Ricci Flow as Spectral Compression: A Latent Interpretation of Perelman's Proof

We develop a spectral interpretation of the Hamilton-Perelman Ricci flow program for 3-manifolds. We show that: (i) the Ricci flow acts as a spectral compression operator, driving the Laplacian eigenvalues toward the maximally degenerate spectrum of the round \(S^3\); (ii) neck-pinch singularities correspond to spectral gap collapse events where a cluster of eigenvalues coalesces; (iii) Perelman's surgery procedure has a natural interpretation as spectral truncation and reconnection; and (iv) the \(\mathcal{W}\)-entropy functional is a spectral compression measure whose monotonicity is equivalent to a log-Sobolev inequality for the spectral distribution. We formalize these correspondences and identify the Ricci flow as an instance of the Latent framework's principle: geometric complexity is compressed into a finite spectral representation, and the flow drives the system toward its minimal Latent.