Golden Lorentz Update

Abstract

A Fibonacci update negates the golden Lorentz form, while two updates preserve it.

Theorem 1.1 (One update exchanges sectors and two preserve the form).

$$let{Q}_{\varphi }(x,y)=x^2-x\cdot y-y^2;letF=matrix2\left(1,1,1,0\right);\left(\left(\left(\forall v\in \mathrm{Vector}\left(\mathrm{Real}\left(\right),2\right),\left(Q_{\varphi }\right)\left(\mathrm{mulVec}\left(F,v\right)\right)=-\left(Q_{\varphi }\right)\left(v\right)\right)\wedge \left(\forall v\in \mathrm{Vector}\left(\mathrm{Real}\left(\right),2\right),\left(Q_{\varphi }\right)\left(\mathrm{mulVec}\left(F^2,v\right)\right)=\left(Q_{\varphi }\right)\left(v\right)\right)\right)\wedge \left(\forall v\in \mathrm{Vector}\left(\mathrm{Real}\left(\right),2\right),0<\left(Q_{\varphi }\right)\left(v\right)\Rightarrow \left(Q_{\varphi }\right)\left(\mathrm{mulVec}\left(F,v\right)\right)<0\right)\right)\wedge \left(\forall v\in \mathrm{Vector}\left(\mathrm{Real}\left(\right),2\right),\left(Q_{\varphi }\right)\left(v\right)<0\Rightarrow 0<\left(Q_{\varphi }\right)\left(\mathrm{mulVec}\left(F,v\right)\right)\right)$$

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

Source. Repository-derived.

Commentary.

The quadratic form is constructed on the real two-dimensional carrier as Q_phi(x,y)=x^2-xy-y^2. The update is the repository’s canonical real Fibonacci matrix with rows (1,1) and (1,0).

Direct expansion gives the one-step negation identity. Applying that identity twice cancels the two signs and proves exact preservation under the squared update.

The last two public clauses spell out the sector consequence: positive values become negative and negative values become positive after one update.

References

• Truth anchor: D5/S3/Observer/GoldenCoding/GoldenLorentzUpdate.golden_lorentz_update
• Dependency: D5/S1/Scale/FibonacciEigen