Exact Replica Mechanics and Born-Record Large Deviations in Monitored Quantum Circuits

We develop an exact finite-dimensional theory of the replica mechanics underlying measurement-induced entanglement transitions in monitored quantum circuits, separating three structures that are often conflated: linear outcome-summed replica channels, normalized conditional trajectories, and asymptotic large deviations. For fixed-basis projective measurements we determine the minimal invariant unital algebra of the one-wire feature space and prove that independently Haar-averaged two-site gates on a perfect matching close the \( n \)-replica permutation space exactly for every replica number, with an explicit Weingarten bond channel. The resulting one-period transfer operators have real, positive, semisimple nonzero spectra that compress to bond-label space; at \( n=2 \) the model is exactly the anisotropic triangular-lattice Ising model of Bao, Choi and Altman, with a closed-form critical point and a closed free-fermion spectrum. Carrying the same computation through \( n=3,4,5 \) locates the annealed critical points \( p_c^{(2)}=0.2338 \) <\( p_c^{(3)}=0.305(3) \)<\( p_c^{(4)}\approx0.383 \)<\( p_c^{(5)}\approx0.47 \) with the Potts-class exponents at \( n=3,4 \), and supports, at the three-size level, the first-order prediction of the $q=n$-Potts identification at $n=5$: an exact three-size test ($L=4,6,8$) on a symmetry-reduced orbit basis of dimension 3,865 over the $2.1\times {10}^8$ bond-label space shows the two-phase gap closing faster than the continuous baseline at every size, with the scaled gap falling below the continuous envelope at the matched sizes and dual effective closing rates ordered consistently with the continuous, marginal and first-order behaviours at matched sizes---a directional trend, with the registered conditional that completes it: if the closing is genuinely exponential, a fourth size ($L=10$) will show the effective closing rate stabilizing at a positive interface tension, and if the asymptotic is a slowly decaying power law, that rate will continue to decay---and at local dimension $d=3$ the same three-size test keeps the gap level below the $d=2$ values at every size. We further prove a no-freezing theorem for finite-reachable record processes, establish the worst-case complexity of conditional-trajectory decision, derive the exact replica interpolation identity with its measurable self-averaging criterion, and give the first numerical quenched record scaled cumulant generating functions for Haar circuits, which exhibit genuine record multifractality; the Clifford record-count disorder statistics are closed exactly through a verified conjugation 3-design.