Directed Causality, Bidirectional Stability, and the Type Theory of Causal Topological Quantum Field Theories (CTQFTs)

The Spectral Realization program establishes that a directed causal tower of cohesive slice $\infty$-toposes natively generates the Batalin--Vilkovisky (BV)--BRST complex and its BV pushforward. This directed ordering appears to conflict with the invertible $\infty$-groupoid structure required by the differential cohomology hexagon. This paper, however, resolves the conflict entirely within standard Differential Cohesive Homotopy Type Theory (DCCT), without invoking Directed Homotopy Type Theory (dHoTT). The Bidirectional Stability Theorem shows that a causal forward transition generates a topological obstruction realized as an uninhabited second-order identity type. Spectral Realization, by projecting the obstruction through the flat modality, dynamically extracts a discrete physical spectrum that inhabits it, restoring the $\infty$-groupoid structure at every tower step. The Charge-as-Witness Theorem makes this explicit via the parametrised Pontryagin--Thom theorem, and a further type-evaluation theorem proves the construction is genuinely constructive in the Brouwer--Heyting--Kolmogorov sense, with no appeal to non-constructive principles such as the Weak Limited Principle of Omniscience.
        
        Categorically, the resulting structure implements a bialgebraic duality of automata. The forward causal slice acts as an initial algebra (controllability), and the computed spectrum as a terminal coalgebra (observability), while the generative tower composed with Spectral Realization forms a lax functor whose extracted spectrum is the coherence laxator. The Causal Topological Quantum Field Theory (CTQFT) admits no free topological caps or cups, but is physically conditioned dualisable, with caps and cups realized exclusively as physically computed morphisms. This restriction forbids closed timelike curves while preserving the hexagon's exactness, and furnishes a constructive physical model of the Rezk condition (Directed Univalence) of dHoTT. This paper completes the type-theoretic and categorical foundation of the Spectral Realization framework.