Dual-Number Transfer Along the Metallic Cycle

Abstract

The metallic reversed-denominator recurrence over integer dual numbers has an explicit linear monodromy at each complete cycle.

Definition 1.1 (The cyclic dual-number coefficients).

Lean statement: D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnboundedTransfer.jetWord

Formalization. D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnboundedTransfer.jetWord (✓ 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.

Work in the integer dual numbers, written a + b epsilon with epsilon squared equal to zero. For n equal to one, the coefficient word is [(1+epsilon,-1), (1+epsilon,1), (-1+epsilon,-1)]. For every other nonnegative integer n, take the metallic cycle in its given order. At each state with fraction data (k,v,D), the corresponding pair is ([q^{k+1}]D + [q^k]D epsilon, v). Thus the word retains the denominator coefficients at degrees k+1 and k together with the sign of each fraction term.

Definition 1.2 (One reversed-denominator step).

Lean statement: D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnboundedTransfer.jetStep

Formalization. D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnboundedTransfer.jetStep (✓ 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.

For a state (z_1,z_2) of two integer dual numbers and a coefficient pair (a,v) consisting of a dual number and an integer, the next state is (a z_1 - v z_2, z_1). Integers act as dual numbers with zero epsilon coefficient.

Theorem 1.3 (The state after complete cycles).

Lean statement: D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnboundedTransfer.cycle_transfer

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnboundedTransfer.cycle_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 a positive integer, let W be its cyclic dual-number coefficient word, and let a sequence of states start at (1,0) and follow the step (z_1,z_2) to (a z_1 - v z_2,z_1), using the coefficient W at the current index modulo its length. Put lambda = 1 for n = 1 and lambda = 2n+1 otherwise. After k complete cycles, for every nonnegative integer k, the state is (1 - k lambda epsilon, 2k lambda epsilon). This includes the separate three-term golden word and the metallic words for every n at least two.

References

• Truth anchor: D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnboundedTransfer.cycle_transfer
• Truth anchor: D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnboundedTransfer.jetStep
• Truth anchor: D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnboundedTransfer.jetWord
• Dependency: D5/S3/Combinatorics/MetallicHankel/MetallicHankelData