Covariant unification of holographic c-functions

We propose a covariant holographic c-function, defined directly in a top-down background and constructed from the extrinsic curvature of codimension-two slices of the bulk geometry. The definition does not rely on a special choice of coordinates or on the existence of a consistent dimensional reduction. We show that it unifies previous foliation-based holographic c-functions into a single covariant formula, reducing to them in the appropriate limits. We evaluate the covariant expression in a range of top-down string backgrounds, including conformal models, confining geometries, flows across dimensions, and the Klebanov-Murugan geometry, in which the holographic radial direction mixes with internal coordinates and which is not the uplift of a lower-dimensional solution. In all cases, the c-function behaves as expected: it interpolates monotonically between AdS fixed points when they are present and decreases towards zero in gapped infrared regions, while in the Klebanov-Murugan case we recover the correct fixed-point values and find evidence for monotonicity. We highlight open conceptual issues, including: the lack of a universal covariant definition of the holographic radial direction in the presence of a nontrivial internal manifold; the derivation of the flow from a bulk action; and the relation to the entanglement c-function.