The Interacting Wick Identity: transfer positivity, the existence of the unitary partner, and an equivalence display for the two-modes proposal

Companion to "The Windowed Reading", sharpening its regrade of the two-modes proposal at the one point where that note stopped: the Wick identity - one spectral number appearing as Euclidean decay rate and as unitary oscillation frequency - was displayed there on the free spectrum only. This note displays it with interaction, exactly, and adds an equivalence. The positive-type property of the joint kernel - the engine of the link-reflection-positivity theorem - makes the transfer operator of the interacting quartic chain positive semidefinite; hence H = -ln T exists on the support of T as a real self-adjoint Hamiltonian, and the interacting excitation energies exist as theorems of the Euclidean data. They are then extracted twice, independently: as Euclidean decay rates (agreement with the spectrum to 9e-16 - machine precision) and as unitary oscillation frequencies (FFT, agreement at the 1e-3 resolution limit); a matrix-pencil analysis of the Euclidean correlator alone recovers two interacting levels with the correct parity selection (Delta E_1 to 1e-12, Delta E_3 to 5e-7); the identity holds across couplings g = 0, 0.5, 2.0, with the free control an exact benchmark (Delta E_1 = ln 2, derived, passed at 1e-6); the gap is grid-converged to nine digits (Delta E_1 = 1.303200265 at g = 0.5); the Euclidean reading passes into the unitary one as the boundary value of the tube (linear convergence); and the interacting poles sit at s = +-i Delta E_n, on-axis and gapped - the mechanism premise verified at the interacting level in this truncation. The control is the sharpest statement: with the antiferromagnetic joint, the transfer operator acquires a macroscopic negative eigenvalue (-0.391 against 1e-16 noise), -ln T turns complex, and no real Hamiltonian generating these data - no unitary reading - exists in this class. In this class, reflection positivity is thereby displayed as equivalent to the existence of the indefinite face: the second mode stands and falls, exactly and computably, with RP. No number is derived; every claim carries a kill criterion; all computations are supplied as self-contained scripts.