Asymptotic Rigidity and Boundary Structure of Yang's Numerical Invariants for the Bidisk Submodules $[z^k-w^\ell]$

Abstract

Let \[ M_{k,\ell}=[z^k-w^\ell]\subset H^2(\mathbb D^2), \qquad k,\ell\in\mathbb N,\quad k\ne\ell . \] We study the large-index asymptotics of Yang's numerical invariants and the boundary structure of their generating function. Starting from the exact staircase formula obtained in our preceding work, we prove that \[ Σ_j(M_{k,\ell}) = \frac{C_{k,\ell}}{j} + O(j^{-2}), \] where \[ C_{k,\ell} = \int_0^\infty \frac{x^2} {(x+1/k)^2(x+1/\ell)^2}\,dx . \] The first strict descent together with $C_{k,\ell}$ recovers the unordered pair $\{k,\ell\}$, yielding an asymptotic rigidity principle. At the next order we obtain a periodic correction \[ Σ_j(M_{k,\ell}) = \frac{C_{k,\ell}}{j} + \frac{Ψ_{k,\ell}(j)}{j^2} + O(j^{-3}), \] whose least period is \(\operatorname{lcm}(k,\ell)\), and we determine the leading amplitudes of the strict drops. For Yang's generating function \[ \mathcal P_{k,\ell}(t) = \sum_{j=0}^{\infty}Σ_j(M_{k,\ell})t^j, \] we prove that it has radius of convergence one and a logarithmic singularity at $t=1$; hence it is never a polynomial for $k\ne\ell$, giving a negative answer to Yang's polynomiality question within this quasi-homogeneous family. The periodic higher-order corrections generate root-of-unity polylogarithmic boundary modes. The leading logarithmic coefficient together with the second-order boundary support determines the unordered pair $\{k,\ell\}$, while successive renormalized boundary limits recover the Fourier coefficients of every finite-order periodic asymptotic term.