The Inaccessible Game
Information Isolation and Selected Dynamics
Neil D. Lawrence
2026-05-20
Information Theory Seminar, Centre for Mathematical Sciences (MR5), University of Cambridge
The Munchkin Provision
So there’s a board game called Munchkin — it’s a card game I used to play with my kids. You have cards that battle each other. The interesting thing about this game is that it explicitly says in the rules that the rules can be inconsistent. There’s a set of written rules, and then there are cards which have different rules on them.
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.
Munchkin Card Game
And it says: if the rules are inconsistent, any other dispute should be settled by loud arguments, with the owner of the game having the last word. And it occurs to me that the Munchkin provision is a property of almost every game or activity we engage in — there has to be an arbitrator to settle how the rules go. But you can reverse that and say: a self-governing system cannot refer to external arbitration. That’s a tautology. If you want a self-governing system, it can’t ask the owner of the game what to do next. And the interesting thing I want to explore today is whether there’s anything interesting in that tautology that we can use to say something about self-governing systems.
Rules may be inconsistent … so …
Any other disputes should be settled by loud arguments, with the owner of the game having the last word.
Munckin Rules (Jackson, 2001)
A Tautology
Self-governing systems cannot refer to external arbitration.
Why should we care about self-governing systems? Well, you’d expect the rules of physics to be self-governing. It would be a problem if a rule of physics had to pause and check VAR — an external decision made outside the game. That clearly cannot work for a dynamical system we expect to be self-sustaining and self-adjudicating. So I’m going to try and formalise this tautology and see what mathematical structure it forces on us.
The No-Barber Principle
So I’ve got three papers in this series so far, and the third paper is actually about this. It talks about what I call the no-barber principle, which is inspired by Russell’s paradox — the idea that the barber shaves everyone in the village apart from those who shave themselves, followed by the question: does the barber shave themselves? In set theory it’s a question around the set of all sets that don’t belong to themselves. This leads to what one would call impredicative circularity. If the barber does shave themselves, then the rules say they don’t; if they don’t, the rules say they do.
Russell’s Barber Paradox:
Barber shaves all who don’t shave themselves
Does the barber shave themselves?
Paradox: Definition includes itself in scope
No External Adjudicators
Forbidden:
External observer
Pre-specified outcome space/Hamiltonian
Privileged decomposition
External time parameter
No appeal to structure outside the game
That’s the sort of thing we want to forbid. And that’s what motivates the no-barber principle. The no-barber principle basically says we want to be in a situation where there are no external adjudicators. That means no external observer, no predefined outcome space, no external time parameter, and no privileged decomposition in the game. Nothing dictated from outside. And I’ll give you a very disconnected sense of how this manifests. Imagine if you had an external time parameter in the game, and as part of that game an entity’s time parameter was manipulated so it could move in time. A time travel paradox has the same impredicative circularity feel as Russell’s paradox — went back in time, met your mother, do you continue to exist? And that seems to come from someone outside the system playing with the time. If something emerged consistently inside the system, that shouldn’t be able to happen. So the idea is to create a game where you’re not appealing to any structure outside the game.
The core idea actually came to me on the train on the way back from watching Sheffield United lose the playoff final. I was sober enough to think clearly, and this is almost exactly a year ago, though I’ve been thinking about this problem for longer.
The Three Axioms
What they show is that from three axioms — functoriality (information loss adds when you compose processes)
\[F(f \circ g) = F(f) + F(g)\]
Information loss is additive
Compose processes → add losses
Convex Linearity
Convex linearity (if you choose between two processes with probability lambda, the information loss is the weighted sum)
\[F(\lambda f \oplus (1-\lambda)g) = \lambda F(f) + (1-\lambda)F(g)\]
Probabilistic mixture of processes
Linear in probability weights
Continuity
And continuity — from those three axioms alone, the unique form for information loss is the Shannon entropy difference, scaled by some factor.
Small change in process
Small change in information loss
\(F(f)\) continuous in \(f\)
The Main Result
Three axioms \(\Rightarrow\) unique form: \[F(f) = c(H(p) - H(q))\]
Information loss = scaled entropy difference
Shannon entropy emerges from axioms
No other measure satisfies all three
I really like this because it appeals to some idea I’d like to be true: that entropy is more fundamental than probability. Probability always asks what will happen, what does happen. Entropy, especially in things like channel coding, gives you impossibilities. Impossibilities are much more robust than predictions. That’s one reason I’m fascinated by entropy, though I’m probably less expert in it than everyone else in the room.
The Inaccessible Game Setup
So what does the no-barber principle help us with? If I want to avoid external structure, and I now have a mechanism for representing information loss, what I want to do is enforce information conservation. If we have defined a form of information loss, it feels like we can now isolate the game from observation or interaction — we can say that no external observer can extract or inject information. The aim is to build a game in the following way: a representation of information loss and a prohibition of information exchange with the game. I’m certainly not doing this perfectly, but that’s the sort of thing I’m aiming for. It’s like a selection principle for the mathematics I want to use in the game.
Avoid external structure.
Represent information loss
Enforce information conservation
Marginal Entropy Conservation
And then the claim is that the sum of those marginal entropies is conserved. I haven’t seen this type of constraint before. And I sort of pause there — everything falls from that. If it’s flawed in some way, everything else I’ve done collapses, but don’t let that make you shy about criticising it.
\[
\sum_{i=1}^N h_i = C
\]
Isolation: cf energy conservation — but for information
The reason I want it to be this atomic form is a notion of exchangeability amongst the game’s variables. If I start talking about interactions, I think there’s a danger of breaking that. It’s also an extensive property — it feels like the type of property where in the future I could take it to infinity and do something non-parametric. I haven’t done that, but I’m not ruling it out. And I don’t know what the underlying dimension is — that would be an external thing. The aim is to isolate the system, which will prevent barbers from intervening.
So why was I looking for something like this? Well, in these kinds of systems, if you look at how effective rules emerge in things like physics, energy conservation tends to be a thing. There has to be something conserved — some constraint.
The Classical Observer
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 - Inaccessible
Here the observer is blocked from monitoring anything inside the sytem.
Joint Entropy
We don’t see see the outcome space
But we know it has a joint entropy
Now imagine that the observer is not integrated. They cannot see inside the system. They don’t have access to the outcome space. We don’t see the individual outcomes; we just know there’s a joint entropy associated with the system. The observer has a marginal entropy for each variable they interact with, but they don’t have access to the joint. This is the inaccessible part — the inner workings are hidden. The observer has their own perspective, not direct access to the internals of the game.
The \(I + H = C\) Structure
So here’s the key relationship. From \(\sum_i h_i = C\) , with a little algebra — using the fact that for a joint distribution the joint entropy equals the sum of marginal entropies minus the multi-information — I can show that \(I + H = C\) .
\[
\sum_{i=1}^N h_i = C
\]
What does this conservation imply for dynamics?
Analogy to classical mechanics
Energy: \(V + T = E\)
Information: \(I + H = C\)
System “rolls downhill” from correlation to disorder
Think of it like potential energy and kinetic energy: the correlation is potential energy, the entropy is kinetic energy. In equilibrium you’d have an equal mix. But we start in a highly correlated state — that’s what I mean by the origin — and the system evolves to maximize entropy.
So now I have the game. I’ve got \(I + H = C\) . And I want the dynamics of the system to maximise \(H\) . But to set up what those dynamics look like, I need to parameterise the configuration space.
Entropy Configuration Mapping
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.
The Entropy Ladder
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.
In between are all the mixed states. And entropy labels the rungs. The entropy induces a preorder on configurations — not a total order, because many configurations share the same entropy value. If you quotient by the equivalence relation that says two configurations are equivalent if they have the same entropy, you get a poset of entropy levels, which embeds totally ordered in the non-negative reals.
The Exponential Family
\[
p(\mathbf{ y}|\boldsymbol{ \theta}) = \exp\!\left(\boldsymbol{ \theta}^\top T(\mathbf{ y}) - \psi(\boldsymbol{ \theta})\right)
\]
\(\boldsymbol{ \theta}\) : natural parameters\(\psi(\cdot)\) : cumulant generating function\(G(\boldsymbol{ \theta}) = \nabla^2\psi(\boldsymbol{ \theta})\) : Fisher information
The standard exponential family is the right parameterisation for working with entropy on this configuration space. The normaliser is what machine learners would call the evidence — the cumulant generating function. The natural parameters are what you’d use to label where you are in the information geometry. The Fisher information matrix \(G\) is the second derivative of the cumulant generating function with respect to the natural parameters. This is the metric on the configuration space. It tells you the local geometry of the space. And the natural gradient is the direction you move in this space that maximises entropy production most efficiently in terms of Fisher information. This becomes the core of the dynamics.
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.
A choice is axiomatically distinguished if it is uniquely identifiable within the game’s axioms — without external structure such as Hamiltonians, clocks, or coordinates.
Maximum Entropy Production
And the natural gradient of entropy with respect to the natural parameters turns out to be \(-\boldsymbol{\theta}\) . So without the constraint, the dynamics are \(\dot{\boldsymbol{\theta}} = -\boldsymbol{\theta}\) — just exponential decay to equilibrium. That’s the axiomatically distinguished trajectory for an unconstrained system.
Maximum entropy production: unique in Fisher metric
Constraint: marginal entropy conservation
Maximum Entropy Production
Maximise \[
\frac{\text{d}H}{\text{d}\tau}
\] subject to \(\sum_i h_i = C\)
Constrained Entropy Ascent
Entropy via the Log-Partition Function
\[H(\boldsymbol{\theta}) = \psi(\boldsymbol{\theta}) - \boldsymbol{\theta}\cdot\nabla\psi(\boldsymbol{\theta})\]
\(\psi(\boldsymbol{\theta})\) : log-partition function (CGF)\(\boldsymbol{\eta} = \nabla\psi\) : moment parameters \(\eta_k = \mathbb{E}[f_k]\) \(G(\boldsymbol{\theta}) = \nabla^2\psi\) : Fisher information matrix
Natural Gradient of Entropy
\[\nabla_{\!\boldsymbol{\theta}} H = -G(\boldsymbol{\theta})\boldsymbol{\theta}\]
Natural gradient: \[\nabla^{\mathrm{nat}} H = G^{-1}\nabla_{\!\boldsymbol{\theta}} H = -\boldsymbol{\theta}\]
Steepest entropy ascent \(\Rightarrow\) \(\dot{\boldsymbol{\theta}} \propto -\boldsymbol{\theta}\)
Descent in natural parameters — the symmetric part
Constrained Natural Gradient Dynamics
\[\dot{\boldsymbol{\theta}} = -\boldsymbol{\theta} + \nu(\tau)\,G^{-1}(\boldsymbol{\theta}) \mathbf{a}(\boldsymbol{\theta})\]
\(\mathbf{a}(\boldsymbol{\theta}) = \nabla_{\!\boldsymbol{\theta}}\!\sum_i h_i\) — constraint gradient.
\[\nu(\tau) = \frac{\mathbf{a}^\top\boldsymbol{\theta}}{\mathbf{a}^\top G^{-1}\mathbf{a}}\]
When you add the constraint, you get a Lagrange multiplier. And the Lagrange multiplier has to be solved for from the consistency condition. You get a projected natural gradient flow. The projection of \(-\boldsymbol{\theta}\) onto the constraint surface, using the Fisher metric, is the constrained dynamics. And the key thing — which is the reason this connects to GENERIC — is that when you linearise around an equilibrium point, the Lagrange multiplier is a function of \(\boldsymbol{\theta}\) . Its \(\boldsymbol{\theta}\) -dependence, when you differentiate, generates an antisymmetric term in the linearised dynamics. That antisymmetric term is what becomes the reversible Hamiltonian dynamics.
GENERIC-like Structure
The linearised dynamics decompose into a symmetric part and an antisymmetric part. The symmetric part is irreversible and produces entropy. The antisymmetric part is reversible — entropy-conserving rotations.
Linearise around \(\boldsymbol{\theta}^*\)
\(\mathbf{q} = \boldsymbol{\theta} - \boldsymbol{\theta}^*\)
Linearised Flow
\[\dot{\mathbf{q}} = M\mathbf{q}\] where \(M = S + A\)
\(S\) is symmetric and irreversible (entropy production)\(A\) is antisymmetric and reversible (entropy-conserving)
And this structure is exactly what’s called GENERIC: General Equation for Non-Equilibrium Reversible–Irreversible Coupling. It was derived by Öttinger and collaborators in a very different way — from a first-principles non-equilibrium thermodynamics perspective. What’s interesting here is that this structure is being derived from the information-geometric setting of the inaccessible game, not imposed from outside. It’s a consequence of maximum entropy production plus information isolation.
Classical Obstruction at the Origin
Here’s what I think is the most interesting result from the origin paper. We want the game to start from some origin state. Thinking about it like thermodynamics, you start with zero entropy — the most ordered state possible — and the system evolves towards maximum entropy. In the classical world, the origin would be a probability distribution with zero joint entropy, which means a point mass, a delta function. But a delta function has zero marginal entropies too.
So if the marginal entropies sum to a constant, and the constant is \(C\) , then \(\sum_i h_i = C > 0\) . We’ve derived that the marginal entropies must be positive at the origin. But classical probability cannot give zero joint entropy with positive marginal entropies — information sub-additivity means the joint entropy is at least as large as each marginal.
Boundary Condition \[
I = C, \quad H = 0
\]
Conditional Shannon entropies always \(\geq 0\) .
Prohibits \(H=0\) with positive marginals.
Von Neumann Entropy Resolution
The only way out is von Neumann entropy and entanglement. An entangled pure state has zero joint entropy and positive marginal entropies. That is the mathematical necessity of going to von Neumann. Within this programme, the move is forced.
Entanglement leads to negative conditional entropy.
Pure entangled state: \[
S(\rho_{AB}) = 0, \quad S(\rho_A) > 0, \quad S(\rho_B) > 0
\]
The Matrix Exponential Family
In the quantum version, the probability distribution is replaced by a density matrix. The exponential family becomes the matrix exponential family. The natural parameters now label operators, and the state is \(\rho = \exp(K - \psi I)\) where \(K\) is a Hermitian operator. The BKM metric — the Bogoliubov–Kubo–Mori metric — plays the role of the Fisher information.
\[
\rho(\boldsymbol{\theta}) = \exp\!\left(\sum_k \theta_k F_k - \psi(\boldsymbol{\theta})\,\mathbf{I}\right)
\]
\(\boldsymbol{\theta}\) : natural parameters\(\psi(\boldsymbol{\theta}) = \log\,\mathrm{tr}\exp\!\left(\sum_k\theta_k F_k\right)\) : cumulant generating function\(G(\boldsymbol{\theta}) = \nabla^2\psi(\boldsymbol{\theta})\) : BKM metric (quantum Fisher information)
Faithful States
And the faithful states, the ones where \(\rho\) is strictly positive definite, are the interior of this family. The boundary consists of pure states, and the BKM metric diverges at the boundary. This is what makes the pure-state origin special — it is infinitely distant in the BKM metric.
Implies faithful states (full rank \(\rho\) )
Pure states are on boundary of family
BKM Metric is divergent
The LME Origin
There’s a beautiful consequence of all this. The LME origin — the locally maximally entangled state — is the unique pure state where all marginal entropies are at their maximum allowed value. It’s the state that saturates the constraint. When you compute the BKM distance from any mixed state to this pure state, you get infinity — it’s infinitely far away in the quantum information geometry.
But what should the value of \(C\) be? By our axiomatically distinguished principle we should set it to some unique number. For example the maximum value.
Globally pure state: \(S(\rho)=0\)
\(C = C_{\max} = \sum_i \log d_i\) (axiomatically distinguished)Implies each marginal maximally mixed: \(s_i = \log d_i\)
Constraint Saturation and the Gibbs Lock
So the trajectory that maximises entropy production approaches this state only asymptotically as you go backwards in time. The origin is never actually reached; it is a distinguished asymptote. And the saturation diagrams show this: the marginal entropies are constrained to sum to \(C\) , and they can’t both be individually at their maximum unless the system is at the LME state, which would require zero joint entropy. That’s the second-order constraint kicking in: as the system approaches the saturation boundary, a new constraint is activated.
Marginal entropies linked: \(s_1 + s_2 = C\) (conserved sum)
Individual ceilings: \(s_i \leq \log d_i\)
Trade-off: as one marginal rises, the other must fall
Linked Marginal Entropies
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.
Saturation and Second Order
At \(C=C_{\max}\) : every \(s_i = \log d_i\) — each at its individual ceiling
Marginals locked: \(s_i(\tau) = \log d_i\) for all time
Saturation of Constraint
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.
Saturation of Constraint
First-order condition vacuous
Admissible velocities: \(\dot{\boldsymbol{\theta}}\in \ker\nabla^2 \sum_i h_i\)
GENERIC Dynamics at the Origin
At the LME origin, constraint geometry produces a GENERIC decomposition (Lawrence, 2026) :
Reversible (Lax): \(\dot{\rho} = -\mathrm{i}[K,\rho]\) — von Neumann equation emerges
Irreversible (SEA): steepest entropy ascent in marginal-preserving subspace
The Origin is Unreachable
\(\|\boldsymbol{\theta}\|\to\infty\) as \(\rho\to\rho_{\text{pure}}\) Fisher (BKM) metric degenerates at boundary
Infinite Fisher distance — never literally reached
Trajectory distinguished by its asymptotic origin, not a literal start
Entropy Time
One of the things that the no-barber principle prohibits is an external time parameter. So there’s no external clock. Instead, we use entropy as a measure of progress. You can reparameterise the trajectory so that entropy increases by a fixed amount per unit of the new parameter — entropy-time. This is an internally defined clock.
Need: an internal clock — external clocks forbidden by isolation
Game time \(\tau\) : affine parameter, degenerates at origin
Entropy Time
\[\frac{\text{d}S}{\text{d}t} = c \quad \text{(constant entropy production)}\]
1 unit of \(t\) = \(c\) nats of entropy produced
No external clock, temperature, or Hamiltonian
Progress measured by entropy produced
In any reparameterisation, what you really care about is the rate at which entropy is produced per unit entropy-time. Once you do this, you get a very clean separation between the reversible dynamics and the irreversible dynamics, because the irreversible part is the one that produces entropy. The reversible part — the antisymmetric part — produces no entropy. It’s pure rotation.
The Gibbs-lock Condition
\[K(\boldsymbol{\theta}) \approx -\beta(\boldsymbol{\theta})\,H\]
\(H\) : fixed operator\(\beta(\boldsymbol{\theta})\) : smoothly varying inverse temperature
The Hamiltonian Clock
Has an associated Hamiltonian clock
\(\beta(t)\) : conversion between entropy-time and Hamiltonian clock
Conclusions
No barber principle
Information isolation
Axiomatic selection
Emergent effective rules
So to summarise: starting from a tautology — self-governing systems can’t appeal to external arbitration — and applying information theory, you get a surprisingly specific set of conclusions. The no-barber principle rules out external outcome spaces, clocks, Hamiltonians, and observers as primitives. Information isolation gives you a conservation law for marginal entropies. The requirement for a pure-state origin with positive marginals forces you into quantum probability — you can’t stay classical. The GENERIC structure and entropy time emerge from the dynamics. And the idea is that a Hamiltonian would have to emerge conditionally, in Gibbs-locked regions.
There are three papers now: the inaccessible game sets up the framework and derives the GENERIC structure. The origin paper derives the necessity of quantum probability. And the Hamiltonian paper derives the emergence of Hamiltonian dynamics from the Gibbs-locked region. Each paper has a “work in progress” quality — I’m not claiming these are complete. What I hope to have shown is that the tautology is not empty: it has mathematical teeth.
I was told I should finish with a question. Here’s mine: can we show that quantum mechanics — density matrices, von Neumann entropy, unitary evolution, Gibbs states — is, within the assumptions of the no-barber programme, the unique internally consistent language for a self-governing system that enforces information isolation? I don’t know the answer yet, but I think it’s the right question to be asking.
