X72-15: Gauge-Hessian Distortion

Continuing from Round 14 (missing from this folder, but this round's hand-off section records that it already established the critical-quotient carrier Q(t), the nonlinear gauge div(|v|v) = 0 satisfied by the optimal representative v = u + ∇q, and the growth identity (1/3) dQ³/dt + νD = I_Q), this round directly analyzes the constraints the gauge places on ∇²q. Proves a curvature-payment dichotomy (positive gauge curvature must be paid for by physical compression or transverse gauge concavity) and a weighted-trace cancellation ∫r³Δq = 0, showing that only skew gauge curvature drives critical growth; and proves a nonlinear Hodge-gradient Pythagorean identity, E_U^{(M)} = D + H, where H is the non-negative gauge-Hessian distortion energy, yielding a necessary condition for growth, Ξ_Q = Q²H/(ν²D) ≳ 1, while an explicit axisymmetric vortex-field counterexample shows that the gauge condition does not automatically imply A_2 regularity of the weight. This round stops at STOP-C19 (the weighted-gauge-Hessian/quotient-dissipation gap): still missing H ≲ Q^{-2}ν²D or an integrable substitute for it, handing off to the next round's tracking of the dynamics of Ξ_Q via a continuous layer-cake decomposition.