Geometric Weights No-Go

Abstract

No nonzero rational rescaling of geometric weights matches every singleton W weight.

Theorem 1.1 (Geometric weights do not match singleton W weights).

$\neg \exists w_1,\Lambda ,c\in \mathbb{Q},c\ne 0:w_1{\Lambda }^k=c{F}_{k+2}\text{for every }k\ge 0.$

Proof. Machine-checked in Lean as D5/S0/Conventions/GeometricWeights.no_geometric_weights_match_zeckendorf_singletons (✓ std3). ∎

Source. Repository-derived.

Commentary.

The W-digit convention defines wValue k as fib(k+2), so singleton bits have weights 1, 2, 3 at indices 0, 1, 2. The first two equations force the geometric ratio to be 2, while the third requires its square to be 3; the nonzero scale excludes cancellation.

References

• Truth anchor: D5/S0/Conventions/GeometricWeights.no_geometric_weights_match_zeckendorf_singletons
• Dependency: D5/S0/Conventions/WDigits