Variational Backbone and Regime Closures XIV (BH): Horizon boundary readouts and black-hole outputs Validity conditions for mass/entropy readouts and thermodynamic summary

This paper is a boundary-readout paper: it does not introduce new primitive horizonlaws, but studies how black-hole mass, entropy, and thermal outputs become readablewhen a horizon acts as a boundary-localized protocol event. Under the Internal InvisibilityPrinciple, a horizon is treated here not as a new primitive law but as a boundary eventat which readable access loses rank and must be reorganized by a boundary protocol.Part XIV inherits from Part I the declared hidden-to-retained summary discipline: unreadcontent may affect the readable channel only through a licensed summary rule, writtenabstractly as F = ΣH,P (unread content) and realized in the scoped first-order representativeclass by F = DI II . Black-hole mass and entropy are then located as boundary-localizedreadout shadows of that same declared summary instance, rather than as direct laws oftransparent interior bulk. Accordingly, Part XIV should not be read as opening a freshrepresentative branch of the theory: it studies a boundary-event readout instance builton top of an already licensed summary representative, with horizon outputs treated asdownstream boundary shadows rather than new primitive couplings. A horizon is defined asa representation-level event in which the readout map undergoes boundary-localized rankloss (or non-smooth/ill-conditioned behavior), enforcing admissible elimination on a thinlayer around a codimension-one surface. Mass is defined as a spectral-gap readout of theinduced effective operator, while entropy is defined as fiber entropy with respect to a declaredcounting measure on protocol-indistinguishable fibers (and, when admissible, by a spectralproxy such as a log-determinant). Within the class of boundary-localized horizon instances,the entropy output has leading area-type scaling, and the same Schur deformation yields adirect operator-level mass–entropy coupling. In an admissible spectral representative, thiscoupling sharpens to a common-source statement: the mass gap, spectral entropy proxy, andtemperature conversion factor are distinct readouts of the same boundary-induced effectiveoperator path Leff(ε), not independent horizon primitives. The absolute Bekenstein–Hawkingcoefficient is not derived from the Structural Layer; it is isolated as a surface-density targetcondition for later calibration or upper-gate input. This Part does not claim a full microscopicderivation of black-hole interior dynamics; it isolates the readout conditions under whichblack-hole language is licensed in this series. We then extend the output chart to (M, J, Q)and derive a first-law differential form as a conversion identity between outputs, with (T, Ω, Φ)interpreted as protocol conversion factors rather than structural inputs. A central point ofthe paper is that horizon-side quantities must descend to the boundary readout quotient:after declaring a horizon observation instance, mass, entropy, and conversion factors arerequired to be invariant under fiber-preserving re-descriptions of that instance. We makethis transmission explicit by passing from readout-map factorization to spectral and fibermeasure invariance of the retained effective operator and the declared counting measure.Equivalently, the area-law calculation isolates the remaining normalization problem as ahorizon surface-density condition; any later upper-gate or calibration input must fix thatdensity rather than alter the structural layer.