Meaning
The calibrated mixed interaction shadow is the phase comparing an actual mixed surface with its independently certified calibration.
After interaction-validity and banded 2-transport have been certified, the boundary-matched actual surface A□ and independent surface I□ define Π□=tI□⁻¹tA□ in U(1). The calibration fixes what “independent” means on the same marked boundary.
Example (illustrative)
If a certified pair of matched line maps gives tA□=−tI□, then Π□=−1. Under the stated obstruction theorem, this nontrivial phase rules out the specified finite admissible decoupling.
Scope and limits
A phase assigned to an uncalibrated surface is not this invariant. Π□=1 does not prove complete decoupling, since the phase may fail to detect other obstructions.
Source
GAT 0 — Finite Cohomological and Coercive Recognition. GAT 0, Definition 20.3.