The Kappa Identity Tree: Normalized Functional Identities, Factorial Multiplicity Hierarchies, and Characteristic Admission in OFT

This fifth paper in the developed Organizational Field Theory (OFT) sequence establishes a mathematical framework for distinguishing and comparing the derived functionals represented by the organizational symbol κ. Functional identity is specified through effective sectoral support, a normalized scale profile, and the complete state–metric differential on a declared common domain, together with a reference value fixing the additive ambiguity. The study develops multivariate factorial-moment and connected factorial-cumulant hierarchies, separating combinatorial multiplicities from genuine connected correlations and irreducible higher-sector interactions. For a two-scalar derivative benchmark, it explicitly derives the functional response, source-weighted adjoints, Hilbert stress tensor, and principal symbol, and establishes analytic conditions for kinetic positivity and characteristic-cone containment. This calculation corrects a mixed contribution omitted in Paper IV v0.4 and shows that, under symmetric preparation, the healthy transverse mode is luminal. A fixed-energy comparison further demonstrates that a larger organizational descriptor need not imply stronger local null focusing. Exact filtering, physical running, and scalar closure are distinguished, while a matching contract combines calibrated values, accessible responses, finite-range scale tests, identifiability conditions, and error bounds. The results constitute a conditional methodology and a reproducible benchmark; they do not establish a microscopic derivation, empirical confirmation of OFT, or nonlinear black-hole formation.