OTF IV: Causal Admissibility and the Emergence of Definite Order: A Frontier Extension to Higher-Order Quantum Processes

This paper extends the Ontological Totality Functional (OTF) programme into the domain of quantum causal structure. Its central claim is that physical admissibility cannot be exhausted by state-level consistency or dynamical stability alone. It must also constrain the admissibility of causal organization itself. The paper introduces a causal defect functional for higher-order quantum processes and argues that definite causal order can emerge as a limiting case of admissible structure. Working in a finite-dimensional process-matrix setting, the paper defines the causal defect as the trace-norm distance to the set of valid causally separable processes with fixed normalization. It proves that this defect is non-negative, vanishes exactly on the causally separable set, and is convex. A free class of admissible supermaps is then identified: linear maps on processes that preserve validity, preserve causal separability, and are contractive in trace norm. Under these maps, the causal defect is monotone. An exact dual operational characterization is established: the causal defect equals the maximal bounded-test advantage over the causally separable set for Hermitian tests of bounded operator norm. In this sense, bounded causal witnesses arise as certified lower-bound functionals for admissibility. To connect the formalism to concrete physical scenarios, the paper studies a dephased-switch family W_eta = W_diag + eta X, where W_diag is causally separable and X encodes interference between orders. For this family, the defect is bounded above linearly in |eta|, while a witness with positive overlap on X yields a matching linear lower bound. As eta tends to zero, the process approaches definite-order admissibility. Within the OTF series, this paper functions as a frontier extension. Paper I introduced physical admissibility mathematically, Paper II studied admissibility in a tensor-cosmology channel, and Paper III interpreted admissibility ontologically. Paper IV shows that the same programme can be pushed into higher-order causal structure, where admissibility constrains not only which states and channels are physically stable but also which causal architectures can count as physically realized. Companion records in the OTF series:OTF I — DOI: 10.5281/zenodo.19068375OTF II — DOI: 10.5281/zenodo.19075284OTF Master Edition — DOI: 10.5281/zenodo.19077303