Even and odd Chern invariants of compactified passive scattering families

Lossless passive multiport impedances have unitary scattering matrices on the imaginary frequency axis after the impedance-to-scattering Cayley transform, and imaginary-axis reactance poles are encoded as self-adjoint linear relations without leaving the unitary picture. Using this compactified scattering object, we realize on one footing the low-degree integer invariants of complex K-theory. Over a circle, the winding number of the scattering determinant is an integer. Over a closed oriented surface, a phase gap selects a Riesz spectral projection whose first Chern number is an integer. Over a closed oriented three-manifold, the full unitary scattering map carries the odd Chern (Wess-Zumino) number, with no phase gap required. We prove integrality, homotopy invariance, and invariance under unitary changes of port basis in each case; we show that the full Lagrangian graph bundle is always trivial, so the nontrivial topology resides in the phase-gapped eigenbundle (even case) or in the winding of the full unitary (odd case); and we treat singular reactance through the relation-valued compactification. We record the Chern-Simons transgression and a spectral-flow / quantized-pumping reading of the odd invariant. Two lattice families are verified numerically against the exact degree: a two-torus model whose first Chern number takes the values 0 and plus or minus 1, computed by an exactly quantized plaquette algorithm, and a three-torus model whose Wess-Zumino number takes the values 0, 1, and minus 2, computed by a logarithmic link scheme that converges under refinement. A finite-rank and Schatten-class extension path is recorded. The construction places the even/odd Chern hierarchy on the passive scattering object and exhibits the two invariants as complementary faces of the same compactified family.