When Horizon Equations Encode Black-Hole Thermodynamics: Conormal Surface Charges, Action-Sector Transgression, and Extremal First Jets

This version has been superseded/corrected by a later version. Please cite v1.1. Horizon field equations are often converted into a black-hole first law by multiplying a selected scalar equation by a geometric or entropy variation. Such a prescription is useful, but it is not invariant under regular rescalings and recombinations of the equations and can become an identity once an exact solution is substituted. We formulate the problem for finite-dimensional stationary families of exact solutions augmented by a marked codimension-two surface. The physical horizons form a regular constraint submanifold, while the covariant surface-charge first-law defect is a one-form whose pullback to that submanifold vanishes. We prove a local constraint-conormal normal form,\(j_H⊥(α_H)=μ_A\,d_X F^A\),and a first-jet rank criterion that decides whether a chosen collection of reduced horizon equations contains the complete transverse thermodynamic information. In particular, a smooth product\(E_H\,d_X S\)cannot represent a nonzero conormal first jet when\(E_H=0\)on the physical family. We also show that identical reduced equation jets can carry inequivalent thermodynamic defects when their action-normalized surface charges differ. Finally, we formulate action-sector transgression: separate Lagrangian sectors can redistribute quasilocal charge between homologous surfaces even though the complete exact-symmetry charge is conserved. The framework is tested in Kerr-Newman-AdS, including\(V\,dP\)and the canonical finite-radius Lee-Wald pointwise density in the rotation direction; in the new-type black hole of New Massive Gravity; and in warped\(AdS_3\)black holes of Topologically Massive Gravity with nonzero Cotton tensor and Tachikawa entropy. At a regular double root satisfying the stated nondegeneracy conditions, the first-law defect and\(d_X T\)furnish the two conormal directions. The construction does not replace the Iyer-Wald first law; it classifies when, and in what representative-dependent sense, that law admits a local horizon-equation realization. Status: Preprint; this manuscript has not undergone peer review. Version: 1.0. 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.