Asymptotic Typicality of Spectral Gaps for Periodic Quadratic Successor Systems: A Probabilistic Reduction

This paper addresses the uniform spectral-gap problem for periodic quadratic irrational parameters. We do not claim an unconditional proof that the property holds for almost every parameter. Instead, we separate the problem into two layers. First, we define an activation condition on random periodic words involving only finitely many local patterns, and give a complete proof that this condition holds with probability tending to one. Second, the passage from this condition to the existence of the natural grouped return operator is isolated as an explicit external deterministic interface. If the predecessor theory supplies a theorem linking the activation condition to the natural realization package, then asymptotic typicality of the spectral gap follows. The probabilistic part is fully closed within the paper; the operator-theoretic upgrade depends precisely on a named interface rather than on implicit heuristics.