Paper 49C: Operational Coherence Cutoffs from Multi–Rail k-Core Network Inversion Revised mathematical and scope-controlled edition

This paper develops a corrected mathematical framework for determining an operational coherence cutoff in multi-rail transport networks. Rather than treating the cutoff as a universal angle or fundamental constant, the paper defines it as the local threshold required to produce a specified global dropout fraction under an explicit network model. The framework begins with a per-rail failure law, combines multiple rails through a declared redundancy and correlation model, and propagates the resulting logical-link or site survival probability through corrected cavity equations for k-core pruning. Under monotone failure laws, the global dropout fraction is nonincreasing with the local cutoff, allowing a unique operational cutoff to be obtained by generalized inversion whenever the target lies on a continuous monotone branch. The primary outputs are the critical local failure probability, the dimensionless cutoff relative to the local strain scale, the redundancy uplift ratio, and the correction between tree-like and finite clustered networks. Earlier numerical values near 0.055 to 0.075 radians are retained only as model-conditioned historical results pending a corrected computational audit. The paper therefore establishes the analytic and interpretive foundation for Paper 49C-1, which will test bond-primary, site-primary, and mixed transport architectures across correlation models, hazard laws, network topologies, and finite-size limits.