Local Rules, Time Foliations, and Why Gauge Symmetry May Be Necessary

Gauge symmetry isn't just a mathematical convenience — this paper argues it may be the price of admission for time to exist at all.

The core idea is a structural analogy between two seemingly unrelated things: the question of when a spatial tiling can be "read" as a sequence of time steps (like frames of a movie), and the question of why physical field theories have gauge symmetry. Both turn out to hinge on the same underlying condition — that local rules propagate consistently across slices.

On the tiling side, the paper surveys two standard ways to formalize tilings mathematically. The first treats them as symbolic patterns on a grid (Wang tiles, shift-of-finite-type systems). The second treats them geometrically, studying the space of all translates of a given tiling. The key question is: when can you slice a 2D tiling into rows and read each row as deterministically generating the next, like a cellular automaton? The answer is: only when the tiling is "directionally closing" — meaning a half-plane of tiles uniquely fixes the adjacent row by a local rule. Most famous aperiodic tilings (Penrose, Robinson) fail this test in any axis-aligned direction. The paper also notes that this CA-like sliceability is precisely what enables time-translational invariance, and hence energy conservation — connecting computation, time, and energy in a single conceptual package.

On the physics side, the paper walks through how Maxwell's equations, Yang-Mills theory, and general relativity all split into evolution equations plus constraints when you foliate spacetime into time slices. In each case, the constraints are "first-class" in Dirac's sense: they are preserved by the Hamiltonian flow, and they generate gauge transformations. The Gauss law in electromagnetism, the non-Abelian Gauss constraint in Yang-Mills, and the Hamiltonian and momentum constraints in ADM gravity all fit this pattern. The paper traces this back to Noether's second theorem: local gauge symmetry in the action implies differential identities among the field equations, which become constraints in the Hamiltonian picture.

The heuristic thesis the paper is building toward: demanding that a physical theory admit a consistent time foliation — that you can slice the "world-tiling" into Cauchy surfaces and evolve forward deterministically — selects exactly those theories whose constraints close and propagate. That closure condition is gauge symmetry. In other words, gauge symmetry may not be an arbitrary redundancy we choose to impose; it may be forced on us by the requirement that time itself be well-defined. The paper is explicit that this is a heuristic correspondence, not a theorem, but the structural parallel is laid out carefully across both the combinatorial and field-theoretic sides.