Fold Theory: From Causal Parsing to Emergent Lorentzian Geometry

This preprint develops Fold Theory as a speculative but falsifiable framework for the emergence of spacetime from a more fundamental causal structure.

The central proposal is that metric spacetime is not taken as fundamental. Instead, the microscopic level is described by discrete causal relations between elementary events, supplemented by local state variables and, potentially, quantum amplitudes. A FoldTick is defined as a candidate local transition associated with a change of a fold order parameter across an elementary causal link. Such local events acquire physical meaning only when their collective organization survives global causal, geometric and dynamical consistency conditions.

The framework introduces a causal parser that maps microscopic causal data to candidate macroscopic Lorentzian geometry through coarse-graining. The distinction between local FoldTicks and a collective geometric Fold is emphasized: the latter denotes a transition toward a non-degenerate, Lorentzian and geometrically admissible sector rather than a literal folding of space.

The paper further explores Null-Timelike Complementarity, a minimal 1+1-dimensional causal model, the emergence of effective horizons through global reachability, and a possible interface with spinorial formulations of Loop Quantum Gravity and spin-foam models. The recovery of special relativity, Regge geometry, the Holst action and Einstein gravity is formulated as a sequence of conditional reconstruction targets rather than established results.

A set of explicit no-go criteria is provided. The framework fails if it cannot recover a Lorentzian continuum limit, a universal causal speed, stable timelike and null sectors, the correct spin-2 gravitational sector, or a regulator-independent low-energy description compatible with general relativity.

The purpose of this work is therefore not to claim a completed theory of quantum gravity, but to convert the Fold intuition into a mathematical research program with identifiable intermediate constructions, numerical tests and falsification conditions.