Indefinite Causal Order from Failure-to-Glue: Contextual Semantics and Parametric Time

• URL: http://arxiv.org/abs/2601.16494v1
• Date: Fri, 23 Jan 2026 06:44:44 GMT
• Title: Indefinite Causal Order from Failure-to-Glue: Contextual Semantics and Parametric Time
• Abstract summary: Indefinite causal order (ICO) has been studied via higher-order quantum processes.<n>PartI develops a category-theoretic formulation of definite-order explainability as a gluing problem.<n>PartII applies this framework to a quantum-gravity motivated setting.
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• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Indefinite causal order (ICO) has been studied via higher-order quantum processes (e.g.\ the quantum switch), process matrices, and quantum-gravity proposals involving superposed causal structure, yet the meaning of ``indefiniteness'' and its relation to definite-order explanations often remain opaque. Part~I develops a category-theoretic formulation of definite-order explainability as a gluing problem: each definite causal ordering (a partial order/DAG type) is treated as a context, and causal separability amounts to a consistent global section (possibly after convex mixing), whereas causal nonseparability is a failure-to-glue. We also introduce a compact seven-valued contextual classifier -- an intuitionistic elaboration -- that separates variation across contexts from genuine indeterminacy. Part~II applies this framework to a quantum-gravity motivated setting where the fundamental time is a parametric ordering variable $τ$, distinct from geometric (spacetime) time. Adopting a stochastic-quantization perspective on spin-network dynamics (Hilbert space not assumed fundamental) and reading the Wheeler--DeWitt condition as an equilibrium/stationarity constraint, we interpret ICO as indeterminacy of the parametric order of coarse-grained relational interventions, even when the microscopic update process is globally ordered by $τ$. Together, the two parts provide a common language for comparing ICO criteria and for stating precisely what ``no hidden definite order'' means.

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