Hierarchical bounds on limit cycles in disjunctive and mixed nested canalizing Boolean networks

Boolean networks with nested canalizing update rules provide a structured representation of hierarchical regulation, yet structural bounds on periodic dynamics are substantially better developed for homogeneous conjunctive or disjunctive systems than for networks in which these rule types coexist. We develop a synchronous framework for mixed conjunctive-disjunctive nested canalizing Boolean networks and derive certified bounds on their periodic orbits together with exact state-space reductions on invariant trap spaces. The central result constructs a hierarchy of admissibility relations that retains successively deeper canalizing layers and proves that each additional layer can only tighten the corresponding cycle bound, with an exact criterion for strict improvement. On every forward-invariant saturated trap space, the dynamics is conjugate to a reduced Boolean system whose state space decreases exponentially with the number of frozen coordinates. We further give a saturation algorithm and distinguish the polynomial cost of closing a specified seed from the potentially combinatorial problem of enumerating relevant trap spaces. Nonconstant disjunctive and genuinely mixed examples illustrate the theory, while a synthetic five-node example and a scope audit of traceable biological models clarify its present applicability and limitations. The framework thus provides a progressively refinable structural approach to periodic dynamics in mixed canalizing networks while making its computational and biological scope explicit.