Kerr-Planck Cosmology in Loop Quantum Gravity: Three Fermionic Generations from Winding Modes in the EPRL Amplitude of a Kerr-Planck

We derive an effective fermionic mass operator directly from the 4-dimensional Einstein-Cartan action coupled to Dirac fields, using the covariant EPRL spin-foam amplitude with a Kerr-Planck Euclidean core as boundary data. The boundary geometry is a compact manifold S 1 τ × S 2 supporting an SU(2)184 Chern-Simons theory with level k = 184 and N = 212 punctures of spin 1/2. Expanding fermions in winding modes around the Euclidean time circle, we show that the EPRL amplitude reduces semiclassically to an effective action Seff(n) = 4π n + nBCS + δn, where the term 4π/n arises from the classical geodesic length of the Euclidean Kerr core, nBCS is the Witten phase of the Chern-Simons boundary state, and δn is the monopole-induced correction from the Dirac spectrum on S 2 . Minimization of Seff(n), together with the Atiyah-Patodi-Singer index on S 1 × S 2 , selects exactly three stable winding modes n = 1, 2, 3. These modes yield exponentially separated Yukawa couplings mn = ve−Seff(n) , reproducing the observed hierarchy of fermion masses without free parameters or fine tuning. This provides a first-principles derivation of three fermionic generations from the covariant dynamics of loop quantum gravity.