Future-Stable Readout Monitors for Independent Write-Once Sources: Exact State Bounds, Matroid-Rank Structure, and Constrained Projection Extremals

The paper investigates the exact state complexity of deterministic observation interfaces for multiple independent write-once binary sources, where future assignments and protected final outputs must remain distinguishable. It separates the complete semantic source layer from a separately budgeted readout monitor and derives an exact minimum-state formula in terms of coordinate projections of the protected label set, including an essential correction for incomplete projection classes. For marginally surjective protected families, it establishes sharp minimum-state thresholds for two, three, and four protected labels and completely characterizes the four-label minimizers. It further proves that even the complete cube-face dimension vector does not determine readout complexity, identifying information that ordinary face counting fails to capture. Beyond these extremal results, the monitor cost is decomposed into truncated partition-matroid rank functions, yielding submodularity and an exact formula for the additional cost of protecting another label. Its main novelty is the combination of an exact future-stable observation-state invariant, sharp constrained projection extremals, and a matroid-rank characterization of protection costs. These results concern the autonomous readout port, not the total physical cost of the combined source-and-monitor implementation.