Geometric Susceptibility and Dynamical-Symmetry Selection Rules for Floquet Sector Splitting

We develop a representation-theoretic response theory for Floquet-sector degeneracies released by geometric perturbations. For a periodic Hamiltonian system with a finite spatiotemporal symmetry, a reduced-shift construction gives exact covariance of the fundamental matrix and imposes selection rules on period-averaged quadratic observables in both simple and degenerate Floquet subspaces. When two internal sectors remain degenerate on both the symmetry axis and the zero-coupling axis, an analytically tracked signed phase gap factorises locally as Δθ(ε,η)=εηF(ε,η). Its leading coefficient Γ is a linear covector on the tangent space of admissible geometries. The theory is then extended to all geometric orders: if the signed gap carries a one-dimensional character χΔ of a finite symmetry group G, the m-th response tensor is an equivariant map from SymᵐD to the gap character χΔ. Thus representation content determines which perturbative orders are permitted. For the framed lemniscate r(a)=(sin 2a, cos a, sin a), the quarter-period symmetry combines with a reflection-reversal operation to generate a dihedral D4 action on deformation space. A pure Q-odd ellipticity produces linear release, with the exact projector prediction Γpure=2Γone-axis. A real two-dimensional irreducible representation has no linear channel but admits a unique quadratic susceptibility Ξ(x,y)=a(x²−y²). Its axial directions therefore release quadratically, whereas its diagonal directions remain exactly silent because of the full reversing symmetry. An independent modified-monodromy campaign confirms machine-level protection, the factor-of-two law, quadratic onset, the dihedral quadrupole, and finite-amplitude diagonal silence. The same Floquet model therefore realises exact protection, linear release, quadratic release, and exact orientation-dependent null channels solely through the symmetry representation of the geometric perturbation.