Operational Geometry and the Light Substrate: Mathematical Foundation for Invariant Coherence

Operational Geometry (OpGeom) provides the mathematical framework that proves the Light Substrate (LS) generates an invariant coherence rate across all reference frames. We demonstrate that threading--the fundamental generative operation of the LS--produces a tree-like structure of coherence accumulation with a universal invariant: for any coherence field with n particles, sum_{i=1}^{n} frac{A_i}{d_i} = log_2(n) + C, where A_i is agency potential and d_i is threading depth. This invariant holds regardless of how the threading tree branches, proving that the structure generated by the LS is identical for all observers. Consequently, any rate that emerges from threading--including light speed c--must be invariant. We show how OpGeom's tree invariant mathematically encodes the principle that all reference frames are coherence patterns emerging from the same substrate, and therefore must measure the same fundamental rate. This paper establishes OpGeom as the rigorous mathematical foundation for the Fabric ontology.