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Biorthogonal eigenvectors of the Holte carry matrix and cascade-free enumeration

Daniel Andreas Moj

Abstract

For $k$-summand base-$N$ addition, the carry process is a Markov chain on $\{0,\ldots,k-1\}$ whose transition matrix--the Holte matrix $T$--has eigenvalues $\{N^{-j}\}_{j=0}^{k-1}$, all simple and independent of $N$. We give the complete biorthogonal eigenvector system. The left eigenvectors factor as $\sum_i u_j[i] x^i = c_{k,j} (x-1)^j A_{k-j}(x)$, where $c_{k,j} = |s(k,k-j)|/k!$ involves unsigned Stirling numbers and $A_n(x)$ is the Eulerian polynomial. The right eigenvectors satisfy $\sum_i \binom{k-1}{i} v_j[i] x^i = (1+x)^{k-1-j} Q_j(x)$, where the quotient polynomials $Q_j$ have palindrome symmetry $x^j Q_j(1/x) = (-1)^j Q_j(x)$ and converge to $(1-x)^j$ as $k \to \infty$; for $j \le 3$, we give explicit closed forms in terms of $k$. The cascade-free avoidance count satisfies $a(L) = (\sqrt{d})^L U_L(x)$ (Chebyshev polynomial of the second kind) whenever the restricted transfer matrix has dimension $d \le 2$; we prove this is sharp: for $k$-summand addition, Chebyshev form holds for $k = 3$ and fails for $k \ge 4$. The proof uses oscillatory matrix theory to establish non-vanishing of all spectral residues. The characteristic polynomial of the restricted transfer matrix is determined in closed form by a Stirling-weighted Lagrange interpolation at the Holte eigenvalues. Two systems with binary carry state spaces are shadow-equivalent if and only if they share the pair $(N, d)$. The general classification for $k$-state systems reduces to the characteristic polynomial of $T$.

Biorthogonal eigenvectors of the Holte carry matrix and cascade-free enumeration

Abstract

For -summand base- addition, the carry process is a Markov chain on whose transition matrix--the Holte matrix --has eigenvalues , all simple and independent of . We give the complete biorthogonal eigenvector system. The left eigenvectors factor as , where involves unsigned Stirling numbers and is the Eulerian polynomial. The right eigenvectors satisfy , where the quotient polynomials have palindrome symmetry and converge to as ; for , we give explicit closed forms in terms of . The cascade-free avoidance count satisfies (Chebyshev polynomial of the second kind) whenever the restricted transfer matrix has dimension ; we prove this is sharp: for -summand addition, Chebyshev form holds for and fails for . The proof uses oscillatory matrix theory to establish non-vanishing of all spectral residues. The characteristic polynomial of the restricted transfer matrix is determined in closed form by a Stirling-weighted Lagrange interpolation at the Holte eigenvalues. Two systems with binary carry state spaces are shadow-equivalent if and only if they share the pair . The general classification for -state systems reduces to the characteristic polynomial of .

Paper Structure

This paper contains 13 sections, 19 theorems, 59 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The Carry Chain
  3. Transfer matrix for $k$-summand addition
  4. Cascade-free sequences
  5. The Holte Spectrum
  6. The Biorthogonal Eigenvector System
  7. Left eigenvectors: Stirling--Eulerian factorization
  8. Right eigenvectors: binomial-palindromic characterization
  9. The Chebyshev Threshold
  10. Stochastic Classification
  11. Binary state spaces
  12. General state spaces
  13. Open Questions and Further Results

Key Result

Lemma 2.1

The characteristic polynomial $\lambda^2 - N\lambda + \mathfrak{d} = 0$ depends only on $(N, \mathfrak{d})$. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: The carry chain for binary state spaces ($k = 2$). Edge labels are transition probabilities under uniform digits.
  • Figure 2: The moduli space $\Omega$ of shadow-equivalence classes for binary carry chains ($k = 2$), shown for $N \le 12$ and $\mathfrak{d} \le 21$. Filled circles are achievable pairs $(N, \mathfrak{d}) \in \Omega$; open circles satisfy the necessary condition $N^2 \ge 4\mathfrak{d}$ but admit no integer factorization $gt = \mathfrak{d}$ with $g + t \le N$. The dashed curve is the AM--GM bound $N = 2\sqrt{\mathfrak{d}}$.

Theorems & Definitions (48)

  • Lemma 2.1: Transfer matrix invariants
  • Remark 2.2: Background from the companion paper
  • Theorem 2.3: Universality Moj2026
  • proof
  • Theorem 2.4: Chebyshev representation Moj2026
  • Theorem 2.5: Scaling law and Fibonacci bisection Moj2026
  • Lemma 3.1: Uniformity of $S \bmod N$
  • proof
  • Theorem 3.2: Holte matrix spectrum
  • proof
  • ...and 38 more