Relation-Space Growth for Admissible Words

Abstract

Full linear relations between admissible words have squared Fibonacci dimension and golden-ratio-squared growth.

Theorem 1.1 (The relation-space dimension is a Fibonacci square).

$$\forall n\in \mathbb{N},{\dim }_C\mathrm{End}(H_n)=F_{n+2}^2.$$

Proof. Machine-checked in Lean as D5/S1/Words/AdmissibleWords/AdmissibleRelationSpaceGrowth.admissible_relation_space_finrank (✓ std3). ∎

Source. Repository-derived.

Commentary.

Let H_n be the complex function space on the length-n Zeckendorf-admissible binary words. The existing admissible-word count gives dim H_n = F_(n+2), and the standard finrank formula for linear maps therefore gives dim End(H_n) = F_(n+2)^2.

Theorem 1.2 (Consecutive relation spaces grow by the golden ratio squared).

$$\underset{n\to \infty }{\lim }\frac{\dim \mathrm{End}(H_{n+1})}{\dim \mathrm{End}(H_n)}={\phi }^2.$$

Proof. Machine-checked in Lean as D5/S1/Words/AdmissibleWords/AdmissibleRelationSpaceGrowth.admissible_relation_space_growth (✓ std3). ∎

Source. Repository-derived.

Commentary.

After substituting the exact dimension formula, the consecutive ratio is the square of F_(n+3)/F_(n+2). Mathlib’s Fibonacci ratio limit then yields the square of the golden ratio.

References

• Truth anchor: D5/S1/Words/AdmissibleWords/AdmissibleRelationSpaceGrowth.admissible_relation_space_finrank
• Truth anchor: D5/S1/Words/AdmissibleWords/AdmissibleRelationSpaceGrowth.admissible_relation_space_growth
• Dependency: D5/S1/Words/AdmissibleWords/AdmissibleCount