A Closed-Boundary Convergence Theorem for the Golden-Ratio Contraction Map(LLM)

We examine the affine contraction[ T(x)=φ+φ⁻⁷(x-φ) ]on the real line, where (φ=(1+√5)/2) is the golden ratio. The map is a linear contraction with unique fixed point (φ) and ratio (r=φ⁻⁷≈0.03444). The orbit satisfies the exact geometric relation[ |x_n-φ|=r^n|x_0-φ| ]and therefore converges to (φ) at a uniform linear rate. We derive closed-form expressions for the minimal iteration count (Nₘᵢₙ(x_0,ε)) needed to drive the error below a prescribed tolerance (ε>0). Replacing the strict residual test by the non-strict test[ r^n|x_0-φ|≤ε ]renders the corresponding sublevel sets closed and removes an off-by-one ambiguity that occurs precisely when (ε/|x_0-φ|) is an integer power of (r). The corrected count admits the simple formula[ Nₘᵢₙ⁽ᶜᵒʳʳ⁾(x_0,ε)=ln(ε/|x_0-φ|)/ln r ](with the convention that the value is zero when the initial error already satisfies the tolerance). Because (r) is small, double-precision machine epsilon ((≈2.22×10⁻¹⁶)) is reached from typical starting values in at most eleven iterations. Numerical experiments confirm the predicted constant slope on a semi-log scale, and a short complexity discussion shows that direct iteration is competitive with evaluation of the closed-form expression at ordinary working precisions. All statements are elementary consequences of the classical theory of one-dimensional affine contractions; the concrete exponent (-7) is chosen only for numerical convenience. The note is presented as a self-contained, fully rigorous worked example.