Index Matching Between Threshold-Compressed and Hierarchy-Compressed Transport

This preprint develops the first disciplined integration layer between the two compressed transport lines of the \kappa-theory series. On the threshold-compressed side, the local index is

\kappa=\alpha(1-\beta),

where \alpha governs gap closure and \beta governs effective coupling attenuation. On the hierarchy-compressed side, the local index is

\kappa=\frac{m}{2},

where m is the local order of vanishing at a turning point. The shared notation is suggestive, but numerical coincidence alone does not imply that the two lines describe the same local singular structure.

The note distinguishes numerical equality from structural matching, introduces the notion of an index-matching regime, and shows that equality of the two index values acquires structural meaning only after reduction to a common local singular object together with sufficient auxiliary local data. In this setting, the A-line and B-line indices are interpreted as two compressed readings of one and the same reduced local singular structure.

This note is intentionally restrained. It does not claim universal unification, does not collapse both lines into a single complete invariant, and does not yet move to application-facing bridges. Its role is more foundational: to separate numerical coincidence from structural matching and to establish the first reduction-based integration layer between the threshold-compressed and hierarchy-compressed lines of the series.