ORUME VII - Uniform-Visibility Photonic Memory: Gain-Invariant Topological Readout, Optimal Four-Channel Fault Correction, and a Certified Eight-Mode Unitary Construction

ORUME VII presents a self-contained mathematical architecture for reading a stored binary state through the winding of optical interference observables. The construction combines a uniform signal floor, cancellation of unknown input/output port gains, and correction of corrupted transfer channels within one explicitly specified optical network. It connects graph-based error correction, interferometric closure invariants, operator dilation, and an idealized bistable material coordinate, with formal arguments, computer-assisted certificates, executable code, and reproducible synthetic validation. The central optical object is a four-by-four monitored transfer block containing sixteen complex coefficients. At either ideal logical endpoint, every monitored coefficient has constant amplitude 0.30 throughout the entire readout sweep. The stored bit changes an integer winding pattern while preserving this amplitude. An exact Halmos–Julia dilation embeds the monitored block in an eight-by-eight lossless unitary matrix. Four auxiliary modes are both sufficient and necessary for a unitary extension containing this particular strictly contractive block. The construction therefore supplies an explicit model for obtaining strong monitored-channel visibility while retaining the additional optical degrees of freedom required by passivity. A central result is optimal fault correction within the declared gain-quotient code model. Alternating products around cycles cancel arbitrary nonzero, separable input/output port gains at each common measurement angle. An explicit integer winding matrix gives nonzero relative logical signatures on all 204 simple cycles of the complete four-input/four-output bipartite graph. The resulting code attains distance nine, equal to the graph’s cycle rank, and permits unique recovery of the bit despite any four arbitrarily corrupted transfer coefficients whose fault locations remain fixed during a sweep. An explicit five-fault ambiguous observation establishes the limit of this guarantee. The release includes an executable decoder that uses the full cycle information. Continuous passivity is supported by a reproducible computer-assisted certificate. Forty-digit interval arithmetic checks positive leading principal minors at 512 sweep angles; an analytical interpolation bound controls the intervals between them. Together, these establish a global spectral-norm bound below 10/3 for the unscaled transfer matrix, proving strict contraction at scale 0.30 over the entire phase sweep. This certificate supports the unitary completion continuously, beyond a finite numerical grid. A complementary engineering analysis identifies scale 0.29 as an alternative with certified operator-norm headroom of approximately 0.0338793, quantifying a tradeoff between monitored amplitude and perturbation reserve. The optical construction is connected to a conditional write–retain–read–erase model. A bistable scalar coordinate controls a continuous interpolation between the two logical states. During writing, monitored coefficients pass through zeros at the topological transition while the full unitary extension remains well defined. Under the specified bounded-disturbance dynamics, invariant storage basins provide a retained-state monitored-amplitude floor of 0.30/√3, approximately 0.173205. For a quasi-static state in these basins, a sufficient readout certificate uses 160 phase intervals, or 161 complete complex-transfer frames, with clean-channel relative complex error bounded by 10%. Reproduction checks verify all 204 nonzero cycle signatures, seven existing unit tests, the interval certificate, and the engineering-margin calculation. Seeded synthetic tests recover the correct state in 160 of 160 main adversarial-support trials and 80 of 80 additional basin trials, using clean-channel perturbations bounded by 6%. The latter series uses 192 phase intervals. The maximum sampled unitarity residual is approximately 3.44 × 10⁻¹⁵. These finite tests corroborate the implementation under their declared conditions; the analytical 10% error certificate is a separate result. The ideal endpoint energy accounting directs 36% of unit input power to monitored outputs and 64% to auxiliary outputs. The distinctive contribution is the explicit integration of uniform optical visibility, continuous passivity certification, gain-invariant cycle-winding encoding, a sharp fault-correction limit, and a decoder attaining that limit. Closure invariants, graph-cycle theory, and unitary dilation are established methods credited in the manuscript. The proposed synthesis offers a concrete benchmark for programmable photonic meshes, robust optical readout of retained material states, and graph-based diagnosis of interferometric channel faults. More broadly, it provides a framework for studying how calibration uncertainty, adversarial faults, signal visibility, and auxiliary optical modes interact in physical information-processing systems. The guarantees assume factorable nonzero port gains, measurements at a common sweep angle, fixed faulty-channel locations, bounded errors on clean channels, and a quasi-static material state during readout. The bistable dynamics are an assumed mathematical model. The release reports theoretical and synthetic results; physical fabrication, event-triggered material switching, measured retention lifetime, photon budgets, write energy, and device-level performance remain to be established. Potential applications depend on validating these assumptions in hardware. The release contains a standalone 15-page manuscript, editable LaTeX source, Python implementations of the optical model and decoder, interval-certification and reproduction scripts, engineering-margin calculations, seven unit tests, ten figures, machine-readable numerical results, an experimental validation protocol, a claims audit, citation metadata, and SHA-256 checksums. It is intended for researchers in mathematical physics, integrated photonics, optical information processing, topology, and error-correcting codes. Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki