A Unified Local Theory of Singular Transport: Threshold Compression, Hierarchy Compression, and Parity-Aware Reduction

This manuscript develops a unified local theory of singular transport and local singular objects. On the transport side, it identifies a principal-channel law for singular spectral projection transport near simple first-gap closure and derives the threshold-compressed index \kappa_A=\alpha(1-\beta), which separates finite, critical, and ruptured cumulative transport regimes. On the object side, it develops a hierarchy-compressed canonical-object theory with hierarchy index \kappa_B=m/2, together with a parity-sensitive architecture in which odd and even local singularities produce different canonical objects. The paper then shows that equality of compressed indices is not sufficient for structural identification and introduces a reduction-based bridge to a common local singular channel. In a principal-channel prototype regime, it constructs the corresponding quotient-level carrier as a canonical reduction envelope and proves that this envelope lifts to the parity-split canonical-object layer. Finally, it develops a technical control layer including worked realizations, perturbative persistence, admissible representation robustness, a first controlled higher-dimensional extension, and a refined internal criticality taxonomy near the critical zone. The result is a unified local framework in which threshold-compressed transport, hierarchy-compressed singular objects, and parity-aware reduction-based matching belong to one theory.