Abstract

In this talk we will explore a zero-player game based on an information isolation constraint. The dynamics of the game emerge from a “no-barber” selection principle that prohibits external structure. The aim is for the game to avoid impredictive-style inconsistencies. Motivated by the selection principle we will derive a “selected” trajectory in the game that consists of a second-order constrained maximum entropy production along the information geometry.

The Munchkin Provision

Without such consistency, we would require what we might call a “Munchkin provision.” In the Munchkin card game (Jackson, 2001), it is acknowledged that the cards and rules may be inconsistent. Their resolution?

Any other disputes should be settled by loud arguments, with the owner of the game having the last word.

Munckin Rules (Jackson, 2001)

While this works for card games, it’s unsatisfying for foundational mathematics. We want our game to be internally consistent, not requiring an external referee to resolve paradoxes.

Figure: The Munchkin card came has both cards and rules. The game explicitly acknowledges that this can lead to inconsistencies which should be resolved by the game owner.

A Tautology

Self-governing systems cannot refer to external arbitration.

While this is a tautology, we will try to formalise it through information theory. The key question is: what mathematical structure is forced on a system that cannot appeal to external adjudication?

The No-Barber Principle

In 1901 Bertrand Russell introduced a paradox: if a barber shaves everyone in the village who does not shave themselves, does the barber shave themselves? The paradox arises when a definition quantifies over a totality that includes the defining rule itself.

We propose a similar constraint for the inaccessible game: the foundational rules must not refer to anything outside themselves for adjudication or reference. Or in other words there can be no external structure. We call this the “no-barber principle” (Lawrence, 2026a).

The no-barber principle says that admissible rules must be internally adjudicable: they depend only on quantities definable from within the system’s internal language, without requiring e.g. an external observer to define the co-ordinates or a privileged decomposition.

Baez-Fritz-Leinster Characterization of Information Loss

Before introducing our fourth axiom, we need to understand how information loss is measured. Baez et al. (2011) showed that entropy emerges naturally from category theory as a way of measuring information loss in measure-preserving functions. They derived Shannon entropy from three axioms, without invoking probability directly.

The Three Axioms

Let \(F(f)\) denote the information lost by a process \(f\) that transforms one probability distribution to another. The three axioms constrain the functional form of \(F\).

Axiom 1: Functoriality suggests that given a process consisting of two stages, the amount of information lost in the whole process is the sum of the amounts lost at each stage: \[ F(f \circ g) = F(f) + F(g), \] where \(\circ\) represents composition.

Axiom 2: Convex Linearity suggests that if we flip a probability-\(\lambda\) coin to decide whether to do one process or another, the information lost is \(\lambda\) times the information lost by the first process plus \((1-\lambda)\) times the information lost by the second: \[ F(\lambda f \oplus (1-\lambda)g) = \lambda F(f) + (1-\lambda)F(g). \]

Axiom 3: Continuity suggests that if we change a process slightly, the information lost changes only slightly, i.e. \(F(f)\) is a continuous function of \(f\).

The Main Result

The main result of Baez et al. (2011) is that these three axioms uniquely determine the form of information loss. There exists a constant \(c\geq 0\) such that for any \(f: p \rightarrow q\): \[ F(f) = c(H(p) -H(q)) \] where \(F(f)\) is the information loss in process \(f: p\rightarrow q\) and \(H(\cdot)\) is the Shannon entropy measured before and after the process is applied to the system.

This provides a foundational justification for using entropy as our measure of information. It is not just a convenient choice — it is the unique measure satisfying these natural requirements for measuring information loss. This is a theorem (Baez et al., 2011). The quantum analogue — replacing finite probability spaces and Shannon entropy with finite-dimensional noncommutative probability spaces and von Neumann entropy — is established by Parzygnat (2022).

The Inaccessible Game Setup

Inspired by the no-barber principle, we set up the game in a way that attempts to avoid “external structure.” The first two things we need to do this are

1. A representation of information loss
2. A prohibition of information exchange with the game

How do we obtain a representation of information loss without including external structure? We use the axiomatic frameworks of Baez et al (Baez et al. (2011)) and Parzygnat (Parzygnat (2022)). They characterise entropy through category theory frameworks that depend on three axioms. Slight differences in the axioms result in different conclusions. Baez et al conclude that difference in Shannon entropy before and after a process is applied characterises information loss. Parzygnat is inspired by Baez et al but reformulates around a different categorical object which implies von Neumann entropy.

In the game (Lawrence (2025)) we introduce information conservation based on these measures of information loss.

Information Isolation

The first three axioms of the inaccessible game, due to Baez et al. (2011), characterise information loss and justify the use of entropy. For the game itself we introduce a fourth axiom: information isolation. Just as an isolated chamber conserves mass and energy, our game is isolated from external observation. No observer outside the system can extract or inject information.

Under additional requirements of exchangeability and extensivity, information isolation implies that the total marginal entropy is conserved. For any finite sub-group of \(N\) variables the sum of marginal entropies \(\{h_i\}_{i=1}^{N}\) sums to a constant \(C\), \[ \sum_{i=1}^N h_i = C. \] The conservation law is imposed in an exchangeable form across the marginal entropies, so that it applies consistently to any finite partition drawn from a potentially countably infinite collection of variables.

The specific form \(\sum_i h_i = C\) is not an arbitrary choice. Any exchangeable quantity depending only on marginal entropies must take the form \(Q = \sum_i f(h_i)\) with the same function \(f\) for each variable. Extensivity (adding one variable increases \(Q\) by a fixed amount) forces \(f(h) = c \cdot h + \text{const}\). Requiring the law to apply consistently as the subset size varies eliminates the constant term. Setting \(c=1\) gives the unique form \(\sum_i h_i = \text{const}\). The fourth axiom is therefore the unique exchangeable, extensive, information-theoretic conservation law for an isolated system.

Information isolation can be seen as stronger than frame invariance. It eliminates not only preferred reference frames but appeal to external reference structures. All physically meaningful quantities must be internal to the system and relational rather than absolute. The variable partition \(\{X_i\}\) that enters the conservation constraint is a structural choice. It is part of the model specification, analogous to choosing a Hilbert space factorisation in quantum mechanics—rather than an externally privileged decomposition.

In traditional thermodynamics, energy conservation defines a built-in potential. Here, marginal entropy conservation plays the analogous role: it defines an intrinsic potential within the information geometry. The curvature of this potential, encoded in the Fisher information, acts as the metric governing how the system redistributes its informational content.

The Classical Observer

Figure: Here the observer is monitoring the movements of the particles. We’ve plotted the velocities alongside the 1 standard deviation contour of their theoretical distribution.

The Classical Observer - Correlated

Figure: Again the observer is monitoring the movements of the particles, but here their motion is correlated (\(\rho=0.95\)).

The Classical Observer - Anti-correlated

Figure: Here the observer is monitoring the movements of the particles, but here their motion is anti-correlated (\(\rho=-0.95\)).

The Classical Observer - Inaccessible

Figure: Here the observer is blocked from monitoring anything inside the sytem.

When we don’t know what’s going on inside, we can’t express outcomes in the way we could with an observer. But we can still express entropies. This highlights an interesting characteristic of entropies. If we don’t express the probability directly, but just work with the entropies themselves, it feels like we can assess the bounds of possibility without directly expressing what’s going on.

Joint Entropy

While we don’t see the underlying probability, we can capture a class of different distirbutions by considering the mapping to the system’s joint entropy.

Think of joint entropy as a scoring system: every configuration gets a number measuring its uncertainty. Once you have that, you can line them up from least to most disordered.

The \(I + H = C\) Structure

We have established four axioms, with the fourth axiom stating that the sum of marginal entropies is conserved, \[ \sum_{i=1}^N h_i = C. \] This conservation law is the heart of The Inaccessible Game, but to understand its dynamical implications, we need to rewrite it in a more revealing form.

Multi-Information: Measuring Correlation

The multi-information (or total correlation), introduced by Watanabe (1960), measures how much the variables in a system are correlated. It is defined as, \[ I = \sum_{i=1}^N h_i - H, \] where \(H\) is the joint entropy of the full system: \[ H = -\sum_{\mathbf{x}} p(\mathbf{x}) \log p(\mathbf{x}). \]

The multi-information has a nice interpretation:

• \(I = 0\): The variables are completely independent. The joint entropy equals the sum of marginal entropies.
• \(I > 0\): The variables are correlated. Some information is “shared” between variables, so the joint entropy is less than the sum of marginals.
• \(I\) is maximal: The variables are maximally correlated (in the extreme case, deterministically related).

Multi-information is always non-negative (\(I \geq 0\)) and measures how much knowing one variable tells you about others.

Using the definition of multi-information, we can rewrite our conservation law. From \(I = \sum_{i=1}^N h_i - H\), we have: \[ \sum_{i=1}^N h_i = I + H. \] Therefore, the fourth axiom \(\sum_{i=1}^N h_i = C\) becomes: \[ I + H = C. \]

This is an information action principle. It says that multi-information plus joint entropy is conserved. This equation sits behind the dynamics of the Inaccessible Game.

This equation has the structure of an action principle in classical mechanics. In physics, total energy is conserved and splits into two parts, \[ V + T = E, \] where \(V\) is potential energy and \(T\) is kinetic energy.

The analogy for The Inaccessible Game is.

• Multi-information \(I\) plays the role of potential energy. It represents “stored” correlation structure. High \(I\) means variables are tightly coupled, like a compressed spring.
• Joint entropy \(H\) plays the role of kinetic energy. It represents “dispersed” or “free” information. High \(H\) means the probability distribution is spread out, with maximal uncertainty.

Just as a classical system evolves from high potential energy to high kinetic energy (a ball rolling down a hill), the idea in the Inaccessible Game will be that the information system evolves from high correlation (high \(I\)) to high entropy (high \(H\)).

Entropy Configuration Mapping

Figure: Many configurations (density matrices \(\rho\)) map under von Neumann entropy \(S\) to a single real number. Configurations with the same entropy value are isoentropy; they form an equivalence class. The quotient is a totally ordered chain of entropy levels.

Formally, von Neumann entropy first induces a preorder on configurations \[ \rho \preceq_S \sigma \iff S(\rho) \leq S(\sigma). \] Configurations with equal entropy are mutually comparable, so the preorder is not antisymmetric. Quotienting by the induced equivalence relation \[ \rho \sim_S \sigma \iff S(\rho) = S(\sigma) \] produces a poset of entropy levels, which embeds into \((\mathbb{R}_{\geq 0}, \leq)\). This quotient is the formal entropy ladder.

We can picture the structure as a ladder: each rung corresponds to an entropy level \(S = c\), and multiple configurations sit at the same rung. Moving up the ladder means increasing entropy, more mixed, less structured. Moving down means decreasing entropy, more ordered, more pure.

This picture does not require us to know configuration the system is in at any rung, only the system sits on the ladder. We can think of dynamics in the inaccessible game as being expressed as movement along this ladder.

Figure: The entropy ladder: each rung is an isoentropy class. Multiple configurations sit at the same rung. Dynamics move the system up the ladder (entropy increase) subject to the marginal entropy conservation constraint.

Von Neumann entropy assigns a real value to each configuration, inducing a preorder on the space of density matrices. The quotient by isoentropy equivalence is a totally ordered chain of entropy levels embedded in \((\mathbb{R}_{\geq 0}, \leq)\). The Parzygnat (2022) characterisation establishes that von Neumann entropy is the unique (up to rescaling) continuous, functorial measure of information loss in \(\textsf{NCFinProb}\); this is the information-loss functor, which the entropy-level ordering reflects but is distinct from.

The Exponential Family

An important class of distribution is known as the exponential family. These distributions can be written as \[ p(\mathbf{ y}| \boldsymbol{ \theta}) = \exp(\boldsymbol{ \theta}^\top T- \psi(\boldsymbol{ \theta})) h(\boldsymbol{ \theta}) \] where \(\boldsymbol{ \theta}\) is known as the , \(T(\mathbf{ y})\) is known as the and \(\psi\) is the the log partition function, or the and \(h(\cdot)\) is known as the base measure.

For the moment we’ll ignore the base measure as for several of the distributions we’ll consider it’s constant, so we will consider the form \[ p(\mathbf{ y}| \boldsymbol{ \theta}) = \exp(\boldsymbol{ \theta}^\top T- \psi(\boldsymbol{ \theta})). \] This form yields a particularly simple likelihood function \[ L(\boldsymbol{ \theta}) = \boldsymbol{ \theta}^\top T- \psi(\boldsymbol{ \theta}) \] and since the gradient of the cumulant generating function is the first cumulant of the sufficient statistics, the gradient of the log likelihood also has a simple form. \[ \nabla_\boldsymbol{ \theta}L(\boldsymbol{ \theta}) = T- \left\langle T\right\rangle_{p(\mathbf{ y}|\boldsymbol{ \theta})}. \] where $ denotes the expecttion under the distribution \(p(\cdot)\).

Axiomatically Distinguished

{Now I want to say what the dynamics of this game look like. I want to choose the dynamics that maximise entropy production, subject to the conservation constraint. The motivation is the no-barber idea: without external structure, there’s no privileged reference, so I should select the dynamics that most efficiently increase entropy. In the Fisher information geometry, the most efficient direction is the natural gradient of entropy. The direction of maximum entropy ascent is the unique steepest-ascent direction in the Fisher (Riemannian) metric. No external structure — no Hamiltonian, no clock, no spatial coordinates — is needed to specify it. Within the inaccessible game framework, this trajectory is axiomatically distinguished: uniquely identifiable under the stated axioms, without introducing external structure (Lawrence, 2025).

The information relaxation principle says the game evolves by maximising joint entropy production subject to the marginal entropy constraint \(\sum_i h_i = C\). In natural parameter space the joint entropy gradient is \[ \nabla H = -G(\boldsymbol{\theta})\boldsymbol{\theta}. \] We enforce the constraint via a Lagrange multiplier \(\nu(\tau)\), giving constrained dynamics \[ \dot{\boldsymbol{\theta}} = -G(\boldsymbol{\theta})\boldsymbol{\theta} + \nu(\tau)\,\mathbf{a}(\boldsymbol{\theta}), \] where \(\mathbf{a}(\boldsymbol{\theta}) = \nabla\!\sum_i h_i\) is the constraint gradient and \(\nu(\tau)\) is determined by requiring \(\mathbf{a}^\top\dot{\boldsymbol{\theta}} = 0\) (the constraint is maintained). Here \(\tau\) is game time, the affine parameter tracking progress along the trajectory.

Constrained Entropy Ascent

The game’s dynamics are fixed by two requirements: maximise joint entropy production, and maintain the information isolation constraint \(\sum_i h_i = C\) exactly. Working out the explicit equations makes transparent why the linearised flow decomposes into symmetric (dissipative) and antisymmetric (reversible) parts.

For a distribution \(p(\mathbf{x};\boldsymbol{\theta})\) in the exponential family, \[ p(\mathbf{x};\boldsymbol{\theta}) = \exp\!\left(\sum_k \theta_k f_k(\mathbf{x}) - \psi(\boldsymbol{\theta})\right), \] the log-partition function \(\psi(\boldsymbol{\theta}) = \log\sum_{\mathbf{x}}\exp(\sum_k\theta_k f_k(\mathbf{x}))\) is the cumulant generating function (CGF). The moment parameters are its first derivatives, \[ \eta_k = \frac{\partial\psi}{\partial\theta_k} = \mathbb{E}_{p_{\boldsymbol{\theta}}}[f_k(\mathbf{x})], \] and the Fisher information matrix is the Hessian: \(G_{jk}(\boldsymbol{\theta}) = \partial^2\psi/\partial\theta_j\partial\theta_k\).

Shannon entropy is the Legendre conjugate of \(\psi\): \[ H(\boldsymbol{\theta}) = \psi(\boldsymbol{\theta}) - \boldsymbol{\theta}\cdot\nabla\psi(\boldsymbol{\theta}) = \psi(\boldsymbol{\theta}) - \boldsymbol{\theta}\cdot\boldsymbol{\eta}. \]

Differentiating entropy with respect to the natural parameters: \[ \frac{\partial H}{\partial\theta_k} = \frac{\partial\psi}{\partial\theta_k} - \eta_k - \sum_j\theta_j\frac{\partial^2\psi}{\partial\theta_j\partial\theta_k} = \eta_k - \eta_k - (G(\boldsymbol{\theta})\boldsymbol{\theta})_k = -(G(\boldsymbol{\theta})\boldsymbol{\theta})_k. \] So \(\nabla_{\!\boldsymbol{\theta}} H = -G(\boldsymbol{\theta})\boldsymbol{\theta}\).

The natural gradient (steepest ascent in the Fisher metric) is then \[ \nabla^{\mathrm{nat}} H = G(\boldsymbol{\theta})^{-1}\nabla_{\!\boldsymbol{\theta}} H = -\boldsymbol{\theta}. \] Unconstrained steepest entropy ascent is simply descent in the natural parameters: \(\dot{\boldsymbol{\theta}} \propto -\boldsymbol{\theta}\). This is the symmetric, dissipative part of the flow.

The information isolation constraint \(\sum_i h_i(\boldsymbol{\theta}) = C\) introduces a Lagrange multiplier \(\nu(\tau)\). The constrained natural gradient ascent is \[ \dot{\boldsymbol{\theta}} = -\boldsymbol{\theta} + \nu(\tau) G^{-1}(\boldsymbol{\theta}) \mathbf{a}(\boldsymbol{\theta}), \] where \(\mathbf{a}(\boldsymbol{\theta}) = \nabla_{\boldsymbol{\theta}} \sum_i h_i(\boldsymbol{\theta})\) is the constraint gradient and \(G^{-1}\mathbf{a}\) is its natural gradient.

Requiring the constraint to be maintained, \(\mathbf{a}(\boldsymbol{\theta})^\top\dot{\boldsymbol{\theta}} = 0\), gives \[ \mathbf{a}^\top\left(-\boldsymbol{\theta} + \nu\,G^{-1}\mathbf{a}\right) = 0 \quad\Longrightarrow\quad \nu(\tau) = \frac{\mathbf{a}(\boldsymbol{\theta})^\top\boldsymbol{\theta}}{\mathbf{a}(\boldsymbol{\theta})^\top G^{-1}(\boldsymbol{\theta}) \mathbf{a}(\boldsymbol{\theta})}. \] Substituting back gives a projected natural gradient flow: \[ \dot{\boldsymbol{\theta}} = -\boldsymbol{\theta} + \frac{\mathbf{a}^\top\boldsymbol{\theta}}{\mathbf{a}^\top G^{-1}\mathbf{a}}\,G^{-1}\mathbf{a} = -\Pi^G_\perp \boldsymbol{\theta}, \] where \(\Pi^G_\perp = \mathbf{I} - G^{-1}\mathbf{a}\,\mathbf{a}^\top / (\mathbf{a}^\top G^{-1}\mathbf{a})\) is the projector orthogonal to the constraint gradient in the Fisher metric.

Because \(\nu(\boldsymbol{\theta})\) depends on \(\boldsymbol{\theta}\), linearising the Lagrange multiplier term \(\nu(\boldsymbol{\theta})\,G^{-1}(\boldsymbol{\theta}) \mathbf{a}(\boldsymbol{\theta})\) around \(\boldsymbol{\theta}^*\) produces a contribution to the Jacobian whose antisymmetric part seeds the reversible sector \(A\) of the linearised dynamics. The symmetric part of the Jacobian comes from \(-\mathbf{I}\) (unconstrained natural gradient) and the symmetric part of \(\nabla[\nu G^{-1}\mathbf{a}]\); the antisymmetric part is sourced by the Lagrange multiplier’s \(\boldsymbol{\theta}\)-dependence (Lawrence, 2025).

GENERIC-like Structure

To understand the local structure of the dynamics we linearise around any point \(\boldsymbol{\theta}^*\) on the constraint manifold. Writing \(\mathbf{q} = \boldsymbol{\theta} - \boldsymbol{\theta}^*\), the linearised flow is \[ \dot{\mathbf{q}} = M\mathbf{q} + O(\|\mathbf{q}\|^2), \qquad M = S + A, \] where \(S = \tfrac{1}{2}(M + M^\top)\) is symmetric (positive-semidefinite) and \(A = \tfrac{1}{2}(M - M^\top)\) is antisymmetric. The symmetric part drives entropy production; the antisymmetric part generates entropy-conserving, rotation-like redistribution of information. This decomposition is the GENERIC (General Equation for Non-Equilibrium Reversible–Irreversible Coupling) structure of Grmela and Öttinger (1997) and Öttinger (2005), and it emerges automatically from the constraint geometry rather than being imposed by hand (Lawrence, 2025).

The ratio \(\|A\|/\|S\|\) varies across the constraint manifold and determines the local character of the dynamics. When \(\|S\| \gg \|A\|\) the system is in a thermodynamic regime: dissipation dominates and entropy is produced rapidly. When \(\|A\| \gtrsim \|S\|\) the system is in a mechanical regime: conservative, rotation-like dynamics dominate. The information topography — the landscape of the Fisher information \(G(\boldsymbol{\theta})\) — governs how these regimes are distributed.

Information Relaxation Dynamics

Consider a simple two-variable system with binary variables \(X_1\) and \(X_2\):

High correlation state (high \(I\), low \(H\)): \[ p(X_1=0, X_2=0) = 0.5, \quad p(X_1=1, X_2=1) = 0.5 \] The variables are perfectly correlated. Marginal entropies: \(h_1 = h_2 = 1\) bit. Joint entropy: \(H = 1\) bit. Multi-information: \(I = 1 + 1 - 1 = 1\) bit.

Low correlation state (low \(I\), high \(H\)): \[ p(X_1, X_2) = 0.25 \text{ for all four combinations} \] The variables are independent. Marginal entropies: \(h_1 = h_2 = 1\) bit. Joint entropy: \(H = 2\) bits. Multi-information: \(I = 1 + 1 - 2 = 0\) bits.

The system relaxes from the first state to the second, conserving \(I + H = 2\) bits throughout. Let’s visualise this relaxation:

import numpy as np

# Generate relaxation trajectory
n_steps = 100
alphas = np.linspace(0, 1, n_steps)

h1_vals = []
h2_vals = []
H_vals = []
I_vals = []

for alpha in alphas:
    p00, p01, p10, p11 = relaxation_path(alpha)
    h1, h2, H, I = compute_binary_entropies(p00, p01, p10, p11)
    h1_vals.append(h1)
    h2_vals.append(h2)
    H_vals.append(H)
    I_vals.append(I)

h1_vals = np.array(h1_vals)
h2_vals = np.array(h2_vals)
H_vals = np.array(H_vals)
I_vals = np.array(I_vals)
C_vals = I_vals + H_vals  # Should be constant

Figure: Left: Multi-information \(I\) decreases as joint entropy \(H\) increases, conserving \(I + H = C\). The colored regions show how the conserved quantity splits between correlation (red) and entropy (blue). Right: Marginal entropies remain constant throughout, making the system inaccessible to external observation.

The visualisation shows the trade-off: as the system relaxes, correlation structure (multi-information) is converted into entropy. The total \(I + H = C\) remains constant (black dashed line), but the system evolves from a state dominated by correlation to one dominated by entropy.

The marginal entropies \(h_1\) and \(h_2\) stay constant throughout this evolution. An external observer measuring only marginal entropies would see no change—the system is informationally isolated, hence “inaccessible.”

Classical Obstruction at the Origin

We now reach the central mathematical obstacle. Everything so far has been built on classical probability and Shannon entropy. The game’s natural origin — zero joint entropy with positive marginal entropies — is provably impossible in that framework.

The information relaxation dynamics suggest that the game begins at the origin: the state of maximum multi-information, \(I = C\), with zero joint entropy, \(H = 0\). Playing forward, multi-information is relaxed and joint entropy increases until \(H = C\) and \(I = 0\).

This natural starting point — zero joint entropy with positive marginal entropies — cannot be represented by classical Shannon entropy. The problem is the non-negativity of conditional Shannon entropy: for any two classical random variables \(X_1\) and \(X_2\), \[ H_{X_1|X_2} = H_{1,2} - H_2 \geq 0, \] because conditional entropy measures residual uncertainty after conditioning, which cannot be negative. This constraint has an immediate consequence for multi-information.

For a two-variable system, multi-information is \[ I_{1,2} = H_1 + H_2 - H_{1,2} = H_1 - H_{1|2}, \] using the chain rule \(H_{1|2} = H_{1,2} - H_2\). Since \(H_{1|2} \geq 0\) we get \(I_{1,2} \leq H_1\), and by symmetry \(I_{1,2} \leq \min(H_1, H_2)\).

At the origin we need \(H_{1,2} = 0\), which gives \(I_{1,2} = H_1 + H_2\). But this would require \(I_{1,2} > H_1\) whenever \(H_2 > 0\), contradicting the bound. The only escape is \(H_1 = H_2 = 0\), i.e. all marginals are zero — but then \(C = 0\) and there is no game.

More generally, for \(N\) variables the Shannon conditional entropy constraint forces: zero joint entropy is only possible when all marginal entropies are also zero.

The Baez–Fritz–Leinster axioms (functoriality, convex linearity, continuity) uniquely select Shannon entropy (Baez et al., 2011), but Shannon entropy structurally forbids the game’s natural origin (Lawrence, 2026b).

Von Neumann Entropy Resolution

The resolution is to replace Shannon entropy with von Neumann entropy. For a bipartite quantum system with density matrix \(\rho\), the von Neumann conditional entropy \(S_{A|B} = S_{AB} - S_B\) can be negative when the subsystems are entangled. A globally pure entangled state has \(S(\rho) = 0\) while its marginal states \(\rho_A\) and \(\rho_B\) may each have positive entropy — precisely the configuration the game needs at its origin.

An analogous axiomatic characterisation of information loss via von Neumann entropy is provided by Parzygnat (2022). This extends the Baez–Fritz–Leinster framework from classical finite probability to noncommutative probability. Adopting the Parzygnat axioms resolves the origin paradox.

Within this programme, the move to noncommutative probability is not an arbitrary modelling choice: it is forced if we insist on an origin with zero global entropy and positive marginal entropies. Von Neumann entropy and entangled pure states provide exactly what is needed, i.e. a pure global state (\(S=0\)) with positive marginal entropies (\(s_i > 0\)), and this configuration is provably unreachable in the classical framework (Lawrence, 2026b).

Having seen why the state space must become noncommutative, we now need the corresponding information geometry. The exponential family and Fisher information that organise the classical dynamics have a quantum analogue: the matrix exponential family, equipped with the Bogoliubov–Kubo–Mori metric.

The Matrix Exponential Family

The matrix exponential family is the non-commutative analogue of the classical exponential family. A density matrix \(\rho\) belongs to the matrix exponential family if it can be written as \[ \rho(\boldsymbol{\theta}) = \exp\!\left(\sum_k \theta_k F_k - \psi(\boldsymbol{\theta})\,\mathbf{I}\right), \] where \(\boldsymbol{\theta} = (\theta_k)\) are the natural parameters, \(\{F_k\}\) are Hermitian operators playing the role of sufficient statistics, and the scalar \[ \psi(\boldsymbol{\theta}) = \log\,\text{tr}\!\exp\!\left(\sum_k \theta_k F_k\right) \] is the quantum log-partition function, which ensures \(\mathrm{tr}(\rho) = 1\).

The gradient of \(\psi\) recovers quantum expectation values: \[ \nabla_{\theta_k}\psi(\boldsymbol{\theta}) = \text{tr}(\rho\, F_k) = \langle F_k\rangle_\rho, \] just as in the classical case the gradient of the cumulant generating function gives the mean of the sufficient statistic. The Hessian of \(\psi\) is the Bogoliubov–Kubo–Mori (BKM) metric, \[ G_{jk}(\boldsymbol{\theta}) = \nabla^2_{\theta_j\theta_k}\psi(\boldsymbol{\theta}), \] the quantum analogue of the Fisher information matrix. When all \(F_k\) commute this reduces to the classical Fisher information. The von Neumann entropy is recovered from \(\psi\) by a Legendre transform, \[ S(\rho) = -\mathrm{tr}(\rho\log\rho) = \psi(\boldsymbol{\theta}) - \boldsymbol{\theta}^\top\nabla\psi(\boldsymbol{\theta}), \] exactly mirroring the classical relation between the log-partition function and the Shannon entropy.

The structural parallel between the classical and matrix cases is exact, with one crucial difference: the non-commutativity of the operators \(F_k\). In the classical case the sufficient statistics \(T_k(\mathbf{x})\) are ordinary functions and commute freely. In the matrix case the Hermitian operators \(F_k\) need not commute, \([F_j, F_k] \neq 0\), and the matrix exponential \(\exp(\sum_k \theta_k F_k)\) does not factorise over the individual terms. This non-commutativity has two consequences. First, the BKM metric differs from its classical counterpart: it involves a symmetrised operator product averaged against the state \(\rho\). Second, a pure state (\(S(\rho) = 0\)) can coexist with strictly positive marginal entropies, because the marginals of an entangled pure state are mixed. The classical exponential family has no analogue of this: a pure joint distribution has zero entropy everywhere.

A density matrix \(\rho\) is faithful if it is strictly positive definite, \(\rho > 0\), meaning every eigenvalue is strictly positive. Every member of the matrix exponential family is faithful: since the matrix exponential \(\exp(A)\) is positive definite for any Hermitian \(A\), the density matrix \(\rho(\boldsymbol{\theta})\) has strictly positive eigenvalues for all finite \(\boldsymbol{\theta}\).

Faithfulness matters for two reasons. First, the BKM metric \(G(\boldsymbol{\theta})\) is only well-defined for faithful states — it involves expressions of the form \(\int_0^1 \text{tr}(\rho^t A \rho^{1-t} B)\,\text{d}t\) that are singular when \(\rho\) has a zero eigenvalue. Second, faithful states are the interior of the set of density matrices; pure states (\(S(\rho) = 0\), rank 1) lie on the boundary. As a sequence of faithful states approaches a pure state, the natural parameters \(\boldsymbol{\theta}\) diverge and the BKM metric degenerates. This is precisely why the LME origin — a pure state — lies at infinite BKM distance from any interior point and is unreachable in finite game time.

The LME Origin

The game’s natural origin is an axiomatically distinguished pure state. Since von Neumann entropy admits states with \(S(\rho)=0\) and positive marginal entropies, the origin is a globally pure state with maximally mixed reduced states on each subsystem. These are the locally maximally entangled (LME) states.

To single out a canonical origin without introducing external structure we apply an additional axiomatic selection: among all globally pure states, choose those that maximise the conserved marginal-entropy sum. Since each marginal entropy is bounded by \(s_i \leq \log d_i\) (where \(d_i\) is the dimension of subsystem \(i\)), this forces every reduced state to be maximally mixed, fixing \[ C = C_{\max} = \sum_i \log d_i. \] The origin is selected up to local-unitary equivalence (Lawrence, 2026b).

Constraint Saturation and the Gibbs Lock

At the LME origin the marginal-entropy sum is saturated termwise: every \(s_i = \log d_i\) is at its individual upper bound. This has a remarkable consequence. Since each marginal is already at its ceiling, none can increase further and none can decrease without violating the sum. The marginal entropies are locked at their maximum values for the entire trajectory: \[ s_i(\tau) = \log d_i \quad \forall i, \;\forall \tau. \] The constraint gradient \(\mathbf{a}(\boldsymbol{\theta}^*) = \nabla_{\boldsymbol{\theta}} C\big|_{\boldsymbol{\theta}^*} = \mathbf{0}\) vanishes at the origin. The usual first-order tangency condition \(\mathbf{a}^\top\dot{\boldsymbol{\theta}} = 0\) is trivially satisfied for any velocity — it no longer selects admissible directions. The constraint becomes a second-order geometric condition: admissible velocities must lie in the kernel of the constraint Hessian \(\nabla^2 C(\boldsymbol{\theta}^*)\).

Figure: The marginal-entropy conservation constraint \(s_1 + s_2 = C\) in the \((s_1,s_2)\) plane. The dashed lines mark the individual ceilings \(s_i = \log d\). Along the constraint segment, increasing one marginal entropy forces a decrease in the other.

Figure: As \(C\) increases toward \(C_{\max} = \sum_i \log d_i\), the accessible constraint segment (coloured lines) shrinks. At \(C = C_{\max}\) the constraint collapses to the single corner point (red dot), the LME origin, where every marginal entropy is individually at its ceiling and the constraint gradient vanishes.

GENERIC Dynamics at the Origin

The second-order admissible velocity subspace consists of directions that preserve all reduced states to first order: \(\dot{\rho}_i = 0\) for all subsystems \(i\). These include the commutator flows \[ \dot{\rho} = -\mathrm{i}[K_{\text{local}}, \rho], \] where \(K_{\text{local}} = \sum_i K_i \otimes \mathbf{I}_{\bar{i}}\) is a local Hermitian generator. This is precisely the von Neumann/Schrödinger equation form.

The constrained dynamics decompose into a GENERIC-compatible structure (Lawrence, 2026b): \[ \frac{\mathrm{d}\boldsymbol{\theta}}{\mathrm{d}t} \propto \underbrace{-\Pi_{\mathrm{marg}}(\boldsymbol{\theta})\boldsymbol{\theta}}_{\text{dissipative (SEA)}} + \underbrace{\mathrm{ad}_\xi\,\boldsymbol{\theta}}_{\text{reversible (Lax)}}, \] where \(\Pi_{\mathrm{marg}}\) is the Fisher-orthogonal projector onto marginal-preserving directions and \(\mathrm{ad}_\xi\) is the adjoint action corresponding to \(\dot{\rho} = -\mathrm{i}[\xi,\rho]\). The reversible sector has the structure of a Lax equation; the irreversible sector realises steepest entropy ascent within the correlation degrees of freedom.

The Origin is Unreachable

Although the origin axiomatically distinguishes the trajectory, the system can never literally have been there. The origin lies on the boundary of the natural-parameter space: as the state approaches the pure LME state, the natural parameters \(\boldsymbol{\theta}\) diverge, \(\|\boldsymbol{\theta}\| \to \infty\). The Riemannian (BKM/Fisher) metric degenerates in the direction of diverging parameters, so the origin is at infinite Fisher distance from any interior point.

The game is axiomatically directed towards an origin that acts as a limit, not a starting point that was literally occupied. The trajectory is uniquely distinguished by its asymptotic direction — the steepest-entropy-ascent direction in the Fisher metric — even though the origin itself is unreachable (Lawrence, 2026b).

Entropy Time

The natural parameter space carries a preferred affine parameter, game time \(\tau\), which tracks progress along the constrained flow. However, as the system approaches the LME origin, the natural parameters diverge (\(\|\boldsymbol{\theta}\|\to\infty\)) and the entropy production rate with respect to game time tends to zero. The origin is at infinite affine distance. This creates a degenerate parametrisation: infinite game time elapses while the system covers a finite range of entropy values.

The resolution is an axiomatically distinguished reparametrisation. An external clock is forbidden by information isolation. But there is one quantity internal to the game that provides a natural measure of progress: the entropy itself. We define entropy time \(t\) by the condition that entropy production is constant (Lawrence, 2026b).

Entropy Time

Entropy time \(t\) is defined by the condition \[ \frac{\text{d}S}{\text{d}t} = c, \] for a fixed constant \(c > 0\), where \(S = -\mathrm{tr}(\rho\log\rho)\) is the von Neumann entropy. Different choices of \(c\) correspond to affine rescalings of \(t\) and leave the integral curves unchanged.

This is much stronger than entropy being monotonically increasing. Here the time parameter is defined by entropy production: one unit of \(t\) corresponds to \(c\) nats of entropy produced. \(t\) is the unique reparametrisation (up to affine shift) for which entropy grows linearly.

The relationship to game time is \[ \frac{\text{d}\tau}{\text{d}t} = \frac{c}{\boldsymbol{\theta}^\top G(\boldsymbol{\theta})\Pi_{\mathrm{marg}}(\boldsymbol{\theta})\boldsymbol{\theta}}, \] which diverges as the origin is approached (\(\boldsymbol{\theta}\to\infty\), metric degenerates), mapping the infinite affine-time approach to the boundary into a finite entropy-time interval (Lawrence, 2026b).

Within the inaccessible game framework, the entropy time parametrisation is axiomatically distinguished: it is uniquely identifiable from within the game’s axioms, without introducing any additional external choice. Information isolation forbids a background Newtonian time, a temperature scale, spatial coordinates, or a Hamiltonian. Entropy time requires none of these — it is defined purely by the constrained information dynamics and the BFL/Parzygnat information-loss functionals.

Crucially, the game does not assume an external time and then derive that entropy increases monotonically. Instead, it singles out a preferred parametrisation of the information flow in which the irreversible (dissipative) sector is uniformised. This is Jaynes’s programme taken to its logical conclusion Jaynes (1980): take the conservation law exact, let the dynamics emerge, and measure progress by the entropy produced.

From Information Geometry to Hamiltonian Mechanics

The inaccessible game has no Hamiltonian in its axioms. Its dynamics are governed entirely by the information-geometric structure: maximum entropy production under marginal entropy conservation, with time measured by entropy production (entropy time). Yet Hamiltonian structure can emerge in metastable regions of the trajectory, as a local effective description.

The key condition is a structural alignment of the modular generator \(K(\boldsymbol{\theta}) = \log\rho(\boldsymbol{\theta})\) (in trace gauge) with a fixed operator direction. When this alignment holds, we say the system is in a Gibbs-locked region (Lawrence, 2026c).

The Gibbs-lock Condition

The Gibbs-lock condition is that the modular generator aligns with a fixed effective Hamiltonian direction \(H\): \[ K(\boldsymbol{\theta}) \approx -\beta(\boldsymbol{\theta})\,H, \] where \(\beta(\boldsymbol{\theta})\) is a smoothly varying scalar (effective inverse temperature) and \(H\) is approximately fixed along the trajectory. The exact locked point is the Gibbs thermal state \(\rho_0 = e^{-\beta_0 H}/Z\), where the reversible (Lax) sector ceases to evolve and \([K_0, H] = 0\).

This condition reverses the usual explanatory order: rather than postulating time-translation symmetry and deriving energy conservation, the game’s constraint geometry selects an approximately conserved direction \(H\) from the modular generator. The effective Hamiltonian is not an axiom — it is an emergent feature of metastable Gibbs-locked regions (Lawrence, 2026c).

The Hamiltonian Clock

Within a Gibbs-locked region, entropy time \(t\) can be converted into a Hamiltonian-frame clock \(\tau_H\) by \[ \text{d}\tau_H = \beta(t)\,\text{d}t. \] This Hamiltonian clock is a derived conversion of entropy time, not an independent temporal structure. The effective inverse temperature \(\beta(t)\) acts as a conversion factor between information-geometric progress (entropy produced) and mechanical phase (Hamiltonian time). Within the locked frame the constrained dynamics take the familiar form of a Lindblad master equation: a reversible Lax sector (the von Neumann equation) plus a dissipative sector that uniformly dephases all off-diagonal coherences in the locked eigenbasis.

These results are conditional on the trajectory entering a Gibbs-locked region. The threshold for these conditional results to carry observational content is \(W_{\mathcal{R}} \gg 1\), a Boltzmann-style multiplicity of action-resolution cells inside the locked basin — essentially the regime in which equilibrium statistical mechanics applies empirically (Lawrence, 2026c).

Conclusions

We began with a tautology — self-governing systems cannot refer to external arbitration — and asked what mathematical structure it forces. The answer, obtained by applying the no-barber principle to information theory, turns out to be surprisingly rich.

No-barber principle. Formalised through the axiomatic frameworks of Baez et al. (2011) and Parzygnat (2022), the requirement of internal adjudicability prohibits outcome spaces, Hamiltonians, clocks, and external observers from appearing as primitives. The game must define everything it uses from within (Lawrence, 2026a).

Information isolation. The no-barber principle suggests the marginal entropy sum \(\sum_i h_i = C\) as a global conservation law. Question: is information isolation a fourth axiom or a derived necessity?

Axiomatic selection. We’ve suggested that the game structure requires a pure-state LME origin with positive marginal entropies — a configuration impossible under Shannon entropy (Lawrence, 2026b). Is this necessary or sufficient?

Emergent effective rules. With no Hamiltonian, no clock, and no spatial structure in the axioms, we look for emergence. Entropy time provides an internal clock, axiomatically distinguished because it is the unique reparametrisation that uniformises entropy production. Next we’re studying Gibbs-locked regions an effective Hamiltonian emerges from the modular generator(Lawrence, 2026c).

The game suggests a quantum-mechanical structure — density matrices, von Neumann entropy, unitary evolution, Gibbs states. Can we show that this is, within the assumptions of the no-barber programme, the internally consistent language for a self-governing system that enforces information isolation and avoids external adjudication?

Thanks!

For more information on these subjects and more you might want to check the following resources.

• company: Trent AI
• book: The Atomic Human
• twitter: @lawrennd
• podcast: The Talking Machines
• newspaper: Guardian Profile Page
• blog: http://inverseprobability.com

References