Exact Control and Encoded Readout in a Native Recognition Register

Can fixed recognition interactions supply the control needed to write an exact record? We prove full unitary controllability on the 1,536-dimensional neutral sector of two source–reference pairs. The allowed controls are simultaneous sums of explicit local braid and paired-contact Hamiltonians, with adjustable positive strengths and durations; the original holding and positive scalar energy charges remain present. The main algebraic step converts effective control on a 1,024-dimensional matched subspace into control supported there, then uses a contact with an explicit right inverse to reach the remaining coordinates. Compact-group control gives actual finite words. An exact eight-label coherent readout follows, together with arbitrary encoded interventions and preparation of each label from one encoded blank. The four active cells sit in a fourteen-cell register: the other ten evolve by a retained unitary, with the same identity valid for every finite external reference. The result establishes exact existence within the prepared encoding, without a uniform synthesis-time or pulse-count bound. The paper and source archive, including verify_certificates.py, are available at https://doi.org/10.5281/zenodo.23875346.