The sum nobody bounded: an optimal 15j inequality, a directional decay law, and the one direction in which absolute majorants fail for the Lorentzian EPRL vertex

Abstract

Every Lorentzian spinfoam amplitude computed numerically truncates the sum over virtual SU(2) spins l_f >= j_f at a shell depth Delta l; the convergence of that sum has not been proved. This paper bounds the sum by the direct absolute-value route and reports how far it reaches. (1) For the first-kind 15j symbol of the vertex assembly we prove |15j| <= J, a closed form in the fifteen labels, with constant 1 proved optimal (saturated at a doubled-label tuple where both sides equal sqrt(3)/6). (2) Assembling J with rigorous booster envelopes under one shared rapidity, and under the explicit hypothesis that the internal 15j label and the four auxiliary intertwiners carry no growth rate, we derive an exact direction-resolved decay law for the envelope: coordinate directions decay with exponent at least 17/6, the law is piecewise affine on a 13-hyperplane arrangement (7,377 cells), and the envelope fails to be summable on a proper subcone around the isotropic diagonal with maximal deficit 7/18. (3) Over every choice the construction admits (1,458 members per booster) the best achievable exponent coincides with the realized one. VERSION 2 CORRECTS THE SCOPE OF (2)-(3): the auxiliary intertwiners have key-bounded cardinality but their values track the shell, so the decay law describes the envelope under the stated hypothesis and not the vertex tail; on the tail's actual support the absolute route fails by more than stated (crude exponent at the isotropic point at most -1/3 rather than 11/3), and the summand is supported only on the cone where, at each edge, the largest virtual spin is matched by at least one other. Theorem 1 is unaffected. All results are exact rational or symbolic statements at the stock convention rho = gamma (j+1), gamma = 6/5.