Black Holes in Induced E8 Gravity: the Inherited Area Law, Three Uniquenesses of Ten Dimensions, and the Horizon's Ledger

Abstract

In a framework where gravity is induced — Newton's constant manufactured by the one-loop vacuum of an underlying E₈ field content, G_N = g*/M_UV² with g* = 0.038 and the compactification radius locked to the same scale — black-hole thermodynamics changes epistemic status: the Bekenstein–Hawking 1/4 stops being a postulate and becomes a quotient. We show that the same heat-kernel integral that generates 1/G (its smooth R-term) also generates the horizon entanglement entropy (its conical term), so the species content cancels and S = A/4G holds identically — species by species, including the gauge sector once Kabat's contact term is kept (c_S = c_G for scalars +1, Dirac fermions +2, gauge vectors = −4 = +2_bulk − 6_contact). The 4D sign problem of a vector-dominated Sakharov sum (Σc = −992 for 248 vectors) is declared and resolved by dimension: the vector coefficient in scalar units is exactly D − 8, and for the super-Yang–Mills multiplet the ledger has the closed form c_SYM = 3/2·(D − 6) = 3/2·(dim A − 4) over the division-algebra dimensions D = 3, 4, 6, 10 where SYM exists: negative for R,C, exactly zero for H, positive only for the octonions — a new, internal answer to "why ten dimensions" (positivity of the gravity factory), stated within its declared frame. Two further uniquenesses of D = 10 follow: the area quotient is invariant under compactification (A₈ = A₄·Vol₆, G₄ = G₁₀/Vol₆ — a known general result, replicated here on this background, not claimed as new), with the internal volume exact, v₆ = Vol_NK(SU(3)/T)/R⁶ = π³/2 (the nearly-Kähler normalization coincides with the −(1/2)tr metric at R = 1; the residual factor 1.70 against the loop-side g* = 3π/248 is adjudicated internally as an exchange rate between accountings, k = M₁₀/M_UV = 1.0684); and the horizon ledger splits exactly into bulk (D − 2 per vector) and a dimension-independent edge/contact share (−6 per vector, the Donnelly–Wall edge modes), with |edge| = net — edge modes carrying exactly half the gross ledger — if and only if D = 10. The entanglement spectrum is then computed from the exact scalar tower of the flag manifold by pure representation theory (lowest mode the adjoint, m_KK = 2√3/R; the fundamental is excluded by zero-weight multiplicity), reproducing the internal volume and curvature to 0.001% as a third independent machinery, measuring the next heat-kernel coefficient (a₄ = 0.092, hence |Riem|² = 1.91 ± 16% "heard" from the spectrum), and yielding a sober accounting: the Standard Model alone contributes +1 (marginal) to the stiffness ledger — in induced gravity, ~99.8% of a horizon's entropy is paid by the ultraviolet tower. A pre-registered integer-census hypothesis (a privileged number of states per horizon cell) fails its own null (e^(62/3π) = 719.39: chance) and is published as such. Finally, the reading map for the framework's logarithmic fingerprint c_log = 31·dim(E₈)/45 = 170.84 is drawn honestly: astrophysical black holes are dead channels (10⁻⁷³–10⁻⁸⁶), the only physical window is the last ~1,200 quanta of an evaporating primordial black hole (S* = c_log·ln S*, M* ≈ 9.8 M_Planck) — alive, conditional, undated — and condensed-matter E₈ realizations (the Zamolodchikov masses observed in CoNb₂O₆) read the algebra, not the gravitational coefficient: the distinction is declared. All claims are classified; scheme dependence is stated; all numerics are script-verified with materials available on request.