Parity families and a kernel-averaged L-function for near-Ramanujan signings

For a signing $\sigma$ of a $d$-regular graph, the spectrum of $A_\sigma$ depends only on the signs of cycles.

We study the affine $\mathbb F_2$ family of signings making every short even cycle unbalanced, and show that averaging over it converts the sign problem of the Bilu-Linial conjecture into a counting problem: a master identity expresses the family-averaged trace as a parity-weighted sum over wrap classes confined to the span $W$ of the constraint cycles, and the family-averaged Ihara $L$-function diagonalizes so that every prime whose parity escapes $W$ contributes the Ramanujan rate $\sqrt{d-1}$ automatically.

Uniform averaging over all signings, by contrast, provably cannot certify a spectral radius below the Kesten profile.

We prove matched upper and lower bounds for the confined walk counts, a doubling injection from below, and from above an ear-decomposition encoding in which the number of fresh runs of a non-backtracking walk equals the cycle rank of its support, combined with a window lemma for bicycle-free graphs and a rank bound via the Moore bound for irregular graphs.

Consequences include $\varepsilon$-versions of the Bilu-Linial conjecture: every $d$-regular graph that is subcritical at scale $\log n$, and every $d$-regular graph bicycle-free at radius $C\log\log n/\delta$, admits a signing in the parity family with $\rho(A_\sigma)\le2\sqrt{d-1}(1+C\delta\log(1/\delta))(1+o(1))$.

We further identify the necessary hypotheses exactly ($K_d$-trapping; tree-burst gadgets), give an exact certificate on the hypercube, and record a decisive obstruction to two-sided interlacing: $\mathbb E_\sigma\det(xI-A_\sigma^2)$ is not real-rooted, already for the quadrilateral, where it equals $(x^2-4x+2)^2+4$.