We introduce and study a Fisher information metric gτF associated to the conjugate heat kernel of a Ricci flow (Mn,gt)
Nearby in the stack
. This tensor measures, at a fixed scale, how the pointed heat-kernel measure changes when the base point is moved. We prove that
gτF
is monotone in scale and satisfies
gτF≤gt
. We relate its trace to the pointed Nash entropy and prove a matrix square identity for the Fisher defect
gt−gτF.
This identity gives a rigidity theorem for the equality case; on closed connected flows one has the strict inequalities
0<gτF<gt
at every positive scale, while in the complete case equality forces a Euclidean splitting. We also develop several consequences of this point of view. These include a sharp reverse Poincaré inequality for the heat semigroup, a contraction formula for
φ
-divergences along conjugate heat flow, and a canonical construction of heat-kernel splitting maps from large eigenvalues of the Fisher endomorphism. As applications, we relate pointed Nash entropy close to
0
to small Fisher deficit at comparable scales, and we obtain a codimension-one Fisher-metric criterion for applying Bamler's