PAAMcm — a complex-cell layer over the PAAM core

This note is a skeleton layer over PAAM 6.5: it adds structure without altering the core. The substrate remains a real diffusion, and the transverse monodromy \(A = DP\) remains a real \(2\times 2\) matrix by construction. The added observation is that a real \(\mathbb{Z}_N\)-equivariant ensemble decomposes, without extra hypotheses, into \(2\times 2\) blocks with complex cells, and that the core’s Part III apparatus — spectrum, stability region, margins, and display — admits an exact complex extension that restricts to the real section already treated in the core. The Jury triangle appears as the real section of a Schur–Cohn region; ledger records are unchanged. Three consequences are not restatements. First, the complex loop gain acquires a phase that is a gauge invariant absent from the real theory; the phase is non-zero precisely when the coupling is both directed and crosses the agent/medium split. Second, the discriminant locus, a separating curve in the real plane, has real codimension two in the complex plane: node and spiral are not complex-analytically distinct, and the two multipliers braid around the locus — the phenomenon measured by the core’s conditioning number. Third, the fold / flip / Neimark–Sacker trichotomy is an artefact of the real form; with complex cells there is one circle rather than three lines. Version 0.2 develops two of these points. Transverse modes are identified as fragmentation waves with a definite drift; the drift is fixed by the linearisation only when the coupling is directed, so a measured non-zero drift certifies non-reciprocity — the first falsifiable claim produced by this layer. The Pauli basis is identified as the isotypic decomposition of the return operator under the whitened gauge group (dilatation / vorticity / shear), with unequal status of the two tensor factors: the member factor is gauge, the internal factor is not.