Contracting Axis Sign and Depth

Abstract

Contracting-axis powers split into parity sign and inverse-golden depth.

Theorem 1.1 (Contracting powers separate sign and depth).

$$\forall n\in \mathbb{N},{\mathrm{contractingEigenvalue}}^n=(-1{)}^n{\phi }^{-n}$$

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

Source. Repository-derived.

Commentary.

The contracting eigenvalue is minus the reciprocal golden ratio. Its nth power therefore factors into the parity sign (-1)^n and inverse-golden magnitude phi^(-n).

The proof is a thin normalization wrapper over the standard power lemmas for negation, inverses, and integer exponents.

This is a partial closure of the contracting-axis sign-reversal clause. The expanding-axis assignment and global spiral interpretation remain open.

References

• Truth anchor: D5/S1/Eigenstructure/ContractingAxisSignDepth.contracting_axis_power_sign_depth
• Dependency: D5/S1/Scale/FibonacciEigen