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 horizon
laws, but studies how black-hole mass, entropy, and thermal outputs become readable
when a horizon acts as a boundary-localized protocol event. Under the Internal Invisibility
Principle, a horizon is treated here not as a new primitive law but as a boundary event
at 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: unread
content may affect the readable channel only through a licensed summary rule, written
abstractly as F = ΣH,P (unread content) and realized in the scoped first-order representative
class by F = DI II . Black-hole mass and entropy are then located as boundary-localized
readout shadows of that same declared summary instance, rather than as direct laws of
transparent interior bulk. Accordingly, Part XIV should not be read as opening a fresh
representative branch of the theory: it studies a boundary-event readout instance built
on top of an already licensed summary representative, with horizon outputs treated as
downstream boundary shadows rather than new primitive couplings. A horizon is defined as
a representation-level event in which the readout map undergoes boundary-localized rank
loss (or non-smooth/ill-conditioned behavior), enforcing admissible elimination on a thin
layer around a codimension-one surface. Mass is defined as a spectral-gap readout of the
induced effective operator, while entropy is defined as fiber entropy with respect to a declared
counting measure on protocol-indistinguishable fibers (and, when admissible, by a spectral
proxy 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 a
direct operator-level mass–entropy coupling. In an admissible spectral representative, this
coupling sharpens to a common-source statement: the mass gap, spectral entropy proxy, and
temperature conversion factor are distinct readouts of the same boundary-induced effectiveoperator path Leff(ε), not independent horizon primitives. The absolute Bekenstein–Hawking
coefficient is not derived from the Structural Layer; it is isolated as a surface-density target
condition for later calibration or upper-gate input. This Part does not claim a full microscopic
derivation of black-hole interior dynamics; it isolates the readout conditions under which
black-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 of
the paper is that horizon-side quantities must descend to the boundary readout quotient:
after declaring a horizon observation instance, mass, entropy, and conversion factors are
required to be invariant under fiber-preserving re-descriptions of that instance. We make
this 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 a
horizon surface-density condition; any later upper-gate or calibration input must fix that
density rather than alter the structural layer.