Abstract

The supplied Family 069-01 source claims an unramified de Rham quantum geometric Langlands equivalence at irrational shifted levels. We retain that claim as a documentary record, not as a verified premise. Its categorical equivalence does not itself specify a positive state space, accessible registers, a decoder, or a reference-preserving channel.

We complete a separation-preserving finite benchmark independently of that source. On the elliptic curve $E=\mathbb{C}/(\mathbb{Z}+i\mathbb{Z})$, we construct the category of unitary local systems whose monodromy factors through a specified finite cyclic quotient. Spectral decomposition gives an explicit dagger equivalence with finite graded Hilbert spaces and a constructive Fourier realization. A named rank-one local system carries a qubit multiplicity subsystem and an arbitrary gauge factor. We construct an access-restricted decoder, prove uniform recovery with arbitrary references and gauge correlations, and determine sharp error floors after removing coefficient legs. The missing-leg interface still recovers a named effect, illustrating the distinction between observable recovery and complete quantum return.

For the complete implemented channel family, matrix-unit coverage and Choi tomography give simultaneous finite-confidence error intervals, including label corruption, calibration allowances, and independently justified transfer error. We separately prove two-probe coverage of a declared homogeneous conserved free-scalar stress class and an exact finite Ward-defect identity.

The torus benchmark is not a specialization of the connected-simple Family 069 theorem. No comparison between its categorical functor and a four-dimensional Weyl net is constructed. The localization– Whittaker, central-torsor, and collision compatibilities needed for a genuine Family 069 operational bridge remain explicitly named. No gravitational or holographic dictionary is inferred.