Let f be a finite sum of quadratic monomial rotation-symmetric Boolean functions, and let W_f(n) denote the Hamming weight of its induced Boolean function on n variables. Let R(f) be the associated rules matrix, and let lambda_1,…,lambda_N be the characteristic values of R(f), counted with algebraic multiplicity. The Quadratic Easy Coefficients Conjecture asserts that, for the relevant values of n, W_f(n)=2^n-1-frac12∑_j=1^Nlambda_j^,n; equivalently, the coefficient sequence of the weight recurrence is given by the power sums of the characteristic values of the rules matrix.

Quadratic Easy Coefficients Conjecture

Let $f$ be a finite sum of quadratic monomial rotation-symmetric Boolean functions, and let $W_f(n)$ denote the Hamming weight of its induced Boolean function on $n$ variables. Let $\mathcal{R}(f)$ be the associated rules matrix, and let $\lambda_1,\ldots,\lambda_N$ be the characteristic values of $\mathcal{R}(f)$ , counted with algebraic multiplicity. The Quadratic Easy Coefficients Conjecture asserts that, for the relevant values of $n$ , $W_f(n)=2^{n-1}-\frac{1}{2}\sum_{j=1}^{N}\lambda_j^{\,n}$ ; equivalently, the coefficient sequence of the weight recurrence is given by the power sums of the characteristic values of the rules matrix.

References

Primary source

arXiv

Additional references

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

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

An unrefereed preprint claims to prove the conjecture, but no independent verification has been found.

The conjecture concerns coefficient formulas for weight sequences of quadratic rotation-symmetric Boolean functions, connected with de Bruijn matrices and symbolic dynamics. Cusick’s February 15, 2025 paper established substantial special cases but explicitly left the general repeated-root case unresolved.

Known results

Cusick, 2025: quadratic cases satisfy integer-coefficient linear recurrences of orders governed by the minimal polynomial and by $2^T+1$ .
Cusick, 2025: the conjecture holds for quadratic monomial rotation-symmetric functions, corresponding to $m=1$ .
Cusick, 2025: for general quadratic functions, the coefficient formula holds when the minimal and characteristic polynomials coincide, equivalently when the characteristic polynomial has no repeated roots.

September 1, 2026 claimed proof

A new preprint identifies a signed de Bruijn transfer matrix with a finite-type shift adjacency block and derives the conjectured coefficients from a dynamical zeta function. This is presented as a full proof, but it is unrefereed and no independent verification or error analysis was found.

Current status (as of September 2026): A preprint claims a complete proof, but the conjecture remains unverified, especially as no independent confirmation of the new argument has been found.

Sources

No solutions have been posted yet.