Theta-Field Observer-Rendering Theory: A Pre-Geometric Substrate for Rendered Geometry and Quantum Probability

This paper formulates the Theta-field observer-rendering theory as a clean axiomatic structure. The fundamental layer is pre-geometric: it contains distinctions, opposition, implication, agreement, closure, density, and stabilized records, but not primitive space, time, matter, force, particles, or gravity. Physical reality is the observer-rendered presentation of this pre-geometric substrate. The observer does not create the substrate; the observer stabilizes a screen of experience from it. The theory derives rendered dimension as the macroscopic rank of independent opposition families, builds microscopic sector geometry as a product of graded opposition chains, and obtains macroscopic Euclidean behavior through the spectral continuum limit of an underlying nearest-neighbour update graph. The paper also gives a compact quantum-probability stack: Sorkin second-order interference, composite consistency, strong positivity, Gram/amplitude representation, local tomographic rendering, substrate-priority, and Born probabilities by common resolution of one unresolved observer state. Dynamics is expressed by transition-preserving inter-record evolution, yielding unitary flow at the ray level. The result is a unified observer-rendering dictionary in which classical appearance, quantum probability, and macroscopic geometry arise as stabilized renderings of one pre-geometric informational substrate.