The Derivation

3.1 — The defragmentation timescale from {S, B, R, C}

The defragmentation timescale is not a free parameter. It is derived from the axiom system. Step 1: AP24 establishes that the break ε has exactly six independent scalar readings — six faces: G (geometry), c (propagation), α_em (coupling), m_e (mass), the stiffness ratio (fabric), and time (direction). No seventh face exists (KS-R.8a). To fully defragment a record is to dissolve all six face-structures; each face is an independent reading of ε, so each must be independently erased. Step 2: AP28 establishes that the arena has twenty-one independent coupling channels: six faces × three spatial dimensions = eighteen face-projections, plus three actualisation couplings. The channels are independent (AP28 Proposition 2). Step 3: the atomic unit of defragmentation work is the erasure of one face — not one face-projection (face × dimension), and not one face × one actualisation coupling. One face. The face is the structural unit because defragmentation reverses the break, and the break has six independent readings (AP24 Proposition 1), not eighteen projections. Step 4: there are six faces to erase; the arena offers twenty-one simultaneous processing channels. The fraction of the total actualisation budget consumed by defragmentation is therefore:

τ / t_H = 6 / 21 = 2 / 7.

Six faces. Twenty-one channels. The ratio writes itself. No free parameters. You can trace every number back to a derived result: 6 comes from AP24, 21 comes from AP28, both derived. A note on why the Hubble time appears: the Hubble time is not merely the age of the universe — it is the total actualisation budget of the substrate (AP20). Defragmentation occurs within the substrate, not within the black hole as a separate system; inside and outside the event horizon are the same substrate, measured by the same clock. t_H is the only available measure of total elapsed actualisation.

3.2 — The rate equation

Let ρ_DM(t) be the record density still defragmenting at cosmic time t, and Ṙ(t) the rate at which new record density enters black holes. Content enters the defragmenting population at rate Ṙ(t) and exits into the fully-processed reservoir at rate ρ_DM/τ:

dρ_DM/dt = Ṙ(t) − ρ_DM/τ.

This is a standard linear ODE. The solution is a convolution: ρ_DM(t) = ∫₀ᵗ Ṙ(t′) × e^(−(t−t′)/τ) dt′. Each parcel of record density that entered at time t′ has been defragmenting for duration (t − t′); the exponential kernel gives the fraction still in process after that duration.

3.3 — Constant feeding rate approximation

First approximation: Ṙ(t) = Ṙ₀ (constant) — black holes have been ingesting matter at a roughly constant rate over cosmic history. Then ρ_DM(t) = Ṙ₀ τ (1 − e^(−t/τ)). The total dark density created over time t is ρ_dark = Ṙ₀ t. The dark-matter fraction of the dark sector is f_DM = (τ/t)(1 − e^(−t/τ)), and the ratio R = f_DE / f_DM is R(t) = [1 − (τ/t)(1 − e^(−t/τ))] / [(τ/t)(1 − e^(−t/τ))].

3.4 — The result

At the present epoch (t = t_H), with τ/t_H = 6/21: e^(−21/6) = e^(−3.5) ≈ 0.03020. f_DM = (6/21)(1 − 0.03020) = (6/21)(0.96980) = 0.27709. The dark-matter fraction of the dark sector is 27.71%; the dark-energy fraction is 72.29%. Applied to the total budget (dark sector = 20/21):

Dark energy: 0.7229 × 20/21 = 68.85%. Observed (Planck 2018): 68.89%. Error: −0.06%.

Dark matter: 0.2771 × 20/21 = 26.39%. Observed (Planck 2018): 26.07%. Error: +1.22%.

Visible: 1/21 = 4.76%. Observed (Planck 2018): 4.86%. Error: −2.02%.

DE/DM ratio: Model: 2.609. Planck 2018: 2.643. Discrepancy: 1.27%.

The two dark fractions within 1.3% of Planck 2018, the visible channel within ≈2%. The comparison value 4.86 is the Planck 2018 late-universe budget net of radiation and relativistic neutrinos (100 − 68.89 − 26.07 − 0.18 ≈ 4.86). Zero free parameters. The only inputs are 6 (faces, AP24), 21 (channels, AP28), and the exponential decay model (AP09) — all three derived. Sit with that. Three numbers that cosmologists have spent decades measuring with billion-dollar satellites, predicted from a ratio of two integers — six faces divided by twenty-one channels — and an exponential. No fitting, no tuning. The clock reads itself. For comparison, AP28 derives the gravitational constant G to 0.69%; this paper derives the dark-sector partition to 1.2%. Same axioms. Same ε.