Golden Factor Cancellation

Abstract

Golden-normalized real involutions multiply to the standard complex structure.

Theorem 1.1 (The golden normalization cancels in the completed phase).

$$\begin{array}{r}letS:\mathrm{Matrix}\left(\mathrm{Fin}\left(2\right),\mathrm{Fin}\left(2\right),\mathbb{R}\right)=\mathrm{smul}\left(\mathrm{inv}\left(2\cdot \phi -1\right),matrix2\left(1,2,2,-1\right)\right);\\ letC:\mathrm{Matrix}\left(\mathrm{Fin}\left(2\right),\mathrm{Fin}\left(2\right),\mathbb{R}\right)=\mathrm{smul}\left(\mathrm{inv}\left(2\cdot \phi -1\right),matrix2\left(2,-1,-1,-2\right)\right);\\ letJ:\mathrm{Matrix}\left(\mathrm{Fin}\left(2\right),\mathrm{Fin}\left(2\right),\mathbb{R}\right)=matrix2\left(0,-1,1,0\right);\\ S^2=I\wedge C^2=I\wedge \\ S\cdot C=J\wedge C\cdot S=-J\wedge \\ J^2=-I.\end{array}$$

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

Source. Repository-derived.

Commentary.

The two real matrices are the source’s explicit polarization factors. Their common denominator is two phi minus one, which equals the positive square root of five.

Direct matrix multiplication proves that both factors are involutions. Their ordered product is the standard integer-entry complex structure, while reversing the order changes its sign.

The completed matrix squares to minus the identity. Its displayed entries no longer contain the golden normalization carried by the two factors.

References

• Truth anchor: D5/S3/Observer/GoldenCoding/GoldenFactorCancellation.golden_factor_cancellation