Consistency probability of the cycle C_n

Establish that for a randomly generated 2-sparse equation system over the binary field F2, in which each edge of the n-cycle graph C_n corresponds to a uniformly and independently chosen Boolean polynomial in its two incident variables, the consistency probability equals q(C_n) = s^n + t^n − 1/4^n − 1/8^{n−1}, where s = (17 + √97)/32 and t = (17 − √97)/32.

Background

The paper studies the inconsistency/consistency probability of randomly generated sparse polynomial systems over F2, associating each 2-sparse system with a graph where edges indicate equations on the corresponding variable pair. For paths and stars the authors derive explicit formulas for consistency probability and characterize extremal trees and forests.

For cycles C_n, the authors do not provide a proof but instead propose a closed-form expression for q(C_n) in terms of the constants s and t that arise as roots of a characteristic equation used in the path analysis. They report computational agreement for n = 3,…,7 and pose the exact formula as a conjecture.