Agentic Information Theory: Ergodicity and Intrinsic Semantics of Information Processes

Alexandra Jurgens; James P. Crutchfield

REVIEW 3 major objections 4 minor 75 references

Agentic Information Theory: Ergodicity and Intrinsic Semantics of Information Processes

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Information signals inherit ergodicity; meaning equals memory

desk verdict A readable packaging of existing computational-mechanics results, but the central ergodicity proofs for causal-state and pointwise information processes rest on a finite-range premise the objects don't satisfy. read the letter →

arxiv 2505.19275 v3 pith:T2O4KO33 submitted 2025-05-25 cond-mat.stat-mech cs.ITcs.MAmath.ITnlin.AO

James P. Crutchfield , Alexandra Jurgens This is my paper · ORCID

classification cond-mat.stat-mechcs.ITcs.MAmath.ITnlin.AO MSC 60G1037A3094A17

keywords informationprocessescognitiveagentsϵ-machinescomputationalmechanicsstationarityergodicitystatisticalcomplexitymeasurementsemantics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to show that the real-time informational signals produced by a cognitive agent—its moment-by-moment estimates of surprise, stored information, and predictive uncertainty—are themselves stochastic processes with good statistical properties. For a stationary, ergodic environment, and an agent that uses the environment's minimal optimal predictive model (its ϵ-machine) and is synchronized to it, the paper argues that these information processes are stationary and ergodic. It also proposes an intrinsic semantics: the "degree of meaning" of an observation is the information needed to identify the causal state it brings the agent to, and the time-averaged meaning equals the statistical complexity, the information the environment stores in its causal states. If correct, this licenses using such real-time signals for reliable downstream estimation, monitoring, and decision-making, and gives a quantitative answer to what a measurement means.

What carries the argument

The central objects are information processes: stochastic processes formed by applying a self-information function $i[\cdot] = -\log_2 \Pr(\cdot)$ to temporal information atoms built from the past $\overleftarrow{X}_t$, present $X_t$, and future $\overrightarrow{X}_t$ of the environment process. The load-bearing mechanism is the ϵ-machine—the minimal optimal predictive model whose states, the causal states $\sigma_t = \epsilon(\overleftarrow{X}_t)$, are equivalence classes of pasts with identical future predictions. Because the causal-state process is first-order Markov, and because the paper treats each monitored self-information as a finite-range function of the process, Propositions 7–12 transfer stationarity and ergodicity from the environment to the information process. A second mechanism is "degree of meaning" $\Theta(x) = -\log_2 \Pr(\sigma)$, the information in the causal state an observation selects.

What would settle it

Run a synchronized ϵ-machine agent on one long realization of the Even Process (a stationary ergodic, infinite-Markov-order generator) and compare the time average of the pointwise bound-information process $b_\mu(t)$ with the ensemble average computed from the stationary causal-state distribution; if the discrepancy does not vanish with sequence length, the claimed ergodicity of information processes fails.

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Extended reading notes

Core claim

On the paper's own terms: information processes—time series of Shannon information measures such as the pointwise entropy rate $h_\mu(t)$, bound information $b_\mu(t)$, ephemeral information $r_\mu(t)$, and statistical complexity $C_\mu(t)$—are functions of the observed environment process. Proposition 12 states that if the environment is stationary and ergodic, then the self-information processes $I[A|A](t)$ are stationary and ergodic; the argument runs through the causal-state process $\sigma_t = \epsilon(\overleftarrow{X}_t)$ being a stationary ergodic first-order Markov process and the monitored quantities being finite-range functions of it. Theorem 1 states that the total average semantic information, $\langle\Theta(x)\rangle$, equals the statistical complexity $C_\mu = I[S]$, the Shannon entropy of the causal-state distribution. The accompanying examples—biased coin, period-2, Golden Mean, and Even processes—display what these real-time signals look like, including negative pointwise informations, and the misdirected-semantics tables show how an incorrect internal model changes the meaning an agent assigns.

Load-bearing premise

The load-bearing premise is that the monitored quantities—causal states and self-informations conditioned on semi-infinite pasts and futures—are finite-range functions of the environment process, even though they actually depend on the entire infinite history; the ergodicity conclusions likely survive via measurable-function arguments, but the written proof mechanism only covers finite-range statistics.

Editorial extensions

If this is right

If Proposition 12 holds, an agent can use time-averaged estimates of entropy rate, bound information, and statistical complexity from a single long realization, because time averages converge to ensemble averages.
Theorem 1 gives an operational account of meaning: the average semantic content of an optimal observer's interpretations is exactly the environment's stored information, $C_\mu$.
The prediction process and causal-state process inherit ergodicity, so downstream inference over these signals is statistically grounded.
The intrinsic self-information grounded in the ϵ-machine resolves the ambiguity in Shannon's self-information of which probability distribution to use.
The example analyses show that pointwise information measures can be negative, so real-time interpretation requires updating intuitions from average information theory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Empirically, the ergodicity claim is testable on any finite-state generator: estimate the time average of a pointwise information atom over one long realization and compare to the ensemble average; the convergence rate should track the synchronization time.
The written proof that information processes are finite-range functions does not literally cover causal states, which depend on semi-infinite pasts; a measurable-factor-map argument would be needed to make Propositions 9–12 fully rigorous as stated.
For agents with incorrect internal models, the average degree of meaning may equal the entropy of the model's state distribution rather than the environment's statistical complexity, giving a quantitative measure of how wrong a model is; the paper only sketches this via short-word misdirected-semantics tables.
Because the pre-synchronization epoch is nonstationary, practical monitoring of information processes must either discard an initial burn-in period or model the transient separately, a point the paper itself notes in its online-prediction example.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Janet,

Quick take on Crutchfield and Jurgens. The paper does a useful thing: it packages the time series of Shannon information measures—entropy rate, bound information, ephemeral information, and the rest—as stochastic processes ('information processes') and asks when they inherit stationarity and ergodicity from a stationary, ergodic environment. That framing is genuinely, if modestly, new. The worked examples (Golden Mean, Even, misdirected semantics) are clear, the background is solid, and the authors correctly cite Billingsley's theorem on functions of ergodic processes.

The soft spot is exactly the one the stress-test note identifies. Propositions 9–12 claim stationarity and ergodicity for the causal-state process, prediction uncertainty, causal-state uncertainty, and general self-information processes by calling them 'finite-range functions.' That premise is false. Causal states depend on the entire semi-infinite past, and the pointwise self-informations in Table III condition on semi-infinite pasts and futures (e.g., r_mu(t) = i[X_t | past, future]). So Propositions 2–3, which handle finite-range functions, do not apply. The conclusions are probably true—these objects are measurable, shift-equivariant functions of the environment, so a factor-map argument plus Billingsley should work—but the paper does not provide that argument. It also leaves ambiguous whether stationarity holds for the bi-infinite stationary measure or for a finite-start agent with nonstationary transients. The paper concedes the transients but does not resolve the ambiguity in the proofs.

I also agree with the reader that Theorem 1 is definitional. The degree of meaning is defined as −log Pr(causal state), so its average is the entropy of the causal-state distribution by construction. Calling that a theorem is a packaging choice, not a substantive result.

The heavy self-citation is not a flaw here; those references point to real prior work. No code or data shipped, but the examples are specified well enough to reimplement.

Bottom line: this deserves a serious referee, but with the expectation of a major revision. The conceptual framework is useful for people monitoring learning agents or information engines, and the examples are worthwhile. The proof gap in Propositions 9–12 is real and needs to be closed before the ergodicity guarantee underwrites downstream inference.

Referee Report

3 major / 4 minor

Summary. The paper introduces "information processes"—real-time time series of Shannon information measures that a cognitive agent generates while observing a stochastic environment. It develops a framework in which an agent uses the environment's ε-machine as its internal model, then claims that if the environment is stationary and ergodic, the resulting information processes (entropy rate, ephemeral, bound, and semantic information processes, including causal-state processes) are also stationary and ergodic. It further defines a notion of intrinsic semantics via causal states, proves that the average degree of meaning equals the statistical complexity (Theorem 1), and illustrates the framework on four example processes (biased coin, period-2, golden mean, and even processes) with numerical plots of the information processes.

Significance. If the central ergodicity claim is established rigorously, the paper provides a useful conceptual unification: many time-local informational quantities used in statistical mechanics and complex systems are shown to be statistically well-behaved, justifying downstream inference and decision-making. The paper's integration of computational mechanics, Shannon information measures, and a semantic interpretation through causal states is a strength, as is its concrete treatment of illustrative processes with explicit ε-machines. The examples and the distinction between subjective and intrinsic semantics are clear and pedagogically valuable. However, the proof mechanism for the key ergodicity propositions is currently insufficient, and Theorem 1 is a definitional identity rather than a substantive result as stated. With appropriate revisions, the paper would be a solid contribution to the statistical mechanics of information processing.

major comments (3)

[§VI.F, Propositions 9–12] The proofs of Propositions 9–12 rely on describing the relevant quantities as finite-range functions of the environment process, but this is not correct for the objects actually defined. Causal states σ_t = ε(←X_t) (§V.A) depend on the entire semi-infinite past, and the information atoms in Table III such as r_μ(t) = I[X_t | ←X_t, →X_t] and b_μ(t) = I[X_t, →X_t | ←X_t] condition on semi-infinite pasts and futures. Propositions 2 and 3 apply only to finite-range sliding-window functions. The ergodicity and stationarity conclusions are likely salvageable by showing each information process is a measurable, shift-equivariant function of the environment and by invoking the ergodic theorem for such factors (e.g., Billingsley, Theorem 36.4), but the manuscript does not supply the needed measurability, well-definedness, or integrability arguments for pointwise conditional informations on semi-infinite histories. This is a load-bearing gap for the paper's central claim.
[§VI.E, Theorem 1] Theorem 1 states that the average semantic information equals the statistical complexity C_μ, but this follows directly from Definition 8: Θ(x) is defined as −log₂ Pr(σ) where σ is the causal state selected by x. The proof simply recognizes ⟨Θ(x)⟩ = −Σ_σ Pr(σ) log₂ Pr(σ) = H[S] = C_μ. As written, the 'theorem' is a restatement of the definition and does not establish a substantive connection between semantics and complexity beyond the chosen definition. The authors should either reframe this as a definitional identity or, if a deeper claim is intended, state and prove it from more primitive assumptions.
[§V.B and §VII.D] There is an ambiguity about which stochastic process the stationarity claims in Propositions 9–12 refer to. The paper distinguishes the causal-state process (defined for all times) from the recurrent causal-state process (after synchronization), and §VII.D explicitly notes that the Even Process exhibits an infinite-duration transient during which information processes such as h_μ(t) and C_μ(t) are not stationary. The proofs of Propositions 9–12 assert stationarity for the causal-state process without specifying the initial distribution or whether the process is taken from the bi-infinite stationary ensemble or from a finite-start initialization. The latter is nonstationary, as the paper itself concedes. The statement of these propositions must be made precise about the ensemble in question, or the conclusions will be incorrect for the finite-start agent setting that the examples use.

minor comments (4)

[Table III caption] The caption states that "the forward and reverse entropy rates—h+_μ(t) and h+_μ(t), respectively—are equal for stationary processes," but the two displayed symbols appear to be identical; presumably one should be h+_μ(t) and the other h−_μ(t).
[§VII.A, Biased Coin example] The text says the Biased Coin Process with Pr(x=1)=2/3 has entropy rate h_μ = 1 bit per time step. The entropy rate of an IID binary source with p=2/3 is approximately 0.918 bits, not 1 bit. The figure and table values should be checked for consistency with the stated bias.
[§IV.D and Figure 7] The text refers to "the shift operator τ of Eq. (3)" when describing the time evolution of information measures, but Eq. (3) is the definition of the push-forward measure μY and not the shift operator. The reference should be to the shift defined in §III.A.
[Proposition 1 proof, Eq. (3)] The displayed expression for μY(y) contains a double integral over ω and z that is notationally ambiguous and likely not what is intended; the measure μY should be defined by a single push-forward integration, and the intermediate sliding-window construction should be separated more cleanly.

Circularity Check

1 steps flagged · score 6.0 of 10

Theorem 1 restates the definition of 'degree of meaning'; the ergodicity proofs have a finite-range gap but are not circular.

self definitional [Sec. VI.C, Def. 8 (Eq. 34) and Sec. VI.E, Theorem 1]
"Definition 8. The degree of meaning of observing x∈X : Θ(x) = − log2 Pr(σ), where the arrow notation signifies σ∈S is the causal state to which x brings the agent. ... Theorem 1. A process' total average semantic information is its statistical complexity: ⟨Θ(x)⟩ = Cµ. Proof: ⟨Θ(x)⟩ = ∑_{σ∈S} Pr(σ)Θ(x) = −∑ Pr(σ) log2 Pr(σ) = I[S] = Cµ."

Θ(x) is defined as the negative log probability of the causal state selected by x, and Cµ is defined in Sec. V.F as the Shannon entropy of the causal-state distribution (I[S]). Therefore the average of Θ is Cµ by construction: the theorem's one-line proof only expands the definition and performs no independent derivation. The headline semantic result—that total average semantic information equals statistical complexity—is thus a tautology rather than a derived relation between two independently defined quantities.

full rationale

The only genuine circularity is Theorem 1, which follows immediately from the definition of Θ and the definition of Cµ. The ergodicity results (Propositions 7–12) are not circular: they lean on standard external theorems (Billingsley Thm 36.4) and on the well-known Markov property of ε-machines from Ref. [27], a prior mathematical result with stated assumptions not including the paper's conclusions. I find no fitted-input-called-prediction or uniqueness-imported-from-authors pattern. However, Propositions 9–12 contain a real proof gap: causal states σ_t = ε(←X_t) and atoms such as r_μ(t) = I[X_t | ←X_t, →X_t] condition on semi-infinite pasts/futures and are not finite-range functions of the environment, so the sliding-window propositions invoked in the proofs do not apply as written. That is a correctness deficiency, not circularity. Score reflects one central semantic claim reducing by construction while the ergodicity portion retains independent content, even though its written proof mechanism is incomplete.

Assumptions & free parameters 2 free parameters · 8 assumptions · 2 invented entities

The central results rest on standard ergodic theory plus the paper's own computational-mechanics framework. No data are fitted to obtain the propositions. The two structural weaknesses are the false finite-range premise in the proofs of Propositions 9 through 12 and the definitional character of Theorem 1. All example parameters are illustrative and not load-bearing.

free parameters (2)

Example-process parameters (Biased Coin bias p; Golden Mean and Even transition probabilities) = p = 2/3; transition probabilities 1/2, 1/3, 3/4, 1/4
Hand-chosen values for the four illustrative environments. They do not enter the central propositions, and no values are fitted to empirical data to obtain the paper's results.
Estimation window length and sample size for figures = Window of 13 (6 past, 6 future, present); sample length 10^6
Chosen for the illustrative plots in Figures 9 through 12; no error analysis or confidence intervals are attached to the resulting empirical estimates.

assumptions (8)

domain assumption The environment process X is stationary (Definition 1).
Invoked throughout Section III and in Propositions 2, 7, 9 through 12; the paper's results are stated for stationary environments.
domain assumption The environment process X is ergodic (Definition 2).
Required for the ergodicity claims in Propositions 3, 8 through 12, and for identifying time averages with ensemble averages.
domain assumption All processes considered are finitary, meaning finite excess entropy E (Section IVB).
Used to guarantee that all information atoms except those over semi-infinite pasts and futures are finite, a precondition for the stationarity and ergodicity transfer claims.
domain assumption The agent is synchronized to the environment and knows its causal state sigma_t (Section VI preamble).
Load-bearing for the semantic results and for treating causal-state processes as stationary; the paper acknowledges nonstationary transients before synchronization.
domain assumption The agent's internal model is the environment's epsilon-machine, the minimal optimal predictor (Section V).
Established in computational mechanics (Ref. [27]); required for the 'intrinsic' semantics of Section VI and for Theorem 1.
standard math Standard ergodic theorems apply, including Birkhoff's theorem and Billingsley Thm 36.4 for functions of ergodic processes.
Invoked explicitly in Proposition 3 and used implicitly for the time-average equals ensemble-average identifications.
standard math A measurable function of a stochastic process is a stochastic process (Appendix A).
The basis for treating functions of the environment process as new processes; the appendix proof contains a preimage typo but the claim is standard.
ad hoc to paper Causal-state and self-information processes are finite-range functions of the environment process (Propositions 9 through 12).
This premise is false as stated for causal states and semi-infinite-conditioned quantities, which depend on infinite pasts and futures; it is the paper's own proof mechanism and is the weakest load-bearing step.

invented entities (2)

Information process (time series of information measures as a stochastic process)
purpose: The central object class of the paper: entropy-rate, bound, ephemeral, enigmatic, and elusive information series generated by an observing agent.
A definitional construction from existing Shannon information atoms (Refs. [24, 50]); no new physical entity and no falsifiable external handle.
Degree of meaning Theta(x) and meaning content (the causal state selected by an observation)
purpose: Quantifies the semantic content of a measurement relative to the agent's internal model.
Defined as -log2 Pr(state the observation selects); because of this definition, its average equals the statistical complexity by construction rather than by empirical measurement.

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Cite this review

Pith. "Pith review of Agentic Information Theory: Ergodicity and Intrinsic Semantics of Information Processes." pith.science (2026). https://pith.science/paper/T2O4KO33

@misc{pith2026250519275,
  author       = {Pith},
  title        = {Pith review of: Agentic Information Theory: Ergodicity and Intrinsic Semantics of Information Processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T2O4KO33}},
  note         = {Machine review of arXiv:2505.19275}
}

read the original abstract

We develop information theory for the temporal behavior of memoryful agents moving through complex -- structured, stochastic -- environments. We introduce and explore information processes -- stochastic processes produced by cognitive agents in real-time as they interact with and interpret incoming stimuli. We provide basic results on the ergodicity and semantics of the resulting time series of Shannon information measures that monitor an agent's adapting view of uncertainty and structural correlation in its environment.

Figures

Figures reproduced from arXiv: 2505.19275 by the authors.

**Figure 3.** FIG. 3. Measurement semantics: The channel consists of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** FIG. 4. Discrete-time, discrete-value stochastic process as [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (8 more)

**Figure 5.** Figure 5: FIG. 5. Real-valued measurable function of a stochastic pro [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. Sliding window processes: The given stochastic process appears on the top line with the construction of sliding window [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7. Information processes [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: FIG. 8. Example [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]

**Figure 9.** Figure 9: FIG. 9. Biased Coin Process observed time series realization (top) versus several of its information processes (below): entropy [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]

**Figure 10.** Figure 10: FIG. 10. Period- [PITH_FULL_IMAGE:figures/full_fig_p023_10.png]

**Figure 11.** Figure 11: FIG. 11. Golden Mean Process observed time series sample versus several of its information processes: entropy rate [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]

**Figure 12.** Figure 12: FIG. 12. Even Process observed time series sample versus several of its information processes: entropy rate [PITH_FULL_IMAGE:figures/full_fig_p026_12.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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2018 doi

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

In the language of measures, as an agent collects more observations, P/∫hort↕eftarrow−x converges in distribution, with respect to theproduct topologyof the space of sequencesX N [41]

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[2] [2]

This also holds for convergence over word distribu- tions Pr(x1:n)

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[3] [3]

A conditional measure is a ratio of likelihoods, as in Eq. (25)

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[4] [4]

For all measuresµ onX Z and µ-almost every past↼−X ∈X Z, the measuresηℓ[↼−X ] defined by: ηℓ[↼−X ](U0,ω) = Pr µ (ω|x−ℓ+1...x 0) converge in distribution to the measure ϵ[↼−X ] : ηℓ[↼−X ]/∫hortrightarrowϵ[↼−X ], asℓ/∫hortrightarrow∞

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[5] [5]

ϵ[↼−X ] is the predictive state of ↼−X and the func- tion that maps to measures over future sequences— ϵ[↼−X ]/∫hortrightarrowM(X Z)—is theprediction mapping. Anticipating future uses, thecausal states of an input- output process—or communication channel—are similarly defined by equivalence classes of channel input sequences and output sequences. The corr...

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[6] [6]

FinitealphabetX of k symbols x∈X

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[7] [7]

} that consists of transient states whose probabilities vanish andre- current states ˜σ whose probability converges to a constant Pr(˜σ)> 0

Casual state set S ={σ1,σ 2,... } that consists of transient states whose probabilities vanish andre- current states ˜σ whose probability converges to a constant Pr(˜σ)> 0. And

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[8] [8]

subjective

Causal dynamic—set ofk (possibly infinite dimen- sion) symbol-labeled transition matricesT (x), x∈ X: T (x) ij = Pr (σj,x|σi). This defines its own class of optimal representations for a wide range of stochastic processes. That said, we can draw several parallels with existing classes of process representa- tions. The definition here identifies anϵ-machin...

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