Golden Visible-Hidden Hyperbolic Transport

Abstract

Golden inflation expands the visible face and contracts the conjugate residual.

Theorem 1.1 (Golden visible-hidden hyperbolic transport).

$$\forall Phi,P_{parallel},P_{\perp }\in {\mathrm{End}}_{\mathbb{R}}({\mathbb{R}}^6),Phi\circ P_{parallel}=\phi \cdot P_{parallel}\wedge Phi\circ P_{\perp }={\phi }^{\prime }\cdot P_{\perp }\wedge P_{parallel}\circ P_{parallel}=P_{parallel}\wedge P_{\perp }\circ P_{\perp }=P_{\perp }\wedge P_{parallel}\circ P_{\perp }=0\wedge P_{parallel}+P_{\perp }=I\wedge P_{parallel}\circ Phi=Phi\circ P_{parallel}\wedge P_{\perp }\circ Phi=Phi\circ P_{\perp }\wedge \mathrm{finrank}\left(\mathrm{range}\left(P_{parallel}\right)\right)=3\wedge \mathrm{finrank}\left(\mathrm{range}\left(P_{\perp }\right)\right)=3\Rightarrow \forall n\in \mathbb{N},\forall x\in {\mathbb{R}}^6,P_{parallel}(Ph{i}^n(x))={\phi }^n\cdot P_{parallel}(x)\wedge \left(P_{\perp }(Ph{i}^n(x))={\phi }^{\prime n}\cdot P_{\perp }(x)\wedge \left(epsilo{n}_n={\phi }^{-n}\wedge \left(epsilo{n}_n={\left|{\phi }^{\prime }\right|}^n\wedge \left(0<epsilo{n}_1\wedge epsilo{n}_1<1\right)\right)\right)\right).$$

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

Source. Repository-derived.

Commentary.

The ambient carrier is the source six-dimensional real space. Phi, P_parallel, and P_perp are supplied operators: the two projections are complementary, commute with Phi, have rank-three images, and satisfy the stated expanding and contracting spectral equations.

Writing q_parallel and q_perp for the two projection readouts, induction on n proves that q_parallel(Phi^n x) and q_perp(Phi^n x) acquire the factors phi^n and (phi-prime)^n. The transport law is therefore a consequence of the hypotheses on the given Phi, not the reduction of a coordinatewise-defined inflation function.

FibonacciEigen supplies contracting_eigenvalue_eq_goldenConj. Together with the pinned real golden-ratio identities, it identifies epsilon_n with both phi^(-n) and |phi-prime|^n and proves that the one-step scale lies strictly between zero and one.

References

• Truth anchor: D5/S3/Observer/HyperbolicTransport/GoldenHyperbolicInflation.golden_visible_hidden_hyperbolic_transport
• Dependency: D5/S1/Scale/FibonacciEigen