G2 — The Factor of 2

G1 contains a precise geometric scaling relation: the smaller tetrahedra have one-half the edge length of the original. G2 asks what follows if that same construction is repeated.

Status: M — Mathematical consequence of the chosen construction.
The corner tetrahedra are similar to the original regular tetrahedron. Reapplying the same midpoint construction therefore produces the same geometric pattern at one-half its scale again.

Halving inward

If a₀ is the starting edge length, a nested corner tetrahedron obeys

a(n+1) = a(n)/2
a(n) = a₀ / 2ⁿ

Level	Scale	Fully branched corner tetrahedra
0	a₀	1
1	a₀/2	4
2	a₀/4	16
3	a₀/8	64
n	a₀/2ⁿ	4ⁿ

One path is not the whole tree

Along one nested path, scale changes by a factor of two at every step. Following all four corner tetrahedra at every generation produces 4ⁿ descendants. These are different coordinates: scale versus population. The 81018 scale architecture concerns the scale relation, not the number of pieces.

Where “doubling” enters

The geometry gives halving inward. Reversing direction and indexing scales outward from a chosen lower threshold gives doubling outward:

L(n) = L₀ · 2ⁿ

The factor of two is geometrically present in the construction. Using it as a physical scale coordinate is a subsequent modeling choice.

A · Modeling choice

Organizing physical scales by repeated powers of two is not itself a consequence that nature must be discretized in powers of two.

Q · Open question

Does this mathematical scaling correspond to a meaningful structure in physical reality?

What G2 does not establish

The Planck length or Planck time as a physical boundary.
That nature is discretized in powers of two.
That all geometric descendants remain physically active.
The 202nd notation as the present cosmic scale.

Next: G3 — Numerical Scale Map