Temporal Readout Without a Global Clock: Classical Labels, Records and the Arrow of Time

Temporal structure in a non-classical causal theory need not come packaged as a globally readable classical clock. The central distinction drawn here is between parametrization and readout: a theory may admit global time parameters, foliations, or boundary times without thereby containing an exact internal classical label map that reads time off every sector. In the no-global-diagonal framework, any exact global classical-label readout reconstructs a natural causal coaction by a commutative coalgebra. Under the imported No-Coaction Principle, no nontrivial such coaction exists on non-classical sectors. From that general theorem follow three temporal consequences: no exact universal clock readout, no exact global record readout and hence no exact bidirectional global record architecture, and no exact global arrow readout as the companion orientation-label case. What remains admissible is local and patchwise classicality: relational clocks, wedge-local records, and local asymmetries of record formation in decoherence-selected commutative sectors. The resulting picture is compatible with relativistic spacetime geometry and holographic parametrizations, because the obstruction targets exact global classical readout rather than global coordinatization as such. Philosophically, the main upshot is a structural restriction on temporal predicates over totality: some familiar global questions presuppose a classical readout structure the theory does not supply.