When Horizon Equations Determine Black-Hole First Laws: Conormal Rank and Action-Normalized Surface Charges

Local horizon equations are often rearranged into black-hole first laws, but on-horizon equalitydoes not by itself determine action-normalized surface charges. We formulate the missing informationon a finite-dimensional family of stationary exact solutions with a marked codimension-two surface.The physical horizon family is a regular constraint submanifold, and the covariant first-law defectrestricts to its conormal bundle. A local normal form leads to a rank criterion: a fixed collection ofreduced equations recovers all transverse thermodynamic data exactly when its first-order classesspan the horizon conormal space. Identical reduced-equation jets can nevertheless coexist withinequivalent charges, as shown concretely by Einstein gravity and Topologically Massive Gravity onthe BTZ family. For a fixed action split and presymplectic representative, we derive a representativedependent sector-transgression identity that tracks compensating finite-radius charge flow. NewMassive Gravity and nonzero-Cotton warped AdS3 black holes in Topologically Massive Gravityrealize this cancellation sector by sector, including the Tachikawa entropy. For NMG, the nominalhair direction is also shown to be an admissible charge-null stationary tangent under the knownrelaxed AdS3 boundary conditions. For Kerr–Newman–AdS we evaluate canonical Lee–Wald mass,electric, and rotation densities at finite radius and obtain an extended conormal identity. At a regulardouble root, the first-law defect together with dXT spans the two extremal normal directions undera nondegeneracy condition. These results delimit when local horizon equations faithfully encode thecovariant first law and which action, normalization, and representative data remain indispensable Status: Preprint; this manuscript has not undergone peer review. Version: 1.1 This record contains the manuscript and its source files.The associated reproducibility package verifies the symbolic identitiesreported in the manuscript. This work develops a constraint-conormal and covariant surface-chargeframework for determining when horizon field equations encode black-holethermodynamics.