Perelman Entropy Shifts in Ricci Flow Dynamics of Metastable Calabi-Yau Compactifications

Mathematical Applications of Science Fiction We present a rigorous derivation of the monotonicity for-mulas for a modified Perelman entropy functional appliedto non-Kähler deformations of Calabi-Yau threefolds. Byintroducing a motivic shift operator Ξ acting on the derivedcategory of coherent sheaves, we analyze the Ricci flow dy-namics in the presence of metastable vacua. We establishthat the entropy shift compensates for the loss of Kählerityinduced by complex structure modulations, governed bythe Kodaira-Spencer classes. We prove eight fundamentallemmas regarding the spectral stability of the LichnerowiczLaplacian under these deformations and demonstrate thatthe shifted entropy WΞ acts as a Lyapunov function forthe coupled geometric-moduli flow. The results have directimplications for the swampland distance conjecture and thestability of non-supersymmetric string compactifications.