An Anisotropic Entropy for the Rotationally Symmetric Ricci Flow

For a rotationally symmetric Ricci flow $g(t)=ds^2+f(s,t)^2g_{\mathbb S^{n-1}}$, $n\ge3$, we study the scale-invariant curvature anisotropy $A=1-f_s^2+ff_{ss}$ introduced by Angenent--Knopf. At the critical power $p_*=(n-1)/(n-2)$, with $Φ=sgn(A)|A|^{p_*/2}$, we prove the distributional identity \[ (\partial_t-Δ)|A|^{p_*} =-\frac4{n-1} \left|Φ_s+(n-1)\frac{f_s}{f}Φ\right|^2. \] Integration against a conjugate heat kernel gives an anisotropic entropy formula. As an application, we obtain an entropy threshold for Type-I singularities on $\mathbb S^n$: if the entropy associated with a singular-time kernel is at most $1$ at an earlier regular time, every tangent flow at the corresponding singular point is a round shrinking sphere. This criterion follows from the $0/1$ quantization of the singular-time anisotropic density.