Emergence I (Extended Edition with Full Derivations): A Unified Field Theory from Wave Intersections on a Pre-Geometric Canvas

This paper presents a complete, mathematically rigorous unified field theory in which spacetime, quantum mechanics, gauge forces, and gravity emerge from a single mechanism: wave intersections on a pre-geometric canvas. The theory is built from first principles. The fundamental entities are mathematical oscillators (pulsating points) with intrinsic frequency: one time point and three space points. Their analog oscillations generate continuous space waves and time waves. When a space wave and a time wave intersect above threshold, a closed wave — a spacetime particle — forms. These particles collectively form a discrete voxel lattice, which is physical spacetime. Gravity arises from compression of this lattice. From these postulates, this paper derives step by step: · The Einstein field equations (via Regge calculus with a complete convergence proof)· Maxwell's equations (U(1) gauge theory)· Yang-Mills equations (SU(2) and SU(3) gauge theories)· The Schrödinger equation (from slowly varying envelope approximation)· The Born rule (from scale separation and time-averaged intensity)· The Klein-Gordon and Dirac equations· The Standard Model gauge group SU(3)×SU(2)×U(1) (from spatial charge)· The spin-statistics connection (from loop twist topology) The paper is presented in two parts: · Part I preserves the original 2D formulation with 36 postulates and five core equations, capturing the essential physical picture with simplicity and clarity.· Part II generalizes to 3+1 dimensions, introduces the sixth core equation (spatial charge and gauge symmetry), and provides complete derivations. A postulate reduction table demonstrates that the 36 original postulates condense to 7 fundamental postulates, with the remainder becoming derived theorems. The model resolves the measurement problem (threshold collapse), black hole singularities (minimum lattice spacing ℓ_P), and the information paradox (closed wave phases). Open problems are acknowledged honestly. What this paper is: A complete, self-contained derivation of the fundamental equations of physics from a small set of pre-geometric postulates. What this paper is not: A compatibility study. (Compatibility with 50+ phenomena across all of physics is demonstrated in companion Papers II–VIII.) Keywords: unified field theory, canvas model, emergence, pre-geometric, wave intersections, discrete spacetime, lattice gravity, quantum gravity, Einstein equations, Maxwell equations, Yang-Mills, Schrödinger equation, Born rule, spin-statistics, dark matter, black hole remnants