The Explanation of the Cosmic Coincidence: The Common Energy Ceiling, the Final Doubling, and the Law Ω_DE = 4^(−Δk)

That the densities of dark energy and dark matter are of the same order today — the cosmic coincidence (the “coincidence problem” of the literature) — stands in standard cosmology as an unexplained accident. This paper is about that coincidence itself, and its conclusion is a single sentence: it is not a coincidence. Working within the series’ counter formalism, and without using a time variable in any equation, the study reconstructs the problem in doubling geometry (k = log₂ n; the steps by which the diameter doubles) and derives four results. (i) The necessity of a common ceiling: an order-one ratio (Ω_DE/Ω_DM ≈ 2.6) is proof that the two components occupy the same dimension rung (d1); ratios between different rungs are powers of the counter (today d2/d1 = 2πn ≈ 5.3×10⁶¹). (ii) Diameter-driven growth: three physical quantities — energy E = (D/2)ε_P, entropy S = πD²k_B, Planckon count N = πD³/6 — grow with powers of the diameter and acquire factors of ×2, ×4, ×8 respectively at each doubling. (iii) The doubling law of the density fractions: under w = −1 and flatness, the dark-energy fraction is a function solely of the number of doublings remaining, Ω_DE = 4^(−Δk); the doubling upper bound is k_max = k_H + ½log₂(1/Ω_DE) = 202.68, and today’s measurement lies inside the final quarter doubling (Δk = 0.273). The transition span (Ω_DE: 0.1→0.9) is exactly log₂3 ≈ 1.585 doublings. (iv) The measure proposition: in the tick measure the same interval carries probability √0.9 − √0.1 = 0.632 — the “coincidence” is a selection effect produced by looking through the doubling measure. For the numerical apportionment of the density fractions, the series’ residue distribution together with the flatness closure gives Ω_DE = 0.6881 (+0.47σ against Planck 2018).