Pure Temporal Geometry: The Invariant

Volumes I–III of Pure Temporal Geometry established a complete, self-consistent framework for a discrete, relational, 5-dimensional temporal manifold. The framework defined the state vector T_n = (Δ_n, Φ_n)^T, the five stepwise operators, the 4-grid of causal identities, and the dynamics of becoming. However, the framework assumed — but never formalised — the existence of an invariant reference frame. Without an absolute origin, the coordinates of Pure Temporal Geometry float. Signs become relative. The 4-grid loses its fixed meaning. Causality degenerates into local opinion. This treatise corrects the omission. We formalise Axiom Zero: the Anchor Triad, a set of at least three systems permanently occupying the null state (0,0). We verify by explicit matrix calculation that T₀ = (0,0) is the unique fixed point of all five operators. We prove seven structural theorems establishing that the Anchor Triad provides the absolute reference for signs, magnitudes, ordinal structure, and causal ordering. We prove that without the Anchor, the 4-grid collapses, entropy loses its baseline, and causality is impossible. We further prove that the Anchor is the First Cause of all systems within Pure Temporal Geometry — not a temporal cause that precedes events in sequence, but a structural cause that is the foundational ground making causation possible at all. We identify the structural analogues of the Anchor across physics, politics, economics, spirituality, and psychology, and clarify the methodological status of these domain mappings as structural illustrations, not normative prescriptions.