Sparse Coefficient Spectra of Pell Representation Polynomials: Growth, Block Structure, and Failure of Halving Laws

Abstract

We study the set of distinct coefficient values of the polynomials \(G_m(x)=\prod_{j=2}^{m+1}(1+x^{P_j}+x^{2P_j})\), where \(P_j\) are the Pell numbers. A coefficient-pair recurrence identifies the spectra at consecutive odd and even lengths. The resulting spectrum \(V_k\) admits an exact representation by products of blocks drawn from levels of the Calkin--Wilf tree, with additive level cost. We give the bound \(a_k\le1+A(5/3)^k\), where \(a_k=|V_k|\) includes zero and \(A=95367431640625/14169773449216<7\). For every \(\lambda>\varphi\), we also obtain \(a_k=O(\lambda^k)\), and deduce density zero within the natural coefficient-value interval. An atomic factorization argument shows that the upper exponential growth rates of the spectrum, primitive cost layers, first block appearances, and block levels coincide in a number \(\rho\in[2/\varphi,\varphi]\). We identify the block sets with the denominator sets of the Kepler tree, using an explicit level-preserving transformation. Finally, an exact finite certificate for 3847 disproves four precisely defined internal halving properties. Neither the existence of an ordinary growth limit nor the value of \(\rho\) is established. Human-directed AI-authored mathematical research. AI mathematical reviews were completed with explicit independence limitations, including overlapping development and self-review. Human review and formal proof certification were not performed. Classical Calkin–Wilf and Kepler trees and Mathar results are prior art. Novelty is not established; no priority claim is made.

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