Deriving the doublet-tail exponent of Lorentzian EPRL transfer operators: stationary-phase mode counting on the vector-geometry manifold

A companion paper (doi:10.5281/zenodo.21150430, v3 doi:10.5281/zenodo.21164123) showed that the slow large-spin closing of the spectral gap of the Lorentzian EPRL melonic transfer operator is the splitting of a superselected orientation doublet and measured it to be a power law g(j) ~ j^-p with p = 3.45 +/- 0.03 at gamma = 1.2, posing the derivation of the exponent as an open problem. This paper derives it, within the standard two-branch representation of the vertex asymptotics: the coherent-intertwiner resolution writes the splitting as an oscillatory integral over the joint Kapovich-Millson shape space; the SU(2) critical set is the vector-geometry class (Regge configurations live in the Spin(4) lift); at the melon the eight shape directions split as 5 amplitude-transverse Gaussian + 3 vector-geometry tangent, so p = (5 + rank Hess phi|VG)/2; and an exact, reachable one-dimensional critical family of the classical coherent action - verified at the action level - forces nullity k = 1, hence p = 7/2 (predicted 3.50 vs measured 3.45 +/- 0.03), derived with no input from the measured tail. The exponent's Immirzi-parameter independence is structural (a pure SU(2) count), and the stationary structure is real (Im S = 0), which is why the tail is a power law rather than exponential. The subleading 1/j coefficient b(gamma), whose measured sign flip near gamma* ~ 1.40 is a genuinely Lorentzian datum, remains open: the paper reports the derived assembly (the overall vertex NLO cancels between cross and direct terms; the splitting couples to the branch-phase imaginary NLO through a gamma-blind coefficient) and a located mechanism gap (the equilateral vertex amplitude is exactly real, so the certifiable degenerate-datum dressing is null and the b(gamma) physics must enter through the non-degenerate boost sector).