characters is sentence-scale, whereas AEP convergence is asymptotic. The point estimate in Table II sets ρ100 = 0: 2100(HGPT5 −HH ) = 2100(0.74−0.77) = 2−3 = 0.125, (C22)
log10 αj .
(C10)
j=1
IGPT5 + 1 ≈ (5 × 1014 )2−3 = 6.25 × 1013 , (C23) ΛGPT5 = log10 {log10 (6.25 × 1013 ) + 1} = 1.170. (C24)
The repeated-control row uses seven to ten binary improvements:
Equivalently, the finite-length corrected expression is
I + 1 ≈ 27 –210 .
IGPT5 + 1 ≈ (5 × 1014 )2100(0.74−0.77)+ρ100 .
(C11)
Hence m = 7 : L = 7 log10 2 = 2.107, Λ = 0.493, (C12) m = 10 : L = 10 log10 2 = 3.010, Λ = 0.603. (C13) This gives Λ = 0.493–0.603. 4.
Sentence-scale symbolic lift
L = n ∆ℓ log10 2,
(C14)
where n is sequence length and ∆ℓ is the per-symbol structured set-level lift in bits. The sentence-scale calculation in Section IV B 5 instead begins from cardinalities. Louwerse’s finite-resolution combinatorial estimate [26] gives NV ≈ 5 × 1021
(C15)
valid or interpretable English sentence-scale strings. The exemplary human-quality target set is estimated as NG ≈ B⋆ S⋆ ∼ (2 × 103 )(5 × 103 ) = 107 .
The compressed score Λ is relatively insensitive to moderate multiplicative changes, but the raw I should be interpreted only at order-of-magnitude precision. The entropy estimates are in bits per character under the 27symbol alphabet, so the sentence-scale length used here is n⋆ = 100 characters, not 20 words. 5.
For symbolic sequences, Eq. (92) gives
Fluctuation-theorem demon
For an ideal demon that realizes an entropy-reducing trajectory of magnitude ∆S, the fluctuation-theorem scaling gives L = log10 (I + 1) =
(C17) (C18)
Therefore LH = log10 (5 × 1014 ) = 14.699, ΛH = log10 (14.699 + 1) = 1.196.
(C26)
With ∆S = 10−19 J/K and kB = 1.380649 × 10−23 J/K, ∆S = 7242.97, L = 3145.58, Λ = log10 (3146.58) = 3.498. kB (C27) The table rounds this to Λ = 3.498. 6.
NG = 2 × 10−15 , NV NV IH + 1 ≈ = 5 × 1014 . NG
∆S/kB . log 10
(C16)
Thus δ⋆ =
(C25)
(C19) (C20)
Using NG = 106 –108 gives IH + 1 = 5 × 1015 –5 × 1013 and ΛH = 1.223–1.167. For the entropy-rate-corrected GPT-5 calculation, we use an AEP-style support-size approximation with an explicit finite-length slack term, qGPT5 log2 = n⋆ (HGPT5 − HH ) + ρn⋆ . (C21) qH
Velocity-selection demon
For the Maxwell–Boltzmann velocity-selection demon, Eq. (70) gives L≈
N q(ϵ)ϕ(ϵ), 2 log 10
(C28)
where N is the number of particles, q(ϵ) is the fraction above the velocity threshold, and ϕ(ϵ) is the excess dimensionless kinetic energy above the thermal mean. The numerical inputs q(ϵ), ϕ(ϵ), and L are listed in Table I. For N = 100, the endpoint values over ϵ = 2–5 are L = 1.85–9.38, and Λ = log10 (L + 1) = 0.455–1.016. (C29) For 1 mm3 of air at one atmosphere and 300 K, N ≈ 2.44 × 1016 . Table I gives L = 4.50 × 1014 –2.29 × 1015 ,
(C30)
18
The 1 mm3 demon therefore exceeds the single-event
∆S = 10−19 J/K demon on this scale because it aggregates a very large number of microscopic velocity selections.
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Λ = 14.653–15.360.
(C31)