A Scale-Invariant Initial-Data Region for Critical Morrey Coercivity in Four-Dimensional Yang–Mills Flow

The companion paper introduced the critical L1-Morrey curvature density theta(A) = sup over x and r of r^-2 times the integral of |F_A| over B_r(x), for Yang-Mills connections on R4. Under a dynamical smallness trigger, the flow's dissipation controls the full covariant derivative of curvature. This paper propagates that trigger from an explicit scale-invariant region of the initial density and energy plane. An additive Morrey bound, theta(t) <= theta_0 + K_4 Z(t), is coupled exactly with the Weitzenbock differential inequality Y' <= -(1 - alpha theta) Z', where Z(t) is the accumulated squared L2 norm of the covariant curvature derivative, alpha = C_Q C_M, and K_4 = C'_Q omega_4^(1/2) S_4^2. For alpha K_4 > 0, if alpha theta_0 < 1 and Delta_0 = (1 - alpha theta_0)^2 - 2 alpha K_4 Y_0 > 0, then alpha theta(t) <= 1 - sqrt(Delta_0) on every admissible flow interval, and Z(t) <= 2Y_0 / (1 - alpha theta_0 + sqrt(Delta_0)). The abelian case is treated separately. The proof uses a non-increasing coupled functional, with no first-exit argument or continuity assumption on theta(t). The region is optimal for the specified pair of comparison inequalities. A budget-free linear bound, theta(t) <= theta_0 + (1/2) C'_Q Y_0 for all t, obtained by pairing the Duhamel source with the L1 norm of |F|^2, which equals 2Y <= 2Y_0, against the exact mass of the heat kernel on a fixed ball, gives a second, linear region alpha theta_0 + (1/2) alpha C'_Q Y_0 <= 1 - eta. At a common prescribed margin eta, its cap in the density-energy parameter plane is larger than the parabolic cap exactly when 1 - alpha theta_0 + eta < 4 omega_4^(1/2) S_4^2; the asserted region is the union of the two and of the pure-energy ceiling. Dispersed-packet families show that small Morrey density is compatible with arbitrarily large energy, so the energy cap, common to all routes, is a genuine additional restriction. The pure-density propagation problem remains open. Under the companion paper's class-compatibility hypothesis, a global admissible flow with data in this region converges to a flat connection. Independently of that compatibility hypothesis, no non-flat critical connection can arise as a local-L1 curvature limit point of such a global admissible flow. A scaling argument rules out a scale-invariant, kernel-free three-dimensional Navier-Stokes analogue using only global enstrophy dissipation; weighted or localized analogues are not excluded. All three sufficient data regions lie strictly below the ground-state energy in the same normalization, hence inside the energy regime of the Oh-Tataru Threshold Theorem. For data in their topologically trivial homogeneous H1 class, that theorem already supplies global existence and convergence. The contribution here is the explicit Morrey margin and full-covariant dissipation budget; transferring qualitative convergence to every admissible flow considered here retains the stated class-compatibility hypothesis. No new global existence or extension of the sharp energy threshold is claimed. Only the geometric factor omega_4^(1/2) in the source estimate is claimed optimal.