Pullback Reflection Coordinate

Abstract

Golden-square scaling conjugates the pulled-back affine reflection to the classical reflection, with an invariant structural line and a single pointwise fixed point.

Definition 1.1 (Pulled-back reflection).

$$J_{qc}(s)=\frac{1}{{\phi }^2}-s$$

Formalization. D5/S1/Deficit/PullbackReflectionCoordinate.qcReflection (✓ std3).

Source. Repository-derived.

Commentary.

This is the affine reflection obtained by pulling z maps to one minus z back through multiplication by phi squared.

Theorem 1.2 (Conjugacy, invariant line, and fixed point).

$$\forall s\in \mathbb{C},ph{i}^2{J}_{qc}(s)=1-ph{i}^{2}s\wedge (\Re (J_{qc}(s))=s_{star}⟺\Re (s)=s_{star})\wedge (J_{qc}(s)=s⟺s=s_{star})$$

Proof. Machine-checked in Lean as D5/S1/Deficit/PullbackReflectionCoordinate.pullback_reflection_coordinate (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every complex s, golden-square scaling carries the pulled-back reflection to one minus the scaled coordinate. The real-part equivalence proves that the structural vertical line is invariant.

The source calls this vertical line a fixed line. For the displayed holomorphic affine map that wording is false pointwise: solving J_qc(s) = s leaves only the real structuralZero. The theorem records both the valid setwise statement and the corrected fixed locus.

Repository searches found the scaling owner but no conjugacy or fixed-locus theorem. Pinned Mathlib contributes field normalization and complex linear arithmetic only.

References

• Truth anchor: D5/S1/Deficit/PullbackReflectionCoordinate.pullback_reflection_coordinate
• Truth anchor: D5/S1/Deficit/PullbackReflectionCoordinate.qcReflection
• Dependency: D5/S1/Deficit/Beatty/GoldenSpectralCoordinate