The lattice offset of loop quantum cosmology is the Wheeler-DeWitt extension angle: closed-form phase and unimodular tower for negative cosmological constant

In Wheeler-DeWitt (WDW) quantum cosmology with a massless scalar and a dynamical cosmological constant, the unimodular-time Hamiltonian at fixed scalar momentum has, for Lambda<0, a geometric (Efimov-like) tower of Lambda levels whose position is set by an arbitrary self-adjoint extension angle theta. We show that in loop quantum cosmology (LQC) this angle is the lattice offset eps of the superselection sector eps+4Z: each lattice realises one definite theta, and every theta is realised by some eps. For the Ashtekar-Corichi-Singh (sLQC) difference equation, extended to every lattice, the Lambda=0 eigenfunctions are given in closed form by a b-representation integral, and their large-volume phase on the two orientation branches is, exactly for that equation (eps not 0), phi = -+ pi eps/4 - arg Gamma(1+ik) (mod pi), k = omega/sqrt(12 pi G). For the Bentivegna-Pawlowski (BP) operator first-order WKB adds delta = 5/(72k) (plus a fitted O(k^-3) term). Matching to the Bessel function of the WDW region, the Gamma functions cancel and |Lambda_n| = (48 pi K/c_L) exp[2(n pi -+ pi eps/4 + delta)/k] as |Lambda| -> 0; on the integral Hilbert space over lattices the tower becomes a continuum. BP's Lambda levels at omega = 40 and 90 agree with this formula within tolerances fixed before the test. Three of the pre-set criteria failed and are reported. Methods note. Note on versions. Version 1 (10.5281/zenodo.23032491) gave the lattice-offset law numerically and stated that pi/4 was not derived. Version 2 derives it, gives the constant in closed form and the Lambda<0 tower formula, and changes the title and abstract; no version 1 number was changed or withdrawn. Archive: scripts, test protocol with amendments, review logs, claims and findings ledgers, and the complete version 1 source (v1/).