Cross-Scale Critical Synchronisation in Driven Multi-Layer Systems Thresholds, Zero-Parameter Predictions, and Two Cross-Paper Conventions Settled

Abstract When the coupling between two adjacent layers of a driven system crosses a threshold, their critical slowing-down phase-locks: the layers no longer approach instability independently but share a single diverging slow mode. This paper gives the closed-form threshold, a zero-free-parameter fitting relation, three observables immune to common-trend contamination, and one signature that a generic coupled-mode theory cannot produce without the expenditure structure of the companion framework. This version additionally settles two conventions that are shared across the companion papers and that could not be closed within any single one of them. The first is the allocation-exponent-to-tail-index link, established here as a distribution-free monotonicity theorem together with an explicit convexity result that identifies the diminishing-returns behaviour reported by the spatial companion paper. The second is the branch label carried by the signed structural stock, for which we give the definition, prove the invariance this paper actually requires, and state the condition under which the concentration readout is a valid estimator of it. One further repair is made to the closure argument for the margin channels, whose inheritance was previously mis-assigned.