The Quadratic Easy Coefficients Conjecture via Finite-Type Shifts and Zeta Functions

Abstract

We prove the Quadratic Easy Coefficients Conjecture stated as Conjecture 1 in T. W. Cusick, \emph{Recursions for quadratic rotation symmetric functions weights}, Discrete Applied Mathematics 378 (2026), 93--101. For an arbitrary finite sum of quadratic monomial rotation symmetric Boolean functions, we identify the recurrent part of the rules matrix with a signed binary de Bruijn transfer matrix $B$. We then give a one-step presentation of the finite-type shift associated with the Boolean function in the symbolic-dynamics construction of Chirvasitu and Cusick. Fourier transformation in an auxiliary $\F_2$ coordinate decomposes the adjacency matrix of this shift into an unsigned de Bruijn block and the signed block $B$. Consequently the dynamical zeta function is \[ ζ_{X_f}(z)=\frac{1}{\det(I-z\cR(f))}, \] where $\cR(f)$ is the rules matrix. This equality identifies, with their algebraic multiplicities, the characteristic values, supplied by symbolic dynamics, with the roots of the characteristic polynomial of the rules matrix. The desired easy coefficients formula follows from the trace of $B^n$. We also prove nonsingularity and justify the unique backward extension of the weight recurrence.