Golden Dual-Time Renormalization

Abstract

Golden reciprocal time scaling preserves the dual product and is reversed by reflection.

Theorem 1.1 (Reciprocal scaling preserves the product and reflection reverses time).

$$leta={\phi }^2;letR=diag2\left(a^{-1},a\right);letJ=matrix2\left(0,1,1,0\right);\forall delta\in \mathrm{Real}\left(\right),\forall L\in \mathrm{Real}\left(\right),\left(\left(\mathrm{mulVec}\left(R,\mathrm{pair}\left(delta,L\right)\right)=\mathrm{pair}\left(a^{-1}\cdot delta,a\cdot L\right)\wedge a^{-1}\cdot delta\cdot a\cdot L=delta\cdot L\right)\wedge J\cdot R\cdot J=R^{-1}\right)\wedge \left(R\cdot R^{-1}=1\wedge R^{-1}\cdot R=1\right)$$

Proof. Machine-checked in Lean as D5/S3/Observer/HyperbolicTransport/GoldenDualTimeRenormalization.golden_dual_time_renormalization (✓ std3). ∎

Source. Repository-derived.

Commentary.

Set a=phi^2. The update contracts the transverse scale delta by a inverse and expands the observation length L by a, so their product is fixed.

The coordinate exchange J conjugates the diagonal update R to the displayed reverse matrix. Lean also checks both matrix products with that reverse are the identity, making the inverse claim explicit.

The theorem records only this self-contained two-coordinate algebra. It does not assert that every observer duality is golden or derive the separate primitive-unimodular classification boundary.

References

• Truth anchor: D5/S3/Observer/HyperbolicTransport/GoldenDualTimeRenormalization.golden_dual_time_renormalization