Golden Power Logarithmic Scale

Abstract

The logarithmic scale of every natural golden power is integral.

Theorem 1.1 (Natural golden powers have integral logarithmic scale).

$$\mathrm{logb}(\phi ,{\phi }^n)=n$$

Proof. Machine-checked in Lean as D5/S1/Eigenstructure/GoldenPowerLog.golden_power_logb_nat (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every natural number n, the pinned logarithm-power identity and the golden-ratio base identity reduce logb(phi, phi^n) to n.

This is a partial closure of the source bundle’s first scale clause only. Its Zeckendorf-addition and three-gap clauses remain unresolved.

References

• Truth anchor: D5/S1/Eigenstructure/GoldenPowerLog.golden_power_logb_nat