The Dual‑Invariant Equation: Gaussian Rigidity and the Minimal‑Uncertainty Gauge

This primer formalizes the Dual‑Invariant Equation as the representational bedrock of the admissibility framework: a minimal‑uncertainty, Fourier self‑dual gauge anchored by the Gaussian, in which the Heisenberg inequality is sharply saturated only by Gaussians and the Hermite ladder furnishes a stable, orthonormal eigenbasis for the Fourier transform. By fixing this rigid gauge, the note shows how admissibility comparisons—distances, spreads, gradients, and capacities—become basis‑independent and therefore well‑posed across domains. In this anchored metric, the Construct Law Equation composes constraints into a single feasibility ledger, the Master Structure Equation executes constrained evolution with normal‑cone/KKT boundary reactions, and Maximal Admissibility Surfaces (MAS) appear as active load–capacity faces where no smooth crossing is possible and evolution becomes tangential. The result is a concise, mathematically grounded reference that explains why the Gaussian/Hermite structure is not a modeling choice but a rigidity requirement, and how this gauge underwrites consistent admissibility geometry in applications ranging from quantum nonlocality (Tsirelson‑bounded correlations) to horizon physics (Rindler‑anchored modes) and atomic exchange (invariant screening and correlation filters).