Black Holes in Induced E8 Gravity: the Inherited Area Law, Three Uniquenesses of Ten Dimensions, and the Horizon's Ledger
The quarter in the Bekenstein–Hawking entropy is one of the most quoted numbers in physics, and one that no closed framework derives from first principles. This paper reports what happens to that number inside a framework whose gravity is induced rather than fundamental — manufactured at one loop by an underlying E8 field content, with the compactification scale locked to the same ultraviolet scale. In that setting the quarter changes epistemic status: it is shown to arise as a fixed geometric ratio between two faces of one and the same computation, so the framework inherits the area law by construction, species by species, with the known subtleties of the gauge sector incorporated rather than hidden, and the result stable under compactification with the internal volume computed exactly. Three sharp uniqueness statements about ten dimensions follow within the declared frame, including a new internal selection argument for the dimensionality of spacetime — of the four dimensions in which super-Yang–Mills theories can exist, only one allows the multiplet to generate its own gravity — and an exact split of the horizon's entropy budget into a bulk share and a dimension-blind boundary share identified with horizon edge modes, which take exactly half of the gross budget only in ten dimensions. The entanglement spectrum of the horizon is computed from the internal geometry by three independent methods that agree to a part in a hundred thousand, one new spectral coefficient is measured, and a sober accounting shows that in induced gravity almost all of a black hole's entropy is paid by ultraviolet physics rather than by the known light fields. One pre-registered counting hypothesis fails its own statistical null and is published as a negative result. The paper closes with an honest map of observability for the framework's logarithmic fingerprint: astrophysical channels are quantitatively dead; the only physical window is the final stage of an evaporating primordial black hole, stated with its conditions; and laboratory realizations of E8 in condensed matter are carefully distinguished from gravitational measurements. Every claim is classified by its epistemic status; scheme dependence and the framework's premises are declared; internal replications are disclosed as such. All numerical statements are script-verified; materials are available from the author on reasonable request.