Minimal Hilbert completion of retained recognition histories

This research note constructs the unique minimal Hilbert completion of all finite retained recognition histories, with each state identified with its recorded successor. It proves convergence of finite-depth approximations and transports finite-time reads while preserving native probabilities and interference terms. Permanent history records remain sharp and commuting, although their effects on initial preparations need not commute. For the supplied current and terminal retained-sector reads, the generated finite algebra is exactly four full 16 × 16 complex matrix blocks. The note assembles these blocks with the permanent archive and gives a four-outcome repeated-read realization on the full Gauss-constrained matter-field Hilbert space, with unrestricted integer field support and entanglement with earlier records. The deposit contains the 13-page research note.