Tick Graph Theory and Remainder Algebra: From Causal Primitives to Quantum Mechanical Emergence

Abstract Causal set theory proposes that spacetime is a locally finite partial order (Bombelli et al., 1987; Sorkin, 2005). We extend this program by introducing remainder as a third primitive quantity. Each node of the causal graph carries three primitives—order, number, and remainder—and edges are discrete tick events in which remainder propagates between nodes. We define the tick-demand operator T̂ and the remainder operator R̂ with commutation relation T̂, R̂ = iℏτ, and write the TCC Hamiltonian Ĥ = T̂ + V(R̂). This first-order Hamiltonian creates a structural tension with the second-order Schrödinger equation. We resolve this tension by conjecturing a two-component embedding of TCC dynamics—motivated by the physical interpretation that remainder propagates in two directions, accumulating and resolving—and applying the Foldy-Wouthuysen (FW) transformation. The Schrödinger equation emerges as the non-relativistic limit, with ℏτ = ℏ fixed by the correspondence principle. Three correction terms arise at next order: a kinetic energy correction (~∂⁴ρ), a Darwin-type contact term (~∂²V/∂ρ²), and a remainder-demand coupling (~(∂V/∂ρ)·∂ρ) with no analog in standard quantum mechanics. We discuss the measurement problem, causal irreversibility, the Born rule as an emergent property, testable predictions that distinguish TCC from existing programs, and catalog the open problems that must be resolved for TCC to make independent empirical predictions.