Golden Branch Observer Decomposition

Abstract

Golden branch conjugation splits the two-dimensional observation space into trivial and sign channels.

Theorem 1.1 (The two golden branches are the trivial and sign representations).

$$\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(J\left(e_{+}\right)=e_{-}\wedge J\left(e_{-}\right)=e_{+}\right)\wedge P_{ev}=\frac{1}{2}\cdot \left(I+J\right)\right)\wedge P_{odd}=\frac{1}{2}\cdot \left(I-J\right)\right)\wedge \left(\forall v\in V_{br},\left(v=P_{ev}(v)+P_{odd}(v)\wedge J\left(P_{ev}(v)\right)=P_{ev}(v)\right)\wedge J\left(P_{odd}(v)\right)=-P_{odd}(v)\right)\right)\wedge \mathrm{range}\left(P_{ev}\right)=V_{ev}\right)\wedge V_{ev}=\mathrm{span}\left(\mathbb{C},e_{+}+e_{-}\right)\right)\wedge \mathrm{range}\left(P_{odd}\right)=V_{odd}\right)\wedge V_{odd}=\mathrm{span}\left(\mathbb{C},e_{+}-e_{-}\right)\right)\wedge \mathrm{IsCompl}\left(V_{ev},V_{odd}\right)\right)\wedge \left(\forall v\in V_{ev},J\left(v\right)=v\right)\right)\wedge \left(\forall v\in V_{odd},J\left(v\right)=-v\right)$$

Proof. Machine-checked in Lean as D5/S3/Observer/GoldenCoding/GoldenBranchObserverDecomposition.golden_branch_observer_decomposition (✓ std3). ∎

Source. Repository-derived.

Commentary.

The carrier is the complex coordinate space on the two real golden embeddings. Galois conjugation is the canonical bit flip, and the even and odd maps are the half-sum and half-difference projectors.

The proof applies the repository’s general involution decomposition, then computes the two projector ranges as the spans of (1,1) and (1,-1). These spans are complementary; conjugation acts on them with eigenvalues one and minus one.

References

• Truth anchor: D5/S3/Observer/GoldenCoding/GoldenBranchObserverDecomposition.golden_branch_observer_decomposition
• Dependency: D5/S0/Conventions/InvolutionDecomposition
• Dependency: D5/S3/PrimeForms/Splitting/GoldenLocalBranchClassification