Endpoint Obstruction and Explicit Constants for the Critical L¹-Morrey Program in Four-Dimensional Yang–Mills Flow

Two companion papers established a critical L1-Morrey criterion for covariant dissipation in four-dimensional Yang-Mills flow and propagated it from an explicit initial-data region with an energy cap. Propagation from the curvature density alone remains open. This paper gives an obstruction to unrestricted scalar comparison and explicit constants for the positive criteria. For the scalar equation partial_t u = Delta u + c u^2, c > 0, a multiple of area measure on a two-plane in R4 has arbitrarily small positive critical Morrey density, but its second Picard iterate is infinite everywhere at every positive time. An exact Gaussian convolution identity exhibits the logarithmic divergence. Truncation and heat mollification transfer the obstruction to nonnegative Schwartz data in L1 intersect L2 at any prescribed density rho > 0: their quadratic Duhamel term has Morrey density at least C_tr rho^2 log(t/(2 epsilon)). A backward heat-kernel test then shows that maximal scalar lifespans can be arbitrarily short at each fixed positive density. Thus no density-only lifespan bound or endpoint bilinear Duhamel estimate holds on this scalar class. This makes the classical measure-data obstruction quantitative for smooth finite-L2 data; it gives no negative result for Yang-Mills flow and does not exclude scalar theories with additional hypotheses. The paper also proves the endpoint Morrey-cubic inequality integral u^3 <= C_M M_2(u) ||grad u||_2^2 with the explicit constant C_M = 3 log 2/(2 pi^2) < 0.1054, from the sharp constant 1/(4 pi^2) of the heat-kernel Morrey bound (attained by the plane measure), a layer-cake identity and a Frullani integral. For SU(2) with inner product = -tr(XY), the algebraic constants satisfy C_Q, C'_Q <= 4 and K_4 <= sqrt(3)/2, using Talenti's sharp Sobolev constant S_4^2 = sqrt(6)/(8 pi). Every non-flat admissible critical connection consequently has curvature Morrey density at least 2.37. A numerical sufficient condition for the companion invariant region is Y(0) < 1.37 (1 - tau_0)^2, where tau_0 = (6 log 2/pi^2) theta(A(0)) and 0 <= tau_0 < 1. The derived thresholds are not claimed sharp: a Gaussian test gives the bracket [0.0377, 0.1054] for the optimal Morrey-cubic constant. Comparisons of the two companion data regions remain conditional on their constants. All three sufficient data regions of the companion paper lie below the ground-state energy in a common normalization; the explicit margins and budgets do not extend the Oh-Tataru energy threshold, and the class-compatibility hypothesis remains unchanged.