Spectral structure of Lorentzian EPRL transfer operators: exact parity superselection, orientation-doublet gap closing, and certified contraction

We study transfer operators built from the deterministically computed Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) four-simplex vertex amplitude (sl2cfoam-next) at boundary spins 2j = 2-16. Three results. (i) An alternating-sign involution P on the four-valent intertwiner recoupling basis is an exact symmetry of the melonic transfer operator, [T,P]=0 at machine precision at every gate tested: the transfer superselects into two parity sectors, which we identify with the two orientation branches of the vertex asymptotics (the ``cosine problem'' structure). At half-integer boundary spin the leading eigenvalue is an exactly degenerate parity doublet; at integer spin the subleading eigenvalue is exactly the top of the opposite parity sector. (ii) Consequently the slow closing of the transfer spectral gap at large spin, mu* = (lambda_2/lambda_1)^2 -> 1 - which we exhibit across the full gate range - is not a renormalization-group instability: it is the splitting of a superselected symmetry doublet, and one composition step acts on it exactly as mu* mu*^2, i.e. strictly gap-opening. On the clean (integer-spin) data the splitting closes as a power law, g = 1-mu* ~ j^-p with p = 3.4-3.5 at the accessible gates and drifting toward 7/2, rather than exponentially; earlier exponential appearances are a crossover effect. This yields two falsifiable predictions: the tail exponent should be independent of the Barbero-Immirzi parameter gamma (while the prefactor is strongly gamma-dependent), and orientation-fixing vertex proposals (proper vertex, causal vertices) should remove the doublet altogether - a simple leading eigenvalue at half-integer spin and no closing gap. (iii) At every gate tested, within each parity sector the transfer is entrywise positive in its Perron gauge, and we establish certified contraction bounds via the Birkhoff-Hopf theorem with outward-rounded interval enclosures: across a gamma [0.1,5], j 1,32,2 grid, all 25 sector certificates close with projective contraction ratio kappa