From Information Conservation to Orbit Foliation and Generalized Energy

Energy conservation is a theorem, not an axiom — and this paper proves it from a single information-theoretic condition.

The starting point is deliberately minimal: just a set of states and a directed graph saying which transitions are allowed. No Hamiltonian, no action, no metric, no probability. The one thing the paper demands is information conservation — given a complete state, the law, and a time offset (positive or negative), you need zero additional bits to identify the actual next or previous state. In graph terms: every node has exactly one outgoing edge and exactly one incoming edge. No branching futures, no merging pasts.

That single condition turns out to be surprisingly strong. No branching means the dynamics is deterministic. No merging means it's reversible — you can always recover the past. And because any external clock, drive, or controller that picks among branches would have to inject information from outside, the system must be closed and autonomous. Three properties for the price of one constraint. The law is then forced to be a bijection on state space — a permutation if the state space is finite, a countable permutation if infinite. And a bijection partitions state space into disjoint, non-intersecting orbits: complete trajectories that never split or merge.

Here's the payoff. Each state can be labeled by two coordinates: which orbit it's on (call it α), and where along that orbit it sits (the phase τ). Dynamics advances τ and leaves α unchanged. So α is conserved — automatically, for free, without assuming any energy function. The paper calls α the generalized energy: the maximal conserved invariant, in the precise sense that every conserved quantity factors through it. Ordinary physical energy — the Hamiltonian, the Noether charge of time-translation symmetry — is not this label but one coordinate of it, recovered only after you add smooth manifold structure, a symplectic form, and time-translation symmetry. Momentum, angular momentum, and other charges are further coordinates. The map from orbit label to scalar energy is many-to-one: many distinct orbits can share the same energy value.

One thing the paper is careful to separate out: time-reversal symmetry. Information conservation says you can run an orbit backward — the past is recoverable. Time-reversal symmetry is stronger: it says the backward-running orbit is itself a valid forward orbit under a mirror map Θ. That's an additional, optional symmetry of the orbit structure, not a consequence of information conservation alone. The paper also connects dissipation to this framework cleanly — a merging transition destroys exactly one bit of past information, and in ergodic symbolic settings the asymptotic dissipation rate relates to Kolmogorov–Sinai entropy via Brudno's theorem.