How much dissipation can a multiscale layer cascade carry?

Multiscale layer constructions produce finite-time blowup for several fluid equations: for the incompressible porous media equation, for the two-dimensional Boussinesq system, for three-dimensional Euler, and, with fractional dissipation |grad|^alpha of order alpha < (22 - 8 sqrt 7)/9, for the forced Navier-Stokes equations. The exponent in the last of these is small, and it is natural to ask what limits it. We give two upper bounds on the dissipation exponent such constructions can carry. The first is independent of every design constant: a cascade whose layers grow at rate sqrt(A) on a background gradient A cannot exceed alpha = 1/2, and one whose layers grow at rate A cannot exceed alpha = 1, both in the |grad|^alpha convention in which classical viscosity is alpha = 2. Neither mechanism reaches classical viscosity, by factors of four and two respectively. The second bound applies to the smooth-forcing design used for the inviscid Boussinesq and Euler results: its correction hierarchy, seed rule and frequency-ratio rule cap the exponent at alpha <= 1.08e-03, attained when exactly one derivative of the force is controlled, which is 86 times below the exponent already achieved with a force of finite regularity. Both bounds follow from the amplitude system of the construction together with one additional ingredient, the effect of fractional dissipation on a single layer, which is derived and then verified against direct simulation of the nonlinear equations, including the effect of the spatial localization the constructions use. The exponent budget behind the published hypodissipative threshold is reconstructed, the constraint that determines it is identified, and the admissible frequency ratios are shown to form an interval that closes exactly at that threshold, so that cascades with geometrically growing frequencies are excluded at every positive dissipation. Self-published preprint, not peer reviewed. The mechanism is due to Cordoba and Martinez-Zoroa, extended to smooth forcing by Alpoge and Buckmaster; nothing here establishes blowup for any equation or improves any published threshold.