Schrödinger’s Fallacy: Universal N mod 4 On/Off Switch for Macroscopic Quantum Coherence via Toroidal-Inspired Geometric Dressing

We report the discovery of a universal, hardware-agnostic binary switch for macro- scopic quantum coherence in cyclic spin-1/2 chains. By aligning every local trans- verse field exactly parallel to a minimal geometric dressing vector derived from the leading toroidal perturbation in the r ≪ R limit—or equivalently, the exact surface normal of a torus projected onto its minor circle—with minor-to-major radius ratio 0.05 < r/R < 0.6, the many-body ground state exhibits strict N mod 4 commen- surability: N ≡ 1, 3 (mod 4) → giant circulating quantum current + macroscopic cat-like state (coherence ON) N ≡ 0, 2 (mod 4) → near-paramagnetic frustration with strongly suppressed current (coherence OFF) The switch is operated solely by adding or removing exactly one spin from the ring. No pulse shaping, frequency tuning, or physical reshaping is required—only the mathematical mapping. Exact diagonalization (TeNPy + QuTiP) of the full many-body ground state up to N = 33 (233 -dimensional Hilbert space) confirms the effect with extreme regularity and a characteristic factor ∼ 4 suppression of site-to-site variance in the frustrated sectors, providing a built-in experimental witness. The protocol is explicitly distinguished from all prior art in twisted boundaries, synthetic gauge fields, Rydberg dressing, or commensurability en- gineering, including recent work on chiral spin liquids [16] and torus degeneracy [?]. The order parameter ⟨m · d⟩ (local magnetization along the dressing vector) converges to ∼ 0.128 in coherent sectors, akin to Aharonov-Bohm phase shifts in toroidal systems and quantized Berry phases in many-body chains. Use cases include scalable quantum switches for computing and topological sensors for magnetic fields.