The Separatrix Invariance and Z2 Parity Algebra

A broad class of empirical systems exhibits a codimension-one separator (a separatrix) across which drift reverses or transport laws change, together with a latent Z2 ambiguity that flips a handedness marker while leaving an energy-throughput quantity invariant. This work formalizes a universality template: under a Z2 action on a lifted state space, the separatrix becomes a coorientation invariant, and any admissible pair of observables (even throughput, odd marker) induces a portable seam certificate based on an involutive pairing relation.

The main theorems state (i) coordinate-free invariance of the separatrix as a drift-null hypersurface under chart reparameterizations, (ii) exact parity closure of even/odd observables under the Z2 involution, and (iii) stability of the seam certificate under small perturbations of the analysis lens, with explicit failure modes under null operations that destroy parity structure. The results separate what is universal (parity-separatrix structure and its invariants) from what is domain-specific (microphysical closure). The claim pursued here is narrow and testable: when these two structures coexist, there exists a portable diagnostic certificate of “seamness” that is independent of microphysical details.

The certificate is expressed as an involutive pairing constraint on windowed observables: an iota-even diagnostic must close across paired windows, while an iota-odd marker must invert. This paper isolates the separatrix invariance and Z2 parity algebra as the core, then treats concrete spectral constructions (septimal echo kernels, etc.) as corollaries rather than foundations.

The universality-class claim is falsified in any candidate system if (i) no stable Z2 involution can be defined that flips an odd marker while leaving an even diagnostic invariant; or (ii) no separatrix can be located as a repeatable drift-null set (or as a domain wall carrying the required gluing conditions); or (iii) the seam certificate survives phase-scramble or seam-off nulls, indicating that the seam extraction is an artifact of the analysis grammar rather than a physical seam. In addition, the claim is demoted if the PASS/FAIL outcome depends on a single hand-picked lens (window length, smoothing scale, discretization), because a genuine seam invariant must persist under near-lens swaps.