CCFU Proof 1 φ is the unique positive attracting fixed point of C2

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CCFU Proof 1
φ is the unique positive attracting fixed point of C2

05/13/2026

Given.
The second-order linear recurrence with unit coefficients:

    C₂:  x(n+1) = x(n) + x(n-1)

Define.  Rₙ = x(n+1)/x(n),  assuming x(n) ≠ 0 and R₀ > 0.

Step 1 — Ratio dynamics.

    R(n+1) = x(n+2)/x(n+1)
           = (x(n+1) + x(n)) / x(n+1)
           = 1 + 1/Rₙ

Step 2 — Fixed point.
At equilibrium R(n+1) = Rₙ = R:

    R = 1 + 1/R
    R² = R + 1
    R² − R − 1 = 0

Solutions:  R = (1 ± √5)/2.
The unique positive root is

    φ = (1 + √5)/2

Step 3 — Local stability.
Define f(R) = 1 + 1/R.  Then f′(R) = −1/R², so

    |f′(φ)| = 1/φ² = 1/(φ + 1) < 1

where we used φ² = φ + 1.
The fixed point is locally attracting.

Step 4 — Global convergence.
For all R₀ > 0:

    R(n+1) − φ  =  1 + 1/Rₙ − φ
                 =  (φ − Rₙ) / (φ Rₙ)
                 = −(Rₙ − φ) / (φ Rₙ)

Therefore

    |R(n+1) − φ| = |Rₙ − φ| / (φ Rₙ)

For any R₀ > 0, we have R₁ = 1 + 1/R₀ > 1.
Therefore for all n ≥ 1:

    Rₙ > 1,    φ Rₙ > φ > 1

so

    |R(n+1) − φ| < |Rₙ − φ| / φ

The error contracts by a factor smaller than 1/φ per step.
Convergence is geometric after the first iteration.

Therefore  Rₙ → φ  for all R₀ > 0.

Conclusion.
φ = (1+√5)/2 is the unique positive fixed point of the ratio map f(R) = 1+1/R.
It is globally attracting on ℝ₊.
No parameters are chosen. The recurrence C₂ alone forces φ.  ∎