Cross-Scale Critical Synchronization in Driven Multi-Layer Systems Thresholds, Zero-Parameter Predictions, and the Non-Normal Variance Trap

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 takes the threshold formula from elementary coupled-mode theory and supplies what that formula alone cannot: a zero-free-parameter fitting relation valid over the whole range on which the two eigenvalues are real, three observables and an explicit account of what each of them tests, and one signature that a generic coupled-mode theory has no reason to produce, namely a kink in the response of the recovery rate to the renewal rate whose location is set by the three-term expenditure structure of the companion framework. Two conventions shared across the companion papers, and closable in none of them singly, are settled here. The first is the link from the allocation exponent to the tail index, established as a distribution-free monotonicity theorem with an explicit domain and a convexity corollary that identifies the diminishing-returns behaviour reported by the spatial companion. The second is the branch label carried by the signed structural stock, for which a definition, the invariance this paper actually requires, and the condition under which the concentration readout is a valid estimator are all given. V10 makes four corrections and four completions to V9. The sharpened result is the variance trap of Section 6.5: in the negative-coupling region the stationary variance not only fails to track the slow mode, it is not even signed by the coupling product. At a fixed coupling product and a fixed slow mode, the total stationary variance is minimized at the skew-symmetric split of that product, is unbounded above, and crosses the zero-coupling value in between, so the same coupling product can raise the variance nearly threefold or lower it slightly according to a degree of freedom the slow mode cannot see. The ordering variable is the Jacobian’s departure from normality. Two symbol conventions are settled with the companion set, one theorem acquires the domain its statement omitted — stated on the chord slope rather than on the support of the fluctuation — one corollary acquires the reading under which its equivalence holds, and one statement of the parameterization acquires the small-share condition it requires. This paper’s predictions are entered into the shared register as the block P11 to P15, replacing the local labels an earlier draft used, and three statuses within that block are recorded rather than left to be inferred from an absence.