Computation Lives on Foliations: Church--Turing, Gauge Symmetry, and the Mathematical Excess of Physics

Physics contains more than any agent can compute, and the boundary runs exactly where time begins.

The paper's central move is to ask: when does physics look like a computation, and when doesn't it? The answer turns on whether you can slice the world into a sequence of "nows" — a foliation — where each slice determines the next by a local rule. When you can, you get a Cauchy problem, a Turing machine, a brain updating its model. When you can't, you still have a perfectly well-defined mathematical object, but no algorithm generates it. The slogan the paper coins: Turing computation lives on slices; mathematics describes the whole tiling.

The tiling analogy earns its keep here. A Wang tiling assigns colored tiles to an infinite grid so that adjacent edges match. The rules are finite and local. But whether a given ruleset tiles the whole plane is undecidable (Berger, 1966), and some rulesets have tilings that exist — provably, by a compactness argument — yet no algorithm can output them. This is not a curiosity; it is the template. A universe governed by local laws can be globally fixed and globally noncomputable. The paper proves a clean lemma: if a tileset plus a finite seed patch has exactly one valid completion, that completion must be computable. So a unique-but-noncomputable universe requires either infinite boundary data acting as an oracle, or a global selection principle (like least action) that picks one solution from many without any agent being able to run the selection.

The same architecture reappears in Hamiltonian physics. De les Coves and Cubitt showed that a simple 2D Ising-with-fields model is universal: its low-energy sector contains the complete physics of every classical spin system. Universality here is a static containment order — one law structurally includes another, with no dynamics required. Time enters only when you choose a "transfer direction," which is just a foliation by another name. And universality breeds undecidability: the very feature that lets one Hamiltonian contain all others guarantees that some questions about what it contains escape every algorithm. Berger's theorem in combinatorics and the undecidability of the spectral gap in quantum many-body physics are the same fact in different clothes.

Constructor theory (Deutsch–Marletto) gets a precise dissection. Its modal layer — which tasks are possible — is a boundary-value question about the admissible set of configurations, entirely time-free, and it inherits undecidability from universality. But constructors themselves — systems that perform a task and can do it again — require "again" to mean something, which requires a foliation. The modal skeleton of constructor theory lives at the law level; the operational machinery lives on slices. The paper argues this split is sharper than constructor theory's own presentation acknowledges.

The closing move is Tarskian. Gödel showed that provability doesn't exhaust truth in a structure. The paper claims the ontic analogue: algorithmic generability doesn't exhaust structural existence. A noncomputable tiling exists in the sense that the constraint is satisfiable — a fact certifiable in a metatheory — even though no computation outputs it. Agents are finite proof-engines; they inhabit the foliable sectors where computation works. The mathematical excess of the title is the gap between what agents can generate and what the structure simply is.