The Derivation of τ

The defragmentation timescale is τ = (6/21) × t_H = (6/21) × 13.8 Gyr ≈ 3.9 Gyr. Here t_H denotes the present cosmic age ≈ 13.8 Gyr (numerically ≈ 1/H₀ at H₀ ≈ 71); the partition at the present epoch depends only on the ratio t/τ = 21/6 and is unchanged under either reading.

5.1 — Why 6/21?

Matter entering a black hole must have all six face-structures of ε erased. Each face is an independent reading of the break (AP24 Proposition 1); defragmentation is the dissolution of all six readings. The unit of defragmentation work is one face, not one face-projection. A face-projection is one face read through one spatial dimension; there are eighteen of them (6 × 3). But the three projections of a single face are not independent defragmentation targets — they are three views of the same structural reading. When the G-face is erased, it is erased across all three spatial dimensions simultaneously. The face is the indivisible unit because the break has six independent readings (AP24), and defragmentation reverses the break. The arena processes through twenty-one channels simultaneously (AP28); the channels are independent (AP28 Proposition 2). Each face-erasure takes 1/21 of the total actualisation budget — one channel-share of the total processing time. Six faces, twenty-one channels; the defragmentation time is the ratio τ/t_H = 6/21. This is 1:1 + 1×ε applied to time: the Hubble time is the full clock (the 1); the defragmentation timescale is the 6/21 fraction of it that erasing six faces through twenty-one channels consumes.

5.2 — Why this is the same logic as AP28

You have seen this before. AP28 asks why the gravitational coupling sits ~10⁴⁵ below the electromagnetic, and answers: gravity couples through all twenty-one channels simultaneously, so α_G = α_em²¹. The hierarchy is a count. This paper asks why the dark sector splits 68/27, and answers: defragmentation erases six faces through twenty-one channels, so τ/t_H = 6/21. The partition is a count. Same structure. Same ε. Different question.

5.3 — What 3.9 Gyr means

The derived defragmentation timescale τ ≈ 3.9 Gyr is comparable to the age of the solar system (4.57 Gyr) and to the cosmic age at the peak of star formation (z ≈ 2, ~3.3 Gyr after the Big Bang). This is not a coincidence: the defragmentation timescale is set by the same structural constants that govern the formation of the structures that produce black holes. Matter that entered black holes in the early universe has had time to fully defragment; matter that entered within the last 3.9 billion years is still in process.