Relational Time-of-Arrival as a Covariant Field Observable: Integrable Event Currents, Clock Holonomy, and a Continuum Clock Limit

We formulate quantum time-of-arrival (TOA) as a covariant relational field observ-able rather than as a self-adjoint time operator. Detector crossings are describedby a conserved matter current Jμ through a world-tube Σ, while clock readingsare generated by an effective event-count current jμev. The construction sepa-rates conservation from integrability: conservation stabilizes event counts, but ascalar continuum clock T (x) exists only when the coarse-grained event-currentone-form is locally integrable. The obstruction is the event-current vorticityΩevμν = ∂μ(jevν )ℓ − ∂ν (jevμ )ℓ. Integrable currents yield a flux-conditioned TOAdensity; non-integrable currents yield path-conditioned TOA distributions gov-erned by clock holonomy. In a 1 + 1-dimensional Klein–Gordon example witha gapped quadratic clock EFT, Gaussian detector smearing gives the explicitregulated clock variance σ2τ,ℓ = (4π)−1K0(μ2τ ℓ2). This provides a detector-level framework in which TOA statistics depend explicitly on flux, geometry,clock-current admissibility, holonomy, and clock-sector correlators.