Abstract When the coupling between two adjacent levels grows past a threshold, their critical slowing down phase-locks: the two levels no longer approach instability independently but share one diverging slow mode. This paper supplies the threshold in closed form, a zero-free-parameter fitting relation, three observables immune to contamination by a common trend, and one signature that generic coupled-mode theory does not produce without the expenditure structure of this framework.This version makes four corrections and three additions relative to V7(UET). Two of the corrections repair load-bearing arguments, and one of the additions removes an experimental trap that V7 (UET)left open.Correction one, the definition of the inter-level coupling coefficient. V7 mounted it directly on the quasi-potential as a mixed second derivative. That definition is dimensionally incompatible with its own use: the quasi-potential is dimensionless, while the off-diagonal element of a Jacobian whose diagonal carries recovery rates must have the dimension of an inverse time. The two other places where the coefficient is used — the near-gradient identity of §5.3 and the circulation test of §9.3 — already carry the correct form. The coefficient is therefore defined here as the off-diagonal element of the two-level Jacobian, equal in the near-gradient limit to minus the diffusion coefficient times the mixed second derivative of the quasi-potential.Correction two, the status of the threshold. V7 argued that a vanishing determinant means the linearized system loses an isolated fixed point, and used that to exempt the paper from the ergodicity condition that constrains snapshot diagnostics. A linear system with non-zero determinant has the origin as its unique fixed point on both sides of the crossing; what changes at the crossing is stability, not existence. With recovery rates of 1 and 0.4 a coupling product of 0.30 gives eigenvalues of −1.3245 and −0.0755, a stable node, and 0.50 gives −1.4681 and +0.0681, a saddle; the origin is the unique fixed point in both. The exemption is restated on a weaker but sound footing: the threshold is located by sweeping a parameter and does not require waiting for an escape, so it is not limited by the observation window relative to an escape time.Correction three, the kink is a crossing condition and not a closed form. The switch point is the value of the renewal rate at which the stock-direction rate meets the activity-direction margin. That margin itself contains the renewal rate, through the adiabatically eliminated driving contribution, so the equation is implicit rather than explicit, and the margin is not a constant against which the renewal rate is compared. The prediction survives intact and becomes an intersection of two independently measurable curves, which is what an experiment measures in any case.Correction four, the anaesthesia candidate fails a premise rather than an instrument choice. V7 recorded that the waking baseline of the brain is edge-of-chaos and prescribed recovery-time indicators. But the reduction of this paper presupposes that both levels follow the fixed-point script, and a chaotic level does not: its spectrum is replaced by a Lyapunov spectrum with reversed sign. The candidate therefore requires an argument that the relevant anaesthetic depth returns the system to the fixed-point script, and is downgraded accordingly.Addition one, and the most consequential: a variance trap inside the corrected negative-coupling region.V7 corrected the direction of the dominant slow mode in the interval between minus the squared recovery-rate difference and zero: the slow mode rises rather than falls, from 0.400 at zero coupling product to 0.500 at −0.05 and to the arithmetic mean 0.700 at −0.09. That arithmetic is correct and is retained. But V7 then tells the experimenter to read variance and lagged autocorrelation, and the stationary variance does not follow the slow mode there. The slow mode depends only on the coupling product, while the stationary covariance depends additionally on how that product is split between the two coefficients, that is, on the non-normality of the Jacobian. With isotropic noise and the same recovery rates, a coupling product of −0.05 gives a total stationary variance of 3.683, 4.861, or 9.620 according as the coefficients are (0.5, −0.1), (1.0, −0.05), or (2.0, −0.025), against 3.500 at zero coupling. The recovery rate rises while the variance rises with it. Any test in the negative-coupling region must therefore be built on a decay rate, recovery time or the decay of lagged autocorrelation, and never on variance.Addition two, the geometric reading of the instability condition carries a scope limit. That the two levels’ quasi-potential loses positive definiteness in the inter-level direction is derived from the Jacobian equalling minus the diffusion matrix times the Hessian, which holds near detailed balance. The same paper argues that a negative coupling product requires circulation and that a Red-Queen state necessarily carries it, so the geometric reading does not apply in the regime the paper cares most about. It is retained with its condition attached.Addition three, a numerical ledger. Every number quoted in the body is listed with the parameter set that produces it, so that a reader can recompute the paper in a few lines.Attribution is stated honestly.The threshold formula is nothing but the vanishing determinant of a two-by-two Jacobian and belongs to elementary coupled-mode theory; it is not original here. What is original lies in four places: identifying the two recovery rates as those of a fully multiplicative master-equation system read under one canonical definition; the procedure of taking an infimum along enumerable directions; the channel-closure argument, whose inheritance is now stated accurately; and the kink with its optimal renewal rate. Citations of this paper’s core should point to those four rather than to the bare threshold.
Qinfu Li (Sun,) studied this question.
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