The Complete Higher-Shift Quasipolynomial Claim
Abstract
Conjecture C asserts integer-valued rational quasipolynomials for every higher metallic Hankel shift.
Definition 1.1 (All parameters, residues and initial sizes).
Lean statement: D5/S3/Combinatorics/MetallicHankel/MetallicHankelQuasipolynomialDefs.claim
Formalization. D5/S3/Combinatorics/MetallicHankel/MetallicHankelQuasipolynomialDefs.claim (✓ std3).
Source. Repository-derived.
Acknowledgement. Guo-Niu Han, Emmanuel Pedon (2026). Hankel determinants for the (n+2)-shifted q-metallic numbers. DOI: 10.48550/arXiv.2610.36132. URL: https://arxiv.org/abs/2610.36132v1.
Commentary.
For each integer n>=1, there exists an integral formal power series Phi with constant coefficient one satisfying q Phi^2+((1+q^n)(1-q)-q[n]_q)Phi=1, with [n]q the sum of q^i over 0<=i<n. For every such Phi and every shift ell>=n+3, put rho=2n(n+1) for even n and rho=4n(n+1) for odd n. There exists a family Q_r of rational polynomials, for all residues 0<=r<rho, such that Q_r(z) is an integer for every integer z and Delta(r+m rho)^(ell)=Q_r(m) for every nonnegative integer m. The determinant is that of the matrix with entry [q^(ell+a+b)]Phi on indices 0<=a,b<r+m rho. Its empty value is one. The claim includes n=1, r=m=0, all initial sizes and vanishing determinants, with the displayed allowed period and no assertion of minimality or degree. No eventual qualification is imposed.