X72-13: Critical-Quotient Gauge Covariance

Continuing from the question left open by Round 12 (whether the projected entropy gradient PJ_{3/2} is the gradient of some scalar functional), this round first proves the answer is yes — but the true critical dual object is actually the quotient space L^{3/2}/G_{3/2}, whose minimal representative v* satisfies the nonlinear gauge div(|v*|^{-1/2}v*) = 0, so Round 12's explicit Leray defect disappears. A deeper defect immediately surfaces, however: component-wise transport does not preserve the gradient gauge; switching to a gauge-covariant one-form Lie transport does repair the gradient quotient, but introduces a strain-stretching term, producing a transport/gauge-covariance trade-off. A local-correction no-go in the affine case further proves that only pure rigid rotation can simultaneously satisfy gauge covariance and entropy neutrality. This round stops at STOP-C17 (the critical-quotient-gauge-covariance/stretching gap), and points out that the Navier–Stokes equation, after quotienting the gradient, is itself a Lie-transport equation, handing off to the next round's test of the original critical one-form/circulation quotient.