Deterministic cellular automata, probabilistic observations: quotient measurement by embedded observers
Year: 2026
Type: article
Abstract: In the cellular-automaton programme for quantum mechanics (including 't Hooft's Cellular Automaton Interpretation and related Hamiltonian-automaton and stochastic-clock approaches), a core question still awaiting operationalization is how to define, inside the automaton, a subsystem that acts as an observer — how measurement is physically implemented, and how an embedded subsystem can operate at an effective resolution below the global update rate. We define an observer by its registration loop rather than by labelled cells: a local process counts as part of an observer S iff it enters S's registration, comparison, maintenance, or response-selection loop. S is a proper subset of the system it inhabits, with no cost-free access to resources beyond itself; since registering a difference requires holding a current item against a maintained contrast item, the distinctions S can concurrently maintain within one operational step are bounded by a registration capacity C(S). Measurement is then a finite map , inducing a partition on configuration space. A resolution asymmetry follows: S can register, at full grain, structures generated more coarsely than itself, but not structures as fine as itself — including any domain containing S, whose registration is a registration of S's own distinction structure (a second-order, self-referential measurement). Since S cannot internally generate a strict refinement of , on the full-grain self-referential task is necessarily non-injective relative to any finer comparison standpoint: a forced, internally undetectable coarse-graining (a quotient) — S cannot tell from a registered value whether its preimage is one fine-grained configuration or several. Other-directed quotienting is conditional, occurring when the target's grain outruns the observer's resolving grain or when concurrent maintenance demand exceeds C(S); self-directed quotienting is unconditional at the resolution level; and any finer standpoint invoked here is itself finite. Information thus forms only downstream of resolvability and concurrent maintenance: what S holds is either a full-grain registration or a quotient — never the pre-registration event structure itself. Two consequences follow. Probability need not enter the update rule: the substrate remains deterministic, in line with the Cellular Automaton Interpretation, while for the embedded observer probabilistic description appears as a normalized valuation on the quotient outcome space — classes corresponding to preimages it cannot unfold — a natural operational form for registration under insufficient capacity, offering a complementary, registration-level route to stochastic-clock-type behaviour. And a "slower observer" requires no modified dynamics: slowness is a property of registration structure — step size, operational window, concurrent capacity — not of the update rule. A minimal simulation can make the contrast between the two standpoints explicit: the modeller reads the full configuration and lattice structure cost-free within the simulation, while the embedded observer holds only the quotient induced by its capacity — a standpoint unavailable to any physical observer. Measurement is thus constitutive rather than read-out. All results follow from counting and partition arguments, independent of the specific update rule.
Author: Shi, Ruipeng
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225623322
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