The Weight Ledger of Metasurface Susceptibilities. Slice-adapted irreducible parametrization of dipole–quadrupole GSTCs: closed-form Neumann counting, incidence selection rules, exact rotation-equivariant fitting — and why C3 and C4 first part ways at the quadrupole

Abstract

Metasurface modeling by generalized sheet transition conditions (GSTCs) suffers from a parameter explosion: the dipolar susceptibility tensors already carry 36 components, and the quadrupolar extension adds rank-3 objects of 27 components each. The established cure is Neumann's principle — symmetry-invariance conditions that eliminate components, applied by a recursive algorithm that deliberately avoids group theory [5]. This article supplies the complementary organization the field has skipped: the weight ledger. Every DQ-GSTC component carries a definite azimuthal weight m under rotations about the surface normal, with multiplicities (3, 2+2, 1+1) at rank 2 and (7, 6+6, 3+3, 1+1) at rank 3; the ledger is the slice-adapted irreducible decomposition of the framework's tensor calculus [1]. Three results. (T1) Closed-form Neumann counting: a C_N-symmetric metasurface admits exactly the components with m = 0 (mod N), matching the character formula (1/N) sum_k (1 + 2 cos(2 pi k/N))^r machine-exactly; corollary — the dipole block is blind to C3 vs. C4 vs. C6 (3 parameters each), and the first symmetry-resolving components are the m = +/-3 quadrupolar ones (C3: 9 vs. C4: 7). (T2) The irreducible coefficients are the natural fitting parameters: under rotation of the structure each coefficient transforms by the phase e^{i m phi} — fitting is exactly equivariant (8e-17 in the toy model), while Cartesian components mix. (T3) Incidence selection rules: at normal incidence the co-polarized channel reads only m = 0, the cross-polarized channel only m = -/+2; m = +/-1 couples to the normal direction, and m = +/-3 is invisible to dipolar normal-incidence readout — measuring it requires gradient illumination. All claims are verified deterministically (19/19). The article proposes no designs and makes no material claims; the contribution is the ledger, its counting table, and its checkability.