Fibonacci Degree Compensator

Abstract

Even Fibonacci degrees compensate the inverse-square golden contraction.

Definition 1.1 (Golden renormalization).

$$R_{\phi }(\Delta )={\phi }^{-2}\Delta .$$

Formalization. D5/S3/CompletionDynamics/PacketRenormalization/FibonacciDegreeCompensator.goldenRenormalization (✓ std3).

Source. Repository-derived.

Commentary.

One golden renormalization step contracts a real transverse defect by the inverse square of the golden ratio.

Definition 1.2 (Even Fibonacci degree).

$$D_r(n)=F_{2n+r}.$$

Formalization. D5/S3/CompletionDynamics/PacketRenormalization/FibonacciDegreeCompensator.fibonacciDegree (✓ std3).

Source. Repository-derived.

Commentary.

At step n and natural offset r, the integer observation degree is the Fibonacci number with index 2n+r.

Definition 1.3 (Integer compensator).

$$\mathrm{IsIntegerCompensator}\left(R,D,g\right)⟺\forall a\in \mathbb{R},\underset{n\to \infty }{\lim }D(n)\mathrm{iterate}\left(R,n,a\right)=ag.$$

Formalization. D5/S3/CompletionDynamics/PacketRenormalization/FibonacciDegreeCompensator.IsIntegerCompensator (✓ std3).

Source. Repository-derived.

Commentary.

A natural-valued degree sequence compensates a real renormalization when, for every initial defect, the degree-orbit product converges to that initial value multiplied by the prescribed gain.

Theorem 1.4 (Fibonacci degrees compensate golden contraction).

$$\forall r\in \mathbb{N},\forall {\Delta }_0\in \mathbb{R},\forall N:\mathbb{N}\to \mathbb{N},\forall \Delta :\mathbb{N}\to \mathbb{R},(\forall n\in \mathbb{N},N_n=F_{2n+r})\Rightarrow \left((\forall n\in \mathbb{N},{\Delta }_n={\Delta }_0{\phi }^{-2n})\Rightarrow \left(\underset{n\to \infty }{\lim }{F}_{2n+r}{\phi }^{-2n}=\frac{{\phi }^r}{\sqrt{5}}\wedge \left(\underset{n\to \infty }{\lim }{N}_n{\Delta }_n={\Delta }_0\frac{{\phi }^r}{\sqrt{5}}\wedge \mathrm{IsIntegerCompensator}\left(goldenRenormalization,\mathrm{fibonacciDegree}\left(r\right),\frac{{\phi }^r}{\sqrt{5}}\right)\right)\right)\right).$$

Proof. Machine-checked in Lean as D5/S3/CompletionDynamics/PacketRenormalization/FibonacciDegreeCompensator.fibonacci_degree_compensator (✓ std3). ∎

Source. Repository-derived.

Commentary.

Assume the source sequences satisfy N(n)=F(2n+r) and delta(n)=delta-zero times phi to the power -2n. The theorem then exposes both displayed limits as separate conclusions.

Its third conclusion states the final structural clause: the actual even Fibonacci degree sequence is an integer compensator for inverse-square golden renormalization, uniformly over its real initial defect.

References

• Truth anchor: D5/S3/CompletionDynamics/PacketRenormalization/FibonacciDegreeCompensator.IsIntegerCompensator
• Truth anchor: D5/S3/CompletionDynamics/PacketRenormalization/FibonacciDegreeCompensator.fibonacciDegree
• Truth anchor: D5/S3/CompletionDynamics/PacketRenormalization/FibonacciDegreeCompensator.fibonacci_degree_compensator
• Truth anchor: D5/S3/CompletionDynamics/PacketRenormalization/FibonacciDegreeCompensator.goldenRenormalization