The lattice offset of loop quantum cosmology is the Wheeler-DeWitt extension angle of each orientation branch: closed-form sLQC 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 a self-adjoint extension angle theta, an arbitrary function of the scalar momentum. We show that in loop quantum cosmology (LQC), for one unimodular ordering, this angle is set by the lattice offset eps of the superselection sector eps+4Z separately on each of the two orientation branches (a structure found by Bentivegna and Pawlowski): theta = -+ pi eps/4 + delta, up to a fixed, eps-independent convention term. A generic lattice therefore carries two interleaved ladders with theta_+ + theta_- = 2 delta (mod pi, with that convention term set to zero), which coincide only at eps = 0 or 2. At each fixed momentum every theta is realised on a branch of some lattice; as a function of momentum, however, each branch of a lattice fixes a single function theta(k) = -+ pi eps/4 + delta(k), and the lattices together give only a one-parameter family of such functions, so LQC restricts the extension rather than leaving it free. 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 an O(k^-3) term that is only fitted). 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], asymptotically as |Lambda| -> 0. This uses the exact phase for sLQC, holds to O(k^-3) for BP, needs a WDW matching region and fails at omega = 10. On the integral Hilbert space over lattices the tower becomes a continuum. Lambda levels of BP's operator, computed here at omega = 40 and 90 on lattices not used to derive the formula, agree with it within tolerances fixed before the test. Four of the pre-set criteria failed and are reported: (1) the phase check at omega = 15 on the sLQC rows (branch leakage); (2) the phase check for the self-adjoint sLQC form at omega = 75 and 150 as written (its tolerance 10 e^(-pi k) had no fit-window allowance; the test code applied a floor of 1e-5); (3) the eps-independence bound for the BP correction at omega = 25; (4) the tower test for the sLQC form (6 of 8 extrapolated intercepts above 1e-6), for which a check added afterwards suggests that the extrapolation model and eigensolver round-off, not the formula, are at fault. Methods note. Archive: scripts and outputs, test protocol with amendments, the plan-review log, claims and findings ledgers, a post-publication review summary, the version 1 source (v1/; run logs not deposited) and the version 2 manuscript source (v2/). Note on versions. This is version 3, a text-only revision after a post-publication review by two independent AI (Claude) referees; no computed number, script or output was changed. Version 2 (10.5281/zenodo.23035110) and version 1 (10.5281/zenodo.23032491) stay available. Version 3 withdraws "each lattice realises one definite theta" (the identification holds per orientation branch, and a generic lattice carries two tied ladders), states that "every theta is realised" holds pointwise in k and that LQC restricts theta(k) (each branch of a lattice selects one function theta(k), and the lattices span a one-parameter family), reports a fourth failed criterion (the self-adjoint sLQC phase check at omega = 75 and 150), quantifies eigensolver round-off from the deposited outputs, drops the residual growth exponent as evidence, states the T2 tolerance as registered, restores the credit to Bentivegna and Pawlowski for the (omega, Lambda) pairs and the two-branch structure, adds citations (Henneaux-Teitelboim, Unruh, Ashtekar-Campiglia-Henderson, Barbero-Pawlowski-Villasenor, and a quotation of Bunao's conjecture), and completes the account (itemised in REVIEW_POSTPUB.md) of what version 2 changed relative to version 1. Version 2 had derived the lattice-offset law that version 1 gave numerically; the paper's Note on versions summarises the changes, and REVIEW_POSTPUB.md in the deposit itemises them.