Symmetry-Protected Cancellation and Geometric Release of Floquet Sector Splitting in a Framed Lemniscate

Abstract

We study a closed framed space curve with centreline r(a) = (sin 2a, cos a, sin a) and a periodically driven transverse tube Hamiltonian. The curve exhibits a clean spinorial 4π frame lift, but the corresponding central holonomy does not by itself select a unique two-dimensional sector. The decisive structure is instead a quarter-period improper symmetry Q: it preserves curvature, reverses torsion, induces the canonical transformation S_Q = diag(1, −1, 1, −1), and changes the sign of the transverse angular-momentum observable L⊥ = u p_v − v p_u. Consequently, the symmetric system has vanishing period-averaged Floquet chirality and exact opposite-k quasienergy degeneracy. A one-axis ellipticity deformation breaks Q while preserving closure and regularity. The classical chirality then appears linearly from zero deformation, and positive-Hilbert-space Floquet quantisation of the quadratic transverse dynamics yields a resolvable opposite-k quasienergy splitting under the coupling H_int = η K_3 L⊥. A 9 × 9 parameter scan identifies one connected sampled region, ε ∈ [0.03, 0.07] and η̃ ∈ [0.002, 0.010], in which a vacuum-connected, definite-k, exactly m-degenerate branch remains isolated from all competitors through total-excitation window N_window = 6. Direct Fock-space propagation confirms the metaplectic spectrum at truncations up to N_max = 28. Global finite-multiplicity isolation is nevertheless impossible in the unbounded quadratic Floquet ladder: irrational Floquet angles generate a dense phase set, whereas rational angles produce infinite occupation degeneracy. The result is therefore a geometrically controlled low-excitation Floquet doublet candidate, not a derivation of electron spin or a globally protected qubit.