The Weight Ledger of Metasurface Susceptibilities: Symmetry, Counting and Design Rules

Every component of a metasurface susceptibility carries an azimuthal weight m about the surface normal. This article organizes the generalized sheet transition conditions by that single number and by three binary labels: the mirror parity, the transposition and, for sheets modulated in time, the family parity of the space–time records. The ledger gives the closed Neumann counting per point group, with threefold and fourfold symmetry first parting at the quadrupole; exact rotation-equivariant fitting; and selection rules telling which experiment reads which block. It sorts the chiral and omega couplings by point group and removes the gyrotropy from reciprocal sheets. A reciprocal sheet's first-order spatial dispersion cannot tell threefold from fourfold symmetry. With temporal dispersion the family relation of a modulated sheet pairs each harmonic at the converted frequency; the resonance and the oscillator strength may be modulated, the damping may not. A rotating sheet shifts the frequency by its weight times the rotation rate, so a rotating threefold sheet shows no doubled shift and a fourfold one none up to the quadrupole. A glide-symmetric bilayer is free of bianisotropy at normal incidence. A space–time modulated sheet stays reciprocal under a standing wave timed at half the period, not under a travelling wave; a dual sheet with at least threefold symmetry does not reflect; the same ledger gives the selection rules of harmonic generation; and a space–time crystal is reciprocal only if its space–time group holds a time mirror or the time-reflection glide whose quotient is a Klein bottle. The results give twenty design rules and eleven refutable predictions, and every statement is measured by a deterministic script with inputs on which it must fail.