Symmetry-Protected Zero Modes and Spectral Obstructions to Mixing ( with Applications to Automata, Number Theory, and Holography)

This paper develops a general operator–spectral framework for identifying when mixing mechanisms fail due to the presence of symmetry-protected zero modes. We show that the existence of an eigenvalue λ = 1 (or equivalently a zero-frequency mode) constitutes a spectral obstruction to decay, independent of stochasticity, locality, or dynamical complexity. The framework is first established in a discrete setting using twisted transfer operators associated with digit-sum observables. We prove that parity observables (mod 2) exhibit exact conservation due to involutive character symmetry, while digit-sum observables modulo primes p > 2 undergo exponential mixing because of character orthogonality. This provides a spectral explanation for why carry-mixing arguments fail for parity but succeed for higher mod-p observables, yielding a sharp criterion for restricted Fourier flatness and complexity-theoretic hardness. We then apply the same spectral logic to continuous systems, focusing on the holographic superfluid model of arXiv:1601.00257. A rigorous operator analysis shows that the superfluid Goldstone mode arises from a symmetry-protected kernel of the linearized spatial operator, generated by Lorentz boosts. The equality of superfluid and particle densities (ρₛ = ρ) is identified as an exact consequence of this zero mode. We clarify which aspects of the analogy with discrete transfer operators are exact, heuristic, or invalid, demonstrating that conservation in holography arises from spacetime symmetry rather than operator composition or character involutivity. Taken together, these results isolate a unifying principle: mixing fails precisely when symmetry enforces a zero mode of the relevant operator, while genuine decay requires spectral gaps combined with orthogonality of the observable to the protected subspace. This framework provides a reusable diagnostic tool across discrete automata, number-theoretic observables, and continuous field-theoretic systems.