Golden Fibonacci Series

Abstract

The golden-conjugate weighting of the shifted Fibonacci scale has an exact sum.

Theorem 1.1 (The alternating golden Fibonacci scale sums exactly).

$$\sum _{k=0}^{\infty }\frac{{\psi }^k\cdot F_{k+1}}{{\phi }^{k+2}}=\frac{1}{2\phi }$$

Proof. Machine-checked in Lean as D5/S3/Constants/Limits/GoldenFibonacciSeries.golden_fibonacci_series_has_sum (✓ std3). ∎

Source. Repository-derived.

Commentary.

Mathlib’s Binet formula splits each shifted Fibonacci number into golden-ratio and golden-conjugate powers. After the source weighting is distributed, both parts are summable geometric series. Their closed forms reduce with the quadratic golden-ratio identities to one half of the reciprocal golden ratio.

This partial closure covers the exact alternating-series identity in part two of the source atom and hence its stated r-bar value. It does not formalize the C-zero identity, the Mobius minus-two rule, the claimed value of D at one from below, or any critical-line remainder.

References

• Truth anchor: D5/S3/Constants/Limits/GoldenFibonacciSeries.golden_fibonacci_series_has_sum