A Compact Counterexample to the Unrestricted $O(1)$ Interpretation of Perelman's Remark 9.6

Perelman stated in Remark~9.6 of \\emph{The entropy formula for the Ricci flow and its geometric applications} that a normalized logarithmic conjugate-heat-kernel quantity is $O(1)$ for $(q,t)$ near the pole $(p,0)$. The remark does not specify a relation between the spatial and temporal approach. Xu later restated the claim in a compact-manifold setting and showed, using the explicit heat kernel on noncompact hyperbolic space $\\mathbb{H}^3$, that the associated static formula fails in the complete category. We show that the unrestricted product-neighborhood interpretation of the $O(1)$ assertion already fails on every closed hyperbolic $3$-manifold. The proof uses the image-sum formula for the heat kernel of a compact hyperbolic quotient and a quantitative estimate showing that the nonidentity images, together with their first two spatial derivatives, are superpolynomially small relative to the identity image along an off-parabolic approach to the pole. More precisely, if d(p,qτ)=τβ,0<β<12,d(p,q_\\tau)=\\tau^\\beta, \\qquad 0<\\beta<\\frac12, then (□fˉ+fˉt)(qτ,−τ)=−12τ2β−1(1+o(1)).\\left(\\Box \\bar f+\\frac{\\bar f}{t}\\right)(q_\\tau,-\\tau) = -\\frac12 \\tau^{2\\beta-1}(1+o(1)). Hence (□fˉ+fˉt)(qτ,−τ)⟶−∞(τ↓0).\\left(\\Box \\bar f+\\frac{\\bar f}{t}\\right)(q_\\tau,-\\tau)\\longrightarrow -\\infty \\qquad (\\tau\\downarrow 0). Thus the quantity is unbounded in every ordinary product neighborhood of $(p,0)$. This does not rule out boundedness under a parabolic restriction such as d(p,q)2=O(τ),d(p,q)^2=O(\\tau), and the result has no bearing on the Poincar'e or geometrization conclusions.