The Generative Time Circuit Theorem (GTCT): Complete Proofs, Derivations, and Applications — Ring 5

The Generative Time Circuit Theorem (GTCT) establishes the fifth operator T in the generative chain C→K→F→U→T. Building on the dm³ contact-geometric framework of Principia Orthogona Volumes I–II, this paper proves the spiral return x₀→G⁶⁴(x₀)→G⁶⁴(x₆₄)=x₀′ with x₀′≠x₀, derives the stability radius ε₀=1/3, formalizes the g-series taxonomy, and provides complete derivations of the action-variable dissipation encoded in the contact form α=dz−λ. Includes six graded exercises, three falsifiability conditions, and open problems linking dm³ to TSVF and entanglement swapping. Ring 5 of the Principia Orthogona series. Part of the Principia Orthogona / Generative Contact Mechanics series · G6 LLC, Newark NJ, 2026. Series root: https://doi.org/10.5281/zenodo.19117399 · AXLE: https://github.com/TOTOGT/AXLE · Contact: pgrossi888@outlook.com · g6llc@proton.me · ORCID: 0009-0000-6496-2186