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

TL;DR

This paper develops a category-theoretic framework to understand indefinite causal order (ICO) as a failure-to-glue problem and applies it to quantum gravity, clarifying the meaning of indefiniteness in causal structures.

Contribution

It introduces a novel categorical formulation of definite and indefinite causal order and applies it to quantum gravity scenarios involving parametric time, offering a unified language for ICO analysis.

Findings

Categorical formulation of causal explainability as a gluing problem.

Introduction of a seven-valued contextual classifier for causal variation.

Application of the framework to quantum gravity with parametric time.

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 $\tau$, 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 $\tau$. Together, the two parts provide a common language for comparing ICO criteria and for stating precisely what ``no hidden definite order'' means.