The Arithmetic of the Vacuum Catastrophe: Volume Counting, the Linear Energy Bound, and the 10¹²² Ratio

The ~10¹²²-fold discrepancy between quantum field theory's estimate of the vacuum energy density and the observed dark energy density (the "vacuum catastrophe") is mostly presented in the literature as an unexplained fine-tuning problem. In this work the arithmetic origin of the ratio is reduced to a single sentence: both calculations perform the same legitimate division (energy ÷ volume), but they write into the numerator two quantities from different dimensional rungs. The mode sum puts into its numerator the entirety of the Planck cells of the causal region; this quantity grows like the volume, as n³. The maximum record permitted by gravity, however, is the black-hole bound (E = c⁴L/2G), and it grows like the radius, as n. The ratio of the two numerators is n², which for today's counter value nH = 8. 497×10⁶⁰ gives 7. 56×10¹²¹ ≈ 10¹²²: the ratio is, down to its decimals, the result of a rung mix-up. It is shown that ρceiling = (3/8πn²) ρP — the linear energy bound divided by the volume — is algebraically identical to the Friedmann critical density under the definition H₀ = c/L; the observed dark energy is ΩDE = 0. 6847 of this ceiling, and the lock ΛℓP² = 3ΩDE/nH² = 2. 845×10⁻¹²² is obtained. The bound itself coincides with the Cohen–Kaplan–Nelson bound of the mainstream literature. The epistemic status of the work is explicitly separated: the form of the smallness (1/n²) is a derivation; saturation (sitting at the ceiling) and the amplitude (ΩDE) are, respectively, an open problem and an observational input.