Mathematical Foundations of Multi‑Orbit Identity Theory within the TO/TOGT Operator Framework

This paper formalizes Multi‑Orbit Identity Theory within the operator‑algebraic framework of Topographical Orthogonal Generative Theory (TO/TOGT). Identity orbits are defined as operator‑generated closed trajectories with invariant structure. Multi‑orbit systems arise when several such orbits coexist, interact, resonate, or unify under the U‑, R‑, L‑, and B‑operator families. All constructions are expressed using dynamical systems, categorical invariants, and compositional operator algebra. This work is mathematically independent of speculative physical or cosmological models that use similar terminology (e.g., multi‑orbital particle theories or discrete‑spacetime cosmologies). Here, “orbit” refers strictly to operator‑generated identity trajectories in dynamical systems, not to physical orbital shells or cosmological structures. All results are presented in a “do not trust, verify” manner: every claim is either proved directly or inherited from previously published theorems in the Principia Orthogona series.