Entropic probability and context states

Benjamin Schumacher; Michael D. Westmoreland

REVIEW 3 major objections 5 minor 9 references

Entropic probability and context states

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that any probability distribution over a set of states can be realized, to arbitrary accuracy, as the entropic distribution of an extended eidostate with context states, yielding a generalized free-energy inequality and…

desk verdict A clean extension of their axiomatic framework with a constructive representation theorem, but the generalized Landauer bound depends on an unproved fine-grained reservoir assumption that fails in their own example. read the letter →

arxiv 2412.12430 v1 pith:5J56EE4U submitted 2024-12-17 quant-ph

Benjamin Schumacher , Michael D. Westmoreland This is my paper · ORCID

classification quant-ph PACS 05.70.-a89.70.Cf

keywords entropicprobabilityeidostatescontextstatesreservoirfreeenergyLandauer'sprincipleaxiomaticthermodynamicsinformation

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

Starting from an axiomatic thermodynamics in which entropy is fixed by which state transformations are possible, the paper extends entropic probability from uniform collections of states to arbitrary collections. The extension augments each state with a context state, normally an information state, so that the whole collection is uniform and its entropic probabilities match any prescribed distribution as closely as desired. The same construction, applied to two collections with prior distributions, produces an abstract free energy $F(a)=Q(a)-(\varepsilon/\sigma)S(a)$ and the inequality $\langle\Delta Q_\mu\rangle-(\varepsilon/\sigma)\Delta H\le -\langle\Delta F\rangle$. This relation says that work-like transfers and changes in stored information are paid for by free energy, and it generalizes Landauer's principle to any conserved component of content. If the construction works, probability and free energy are not added to thermodynamics but emerge from its axioms.

What carries the argument

The load-bearing object is the context-state extension: for each state $e_k$ of a uniform eidostate $E$, form the uniform eidostate $\hat E=\bigcup_k(e_k+C_k)$, where $C_k$ are context states, chosen as information states with entropy $\log n_k$. This construction converts a target probability $p_k$ into an entropic probability because the entropy of $\hat E$ becomes $\log\sum_k n_k2^{S(e_k)}$, so $P(e_k+C_k|\hat E)=n_k2^{S(e_k)}/\sum_\ell n_\ell 2^{S(e_\ell)}=p_k$ once $n_k/N$ approximates $p_k/2^{S(e_k)}$. In the free-energy part, the same context states are combined with reservoir ladders $\theta_n$ (entropy step $\sigma$) and mechanical ladders $\mu_n$ (component of content $\varepsilon$) to tune two distributions $p_i$ and $q_j$ while preserving uniformity; the inequality $\langle\Delta Q_\mu\rangle-(\varepsilon/\sigma)\Delta H\le -\langle\Delta F\rangle$ is obtained by comparing $S(\hat A)\le S(\hat B)$ with conservation of $Q$.

What would settle it

Find any model of the axioms in which every reservoir ladder has entropy step $\sigma$ bounded below by a positive minimum; then choose target distributions $p_i$ and $q_j$ whose ratios fall between the available powers of $2^{n\sigma}$, so the Section 7 approximation cannot run, and check whether the inequality (38) still holds in that model.

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Extended reading notes

Core claim

The central claim is that the entropic probability rule $P(e|E)=2^{S(e)}/2^{S(E)}$ is not limited to uniform eidostates. For any finite collection $A$ and any target distribution $p$ over it, one can find a uniform eidostate $\hat A=\bigcup_a(a+C_a)$ whose entropic probability is arbitrarily close to $p$, with $C_a$ information states of entropy roughly $\log(p_a/2^{S(a)})$. The paper then applies the same context-state construction to two collections $A$ and $B$ with a priori distributions $p$ and $q$, and shows that the possibility of transforming $\hat A$ into $\hat B$ is governed by the free energy $F(a)=Q(a)-(\varepsilon/\sigma)S(a)$, leading to $\langle\Delta Q_\mu\rangle-(\varepsilon/\sigma)\Delta H\le -\langle\Delta F\rangle$. The paper reads this as a fully general Landauer principle: erasing one bit costs $\varepsilon/\sigma$, a charge that can be paid from mechanical work or from free energy.

Load-bearing premise

The proof assumes that for every needed probability ratio there is a reservoir ladder with entropy step $\sigma$ small enough to express the ratio as an integer power of $2^{n\sigma}$, an assumption the paper states without proof.

Editorial extensions

If this is right

Landauer's principle holds in full generality: erasing one bit costs $\varepsilon/\sigma$, where $\varepsilon$ is the change in any conserved component of content of the mechanical state and $\sigma$ is the entropy increment of the reservoir ladder.
The inequality $\langle\Delta Q_\mu\rangle-(\varepsilon/\sigma)\Delta H\le -\langle\Delta F\rangle$ governs every process between context-extended collections, giving probabilistic processes the same directional character as deterministic state transformations.
The ratio $\varepsilon/\sigma$ acts as an effective temperature, so the free energy $F(a)=Q(a)-(\varepsilon/\sigma)S(a)$ is available for any conserved component of content, not only energy.
Any prior distribution over states has a concrete physical realization as a memory register, so Bayesian assignments are not extra bookkeeping but part of the physical state.

Reading between the lines

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

The paper leaves implicit that the context states make probability assignment itself a thermodynamic resource; one could develop a resource theory in which changing a prior distribution costs free energy, in analogy with changing a quantum state under a thermal operation.
If fine-grained reservoir ladders exist in all models, then the temperature-like parameter $\varepsilon/\sigma$ is adjustable for every conserved quantity; a testable extension is to ask whether the coin-and-box model with boxes built from larger sets $K$ can produce ratios that are not powers of two.
The approximation via information states suggests that exact probabilistic statements require infinite resources; a finite-size version of the inequality with a correction depending on the largest context $n_k$ is a natural next step.
One could connect this emergent probability to Bayesian updating by treating the addition of a context state as the physical correlate of conditioning on evidence; conditional probabilities would then correspond to composing context states, a direction the paper does not pursue.

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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

Ben, quick take on Schumacher & Westmoreland. The new bit is the context-state construction: any probability distribution over the states of a uniform eidostate can be approximated by entropic probabilities if you attach context states made from information states. That is a neat, explicitly constructive result, and the coin-and-box examples check out. They also state a generalized free-energy inequality, Eq. (38), which they read as a Landauer-type bound for any component of content.

The soft spot is right where the stress test put it. The derivation of Eq. (38) needs reservoir states with arbitrarily small entropy increment σ, so that any positive ratio can be hit by 2^{nσ}. The paper says 'essentially, we assume' this—it is not derived from Axioms I–IX. Worse, in the only concrete model they give, the coin-and-box model, σ = log k for integer k ≥ 2, so σ ≥ log 2 and the ladder is not fine-grained. The paper says later results do not depend on the model, but that does not make the assumption any less extra. As stated, Eq. (38) is conditional on an existence postulate, not a theorem of the axioms. That should be fixed either by constructing a fine-grained reservoir in some model or by stating the result with the assumption made explicit.

Another minor point: the free-energy inequality is mostly a relabeling of the entropy comparison S(Â) ≤ S(B̂) with new names for the terms. That is fine as an interpretive move, but it means the physical content is in the identification of ε/σ as a temperature-like ratio, not in a new inequality.

The representation theorem stands on its own and is worth having. The paper is honest, clearly written, and the internal derivations are consistent once you grant the axioms of [1]. If I were refereeing, I would ask for the fine-grained reservoir issue to be addressed before publication, but I would not desk reject it. It deserves serious peer review.

Who's it for: people working on axiomatic thermodynamics and Landauer's principle who want a minimal set of axioms plus an extra assumption. I would cite it for the context-state representation. Bring it to reading group if you are interested in the program.

Referee Report

3 major / 5 minor

Summary. The paper extends the authors' axiomatic information thermodynamics framework. It defines entropic probabilities for uniform eidostates via S(E)=log Σ 2^{S(e)}, extends the definition to non-uniform sets by uniformization with mechanical states, reservoir ladders, and context states, and shows that context states can tune entropic probabilities to approximate arbitrary distributions. It then defines free energy F=Q-(ε/σ)S and derives the generalized Landauer inequality ⟨ΔQ_μ⟩-(ε/σ)ΔH ≤ -⟨ΔF⟩.

Significance. If the main inequality were an unconditional consequence of the axioms, this would be a notable result: it would unify probability, free energy, and information erasure in a single abstract framework, with Landauer's principle as a corollary. The coin-and-box examples are concrete and the calculations in Sections 5 and 6 check out. The paper is honest about several limitations, but these limitations are load-bearing: the arbitrary-approximation result is only approximate, and Eq. (38) depends on an unproved fine-grained reservoir assumption. The construction of context states also means that the match to p and q is fitted rather than predicted, so the paper's claims about what the axioms 'establish' need to be qualified.

major comments (3)

[§7, Eqs. (30) and (38)] The derivation of the generalized Landauer inequality for arbitrary a priori distributions p_i and q_j assumes that a reservoir ladder θ_n exists with arbitrarily small entropy increment σ, introduced as 'essentially, we assume' in §7. This is not derived from Axioms I–IX, and it fails in the paper's own coin-and-box model: by §5, reservoir states b^K_n have σ=log|K|, so σ≥log2. Hence Eq. (38) is not a consequence of the axioms for arbitrary p and q; it is conditional on an additional existential postulate that is neither stated as an axiom nor shown to be satisfiable in any model.
[§6, Eqs. (25)–(28)] The context-state representation is a fitting construction: the C_k are chosen so that the entropic probabilities reproduce the target p_k. Consequently, the subsequent free-energy inequality (38) is not an independent prediction about arbitrary distributions; it is a consistency relation that follows once the context states have been engineered to realize p and q. The paper's language 'adjust our entropic probabilities' and 'tune' acknowledges this, but the abstract's claim of 'establishing a relation' should be qualified.
[§6 and §7, Eq. (30)] The realization of a priori distributions is only approximate. For arbitrary real p_i and q_j, the equality in Eq. (30) is an idealization: the rational approximation in Eq. (26) means that the entropic probabilities match only up to a finite error. Since Eq. (38) is the main quantitative result, the paper should state whether the inequality holds exactly in a limiting sense or only up to approximation error, and it should specify the sense of convergence.

minor comments (5)

[§4] There is a duplicated word in 'An eidostate represents represents the knowledge of an agent'; it should read 'represents'.
[§5] There is a typo in 'bojx-state reservoir sequences'; it should be 'box-state'.
[§5, Maxwell's demon paragraph] The word 'introducers' should be 'introduces'.
[§7] The phrase 'approximate any positive number by 2^{nσ}' is imprecise for a fixed σ>0: the set {2^{nσ}: n∈Z} is discrete, not dense. The intended meaning should be clarified, presumably that for a given finite set of ratios one can choose σ sufficiently small.
[§7, Eq. (31)] The equation writes Q(θ_k)=kε, which implicitly assumes Q(θ_0)=0. If Q(θ_0) is not zero, the constant should be carried through; it cancels in the averages, but the assumption should be stated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the context-state construction is an explicit representation theorem, and Eq. (38) is a direct consequence of the prior entropy theorem; the fine-grained-reservoir assumption is a limitation, not a circular input.

full rationale

The paper's central derivation is not circular. Section 6 explicitly constructs context states C_k = I_{n_k} from a desired distribution p_k; the equality P(e_k|Ê) = p_k is the content of a representation theorem, not a hidden prediction, and the text says 'Our challenge is to find a set of context states' rather than claiming p_k is predicted from first principles. Section 7 uses the same construction to realize a priori distributions p_i and q_j, so Eq. (30) is an imposed normalization, not a derived output. The free-energy inequality Eq. (38) is obtained by substituting the uniformity condition Eq. (31) and the entropy criterion S(Â) ≤ S(B̂) from Theorem 8 of [1] into this representation; defining F(a) = Q(a) - (ε/σ)S(a) makes Eq. (38) a rewritten entropy comparison. This is a legitimate derivation, not a circular reduction. The load-bearing theorem from [1] is a parameter-free mathematical result with stated axioms that do not include Eq. (38), so citing it is a normal dependency rather than circularity. A genuine limitation is the Section 7 assumption of fine-grained reservoir ladders ('Essentially, we assume that the reservoir states are fine-grained enough...'); it is not derived from Axioms I–IX and is false in the paper's own coin-and-box model, where σ = log k ≥ log 2. This means Eq. (38) is conditional on an extra existence postulate, but that is a soundness/scope gap, not a circular step. No step in the paper reduces its own conclusion to its inputs by definition.

Assumptions & free parameters 1 free parameters · 3 assumptions · 2 invented entities

The central results rest on the full axiom system from the authors' 2018 paper, which is self-cited and not re-verified. The fine-grained reservoir assumption and the context-state construction are introduced in this paper to realize arbitrary distributions; the resulting free energy depends on the chosen ratio ε/σ.

free parameters (1)

ε/σ (energy-entropy scale ratio) = depends on chosen reservoir; no universal value
The free energy F(a)=Q(a)−(ε/σ)S(a) is defined using this ratio, which varies from one reservoir ladder to another. The paper treats it as an arbitrary scale rather than deriving it from first principles.

assumptions (3)

domain assumption Full axiom system of [1] (Axioms I-IX as reviewed in the Appendix), including the entropy existence theorem.
The paper assumes the existence of states, eidostates, the → relation, record/information states, bit processes, demon axiom, mechanical states, and the entropy theorem (Theorem 8 from [1]) without proof.
ad hoc to paper Fine-grained reservoir states exist with arbitrarily small entropy increment σ.
Section 7 assumes reservoir states θ_n with arbitrarily small σ so that any ratio can be approximated by 2^{nσ}; no construction or consistency proof is given.
domain assumption Uniformization is possible for the considered sets of states.
The definitions of uniformizable sets and context state extensions assume that mechanical or reservoir states can be appended to make a uniform eidostate; this is an existence assumption about the state space.

invented entities (2)

Context states C_k
purpose: Appended to elements of an eidostate so the entropic probability over the extended eidostate matches a desired distribution
Section 6 introduces context states and realizes them as information states In_k chosen to reproduce p_k; they have no observable consequences outside the theory.
Reservoir states θ_n
purpose: A ladder of states with entropy increment σ used to uniformize non-uniform sets with a temperature-like scale and to define free energy
Section 5 postulates their properties (θ_n + µ → θ_{n+1}, and θ_k + θ_l ↔ θ_m + θ_n iff k+l=m+n); their existence for arbitrary σ is not independently evidenced.

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Cite this review

Pith. "Pith review of Entropic probability and context states." pith.science (2026). https://pith.science/paper/5J56EE4U

@misc{pith2026241212430,
  author       = {Pith},
  title        = {Pith review of: Entropic probability and context states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5J56EE4U}},
  note         = {Machine review of arXiv:2412.12430}
}

read the original abstract

In a previous paper, we introduced an axiomatic system for information thermodynamics, deriving an entropy function that includes both thermodynamic and information components. From this function we derived an entropic probability distribution for certain uniform collections of states. Here we extend the concept of entropic probability to more general collections, augmenting the states by reservoir and context states. This leads to an abstract concept of free energy and establishes a relation between free energy, information erasure, and generalized work.

Figures

Figures reproduced from arXiv: 2412.12430 by the authors.

**Figure 1.** A simple Maxwell’s demon. Recalling that σ = 1, the entropy is S(Aˆ) = log 2 1 + 21 + 20 = log 5. (21) This yields probabilities Pb(h|A) = 2 5 Pb(t|A) = 2 5 Pb(h + t|A) = 1 5 . (22) As an illustration of these ideas, consider the version of Maxwell’s demon shown in [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗

**Figure 2.** Extraction of work by dividing gas enclosure into unequal volumes.. [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Westmoreland

Austin Hulse, Benjamin Schumacher, and Michael D. Westmoreland. Ax- iomatic information thermodynamics. Entropy, 20(4):237, 2018

work page 2018
[2]

R. Giles. Mathematical Foundations of Thermodynamics. Pergamon Press Ltd., Oxford, 1964

work page 1964
[3]

Lieb and Jakob Yngvason

Elliott H. Lieb and Jakob Yngvason. A guide to entropy and the second law of thermodynamics. Notices of the American Mathematical Society , 45:571–581, 1998

work page 1998
[4]

On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings

Leo Szilard. On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings. Zeitschrift fur Physik , 53:840–856,

work page
[5]

Charles H. Bennett. The thermodynamics of computation—a review. International Journal of Theoretical Physics , 21:905–940, 1982

work page 1982
[6]

C. E. Shannon. A mathematical theory of communication. Bell System Technical Journal, 27:379–423,623–656, 1948

work page 1948
[7]

Cover and Joy A

Thomas M. Cover and Joy A. Thomas. Elements of Information Theory (Second Edition). John Wiley and Sons, Hoboken, 2006

work page 2006
[8]

Landauer

R. Landauer. Irreversibility and heat generation in the computing pro- cess. IBM Journal of Research and Development , 5:183–191, 1961. 18

work page 1961

Show all 9 references

[1929]

(English translation in Behavioral Science 1964, 9, 301–310.)

1964

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Westmoreland

Austin Hulse, Benjamin Schumacher, and Michael D. Westmoreland. Ax- iomatic information thermodynamics. Entropy, 20(4):237, 2018

work page 2018

[2] [2]

R. Giles. Mathematical Foundations of Thermodynamics. Pergamon Press Ltd., Oxford, 1964

work page 1964

[3] [3]

Lieb and Jakob Yngvason

Elliott H. Lieb and Jakob Yngvason. A guide to entropy and the second law of thermodynamics. Notices of the American Mathematical Society , 45:571–581, 1998

work page 1998

[4] [4]

On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings

Leo Szilard. On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings. Zeitschrift fur Physik , 53:840–856,

work page

[5] [5]

Charles H. Bennett. The thermodynamics of computation—a review. International Journal of Theoretical Physics , 21:905–940, 1982

work page 1982

[6] [6]

C. E. Shannon. A mathematical theory of communication. Bell System Technical Journal, 27:379–423,623–656, 1948

work page 1948

[7] [7]

Cover and Joy A

Thomas M. Cover and Joy A. Thomas. Elements of Information Theory (Second Edition). John Wiley and Sons, Hoboken, 2006

work page 2006

[8] [8]

Landauer

R. Landauer. Irreversibility and heat generation in the computing pro- cess. IBM Journal of Research and Development , 5:183–191, 1961. 18

work page 1961