CCFU Proof 8 — Fractal Similarity Dimension D = 1 for φ-Branching Tree

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CCFU Proof 8 — Fractal Similarity Dimension D = 1 for φ-Branching Tree

05/16/2026

Given.
Let φ = (1+√5)/2, so φ² = φ + 1.
A Pythagoras tree with branch scales 1/φ and 1/φ².
Assume the IFS satisfies the open set condition
(no geometric overlaps between branches).

IFS equation for similarity dimension D:

    (1/φ)^D + (1/φ²)^D = 1

Claim.
D = 1 is the unique solution.

Proof.

    (1/φ)¹ + (1/φ²)¹
    = 1/φ + 1/φ²
    = (φ + 1) / φ²
    = φ² / φ²          [since φ + 1 = φ²]
    = 1  ∎

Uniqueness.
Let g(D) = φ^{−D} + φ^{−2D}.
g is strictly decreasing in D (sum of decreasing exponentials).
g(0) = 2 > 1, g(∞) = 0 < 1.
Therefore g(D) = 1 has exactly one solution.
Since g(1) = 1, D = 1 is the unique solution.  ∎

Note.
This proves the similarity dimension. Under the open set condition,
the Hausdorff dimension equals this value.
If geometric overlaps occur in a specific realization,
the Hausdorff dimension may be lower.