Paper 49C-1: Full Hardening of Corrected Multi-Rail Network Inversion Monotone Operational Cutoffs, Redundancy–Correlation Ordering, and Finite-Size Stabilization Across Bond, Site, and Mixed Architectures

This paper strengthens the local-to-network inversion of Paper 49C through exact ordering results,a verified historical finite-network audit, and real contact mechanics on occupied cuboctahedral andface-centered cubic graphs. The historical hardening data reproduce their reported cutoff coverageand late-size diagnostic, including the disappearance of the preliminary flags under both the originaland revised decision rules. A counterexample shows why that diagnostic cannot establish asymptoticconvergence. Direct Gaussian integration resolves the small dependence effect that the earlier sampledcurves could not distinguish: two-sided failure correlation begins quadratically in signed Gaussiancorrelation and is bounded above by its square. A reverse-deletion algorithm computes completefinite-core response staircases and pooled ensemble inverses without interpolating across jumps.For a compatible harmonic preparation, exact contact-variance corrections differ between periodicFCC and an open filled cuboctahedron. Their shared preparation scale does not remove the spatialcorrelations that affect core survival. The fixed cosine Hamiltonian also supplies an all-time sufficientbound on diagnostic dropout through conserved relative energy, with a stronger certificate for fullretention. These results connect operational cutoffs to stated Holosphere states and preparationswhile keeping core retention, transport, and a universal critical angle as distinct questions.