KNOTT-4.2 — Extremal Continued Fraction [1; 1, 1, …], Convergents Fₙ₊₁/Fₙ, & Nested Radicals

The all-ones simple continued fraction Φ = [1; 1, 1, …] has rational convergents cₖ = Fₖ₊₁/Fₖ with exact alternating error cₖ − Φ = (−φ)ᵏ/Fₖ, saturating Hurwitz’s irrationality constant √5 via \(F_k^2 |c_k - \Phi| \to 1/\sqrt{5}\).

Curated Source & Catalog Provenance

Catalog ID	KNOTT-4.2
Curated Source	Dr. Ron Knott, Phi’s Fascinating Figures (#contfract)
Source Attribution	© 1996–2025 Dr. Ron Knott, University of Surrey — Explore Dr. Knott’s original exposition
Exact SymPy Validator	geometor.knott.phigits.construct_convergents_model() (0 sp.Float atoms)
GEOMETOR Concordance	geometor/divine, geometor/wildberger

Constructive Synthesis & Exact Proof

Rewriting \(\Phi = 1 + \frac{1}{\Phi}\) generates the all-ones simple continued fraction \(\Phi = [1; 1, 1, 1, \dots]\) and nested radical \(\Phi = \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}\) cataloged on propsOfPhi.html (KnottPropsPhi).

• Alternating Fibonacci Convergent Lattice: In the \((F_k, F_{k+1})\) plane from origin O, the exact golden-slope ray OP (\(y = \Phi x\)) is straddled alternately above and below by convergent vectors OC1 (\((1, 1)\)), OC2 (\((1, 2)\)), OC3 (\((2, 3)\)), OC4 (\((3, 5)\)), and OC5 (\((5, 8)\)).

• Cassini Sector Triangles & Hurwitz’s √5 Barrier: By Cassini's identity \(F_{k+1} F_{k-1} - F_k^2 = (-1)^k\), consecutive convergent sector triangles OC1C2, OC2C3, OC3C4, and OC4C5 are equal in unsigned area (\(1/2\), zero interior lattice points), yielding exact error \(\frac{F_{k+1}}{F_k} - \Phi = \frac{(-\phi)^k}{F_k}\) and saturating Adolf Hurwitz’s (Hurwitz1891) irrationality constant \(F_k^2 \left|\frac{F_{k+1}}{F_k} - \Phi\right| \to \frac{1}{\sqrt{5}}\).

Machine-Verified Exact Symbolic Invariants

Invariant	Exact Symbolic Value
c5 value	\(\frac{8}{5}\)
c6 value	\(\frac{13}{8}\)
c7 value	\(\frac{21}{13}\)
c8 value	\(\frac{34}{21}\)
hurwitz constant	\(\frac{\sqrt{5}}{5}\)
cassini triangle area	\(\frac{1}{2}\)