Theorem 5: Self‑Similarity selects k = 1 and φ
Fixed point of x = 1 + 1/x and integer split‑count
Statement
In the recurrence xn+1 = 1 + k/xn, atomicity and global cost minimization force k = 1. The fixed point is φ, solving x = 1 + 1/x.
In plain English
Don’t over‑split the job. If you try to chop one recognition into k pieces each step, you pay more overhead for k ≥ 2. The cheapest self‑similar groove is one‑at‑a‑time (k = 1), and the scale it locks onto is the golden ratio φ.
- Why inevitable: atomicity makes k an integer; the unique cost (T4) makes total effort grow with k ≥ 2.
- What it buys: φ as the universal “just‑right” scale for recursive recognition.
Sketch
- Atomic postings imply integer split‑count k ≥ 1.
- Total cost is monotone increasing in k; minimum is k = 1.
- Fixed point of the recurrence yields φ.
Implications
- φ emerges as the universal scaling constant for recognition efficiency.
- Underpins φ‑scaled ladders used across predictions.