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

Abstract

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. The second is the branch label carried by the signed structural stock, for which a definition, the invariance this paper requires, and the condition under which the concentration readout is a valid estimator are all given. V11 makes three completions, one regrading, one correction and one register separation. Three of these were in the version circulated for internal reading and are described first: one completion closes a gap this paper itself declared open, one turns a magnitude reading of the coupling product into a sign reading, and one names a third aggregation operator and its division of labour with the two already in use. The regrading and the correction were added before deposit and are described after them. The gap closed is the number of layers: V10 opened Section 4 by stating that the layer count is not theoretical content and must be supplied by the system under study, and the adaptation-fidelity companion now derives it from requisite variety together with one band per layer, so the two-layer reduction becomes a testable claim about the environment rather than a modelling convenience. The sign reading is the load-bearing one for practice: cutting or distorting the upward feedback channel flips the sign of an off-diagonal entry rather than reducing its magnitude, which carries the system across Table 2 rather than along it, and the signature is that the damped cross-layer oscillation disappears rather than attenuating. The corrections and completions V10 made to V9 stand unchanged and are described in Section 1.3. The regrading concerns what Section 5.2 inherits from, and it moves that section half a grade. V10 corrected an earlier claim that the closure of the direction list follows from the axioms, and assigned it instead to the low-dimensional carrying assumption of the master equation. That correction stands and is not withdrawn; it was, however, weaker than it needed to be. Splitting the generator about the passive baseline gives the income-minus-expenditure shape of the master equation as an identity, so what is carried is one step inside it rather than the equation, and that step admits a one-sided bound. An infimum survives a bound taken on the correct side, which is exactly what take the smallest requires. Section 5.2 therefore now assigns the closure to a one-sided inequality: still not to the axioms, but no longer to an unlabelled reduction. The correction concerns the enumerable directions, and it adds a bottleneck this paper did not previously admit. The stock direction was listed with a diagonal entry equal to the negated total decay rate, which can never be the smallest of the four. That is true only while the stock equation is linear in the stock. Under a gain factor in the income term the entry becomes I·h′(x) − D, which vanishes where the size coordinate folds and is three times smaller on a high-concentration branch than on a low-concentration one at the reference parameters. The kink of Section 9 accordingly has a third bottleneck candidate rather than two, which strengthens rather than weakens the claim made for it, and a new entry CS-3 states the measurable consequence. Assumption A(θ) acquires the qualification that follows from the same change: once the allocation exponent is a function of the state rather than a parameter, the assumption must be checked along a trajectory and not only at a point.