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The Two-Coordinate Metallic Transfer

Abstract

The metallic transfer word returns the pair (1,0) after each cycle and bounds both coordinates by the set {-1,0,1,2}.

Definition 1.1 (The ordered transfer coefficients).

Lean statement: D5/S3/Combinatorics/MetallicHankel/MetallicHankelTransfer.topWord

Formalization. D5/S3/Combinatorics/MetallicHankel/MetallicHankelTransfer.topWord (✓ std3).

Source. Repository-derived.

Acknowledgement. Guo-Niu Han, Emmanuel Pedon (2025). Hankel continued fractions and Hankel determinants for q-deformed metallic numbers. DOI: 10.48550/arXiv.2502.05993. URL: https://arxiv.org/abs/2502.05993v2.

Commentary.

At n = 2 the word of pairs (d,b) is (2,-1),(0,1),(0,-1),(1,-1),(0,-1),(0,-1),(2,1),(-1,-1). For every other nonnegative n it starts with (2,-1),(0,1), contains n-2 copies of the triple (1,-1),(1,-1),(-1,-1), then (0,-1),(1,-1),(0,-1),(-1,-1), then n-3 copies of the same triple, and ends with (1,-1),(1,-1),(0,-1),(2,1),(-1,-1). Repetition counts use natural subtraction truncated at zero.

Theorem 1.2 (Periodicity and bounds of the transfer coordinates).

Lean statement: D5/S3/Combinatorics/MetallicHankel/MetallicHankelTransfer.block_transfer

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MetallicHankel/MetallicHankelTransfer.block_transfer (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Guo-Niu Han, Emmanuel Pedon (2025). Hankel continued fractions and Hankel determinants for q-deformed metallic numbers. DOI: 10.48550/arXiv.2502.05993. URL: https://arxiv.org/abs/2502.05993v2.

Commentary.

Let n be at least two, let W be its transfer word of length L, and let z_p be a sequence of integer pairs with z_0 = (1,0). If W at position p modulo L is (d,b), require z_{p+1} = (d x - b y,x), where z_p = (x,y). Then for every nonnegative p, z_{p+L} = z_p and both coordinates of z_p belong to {-1,0,1,2}. Repeated triples and the two end segments give the return and the bounds at every intermediate position.

References