Alternating Slices of the A2k−1 Theta Series

We study the alternating linear slice of the \(A_{2k-1}\)-theta series arisingfrom integer mass profiles on the cycle graph \(C_{2k}\). The classicaldictionary is\[ \Jac(C_n)\cong \ZZ/n\ZZ, \qquad \ZZ^n/\ZZ\one\cong A_{n-1},\]and cyclic square-gradient energy is the \(A_{n-1}\)-norm. We separate thisbackground from the arithmetic coefficient theory of the alternating slice,and also record that the principal divisor lattice of a cycle is the squareof the augmentation ideal; for prime \(n\) this is Craig's lattice\(A_{n-1}^{(2)}\). For a neutral divisor \(D_q=\sum q_i[i]\) on \(C_n\), with\(P_q(z)=\sum_i q_i z^i\) and\(\widehat q(a)=\sum_i q_i\zeta_n^{-ai}\), we record the Fourier--Abelidentity\[ [D_q] = P_q'(1) = -\sum_{a=1}^{n-1} \frac{\widehat q(a)}{1-\zeta_n^a} \pmod n.\]Equivalently, the Jacobian class is the first augmentation jet\(P_q'(1)\bmod n\) of the neutral charge polynomial. For \(n=2k\), define\[ \Theta_{2k}^{\Alt}(u,y) = \sum_{m\in\ZZ^{2k}/\ZZ\one} u^{E_{2k}(m)} y^{M_k(m)}, \qquad M_k(m)=\sum_i(-1)^im_i.\]Writing \(N_k(E,M)=[u^Ey^M]\Theta_{2k}^{\Alt}\), we prove the exact sliceformula\[ N_k(E,M) = \sum_N R_k(N,M)R_k(E-N,M),\]where\[ R_k(N,M) = \#\{a\in\ZZ^k:\sum_i a_i=M,\ \sum_i a_i^2=N\}.\]Consequently, if \(M=ka+r\), \(0\le r<k\), then the true arithmetic floor is\[ E\ge 2\mu_k(M), \qquad \mu_k(M)=ka^2+2ar+r =\frac{M^2+r(k-r)}{k},\]and\[ N_k(2\mu_k(M),M)=\binom{k}{r}^2.\]The spectral Rayleigh boundary is exactly the residue class \(r=0\), and themaximal integer alternating coherence at moment \(M\) is\[ \frac{M^2}{M^2+r(k-r)}.\] We also refine the alternating slice by graph-Jacobian class. If\[ \alpha_{2k}(x)=\sum_i ix_i\pmod{2k}\]is the Jacobian coordinate of a difference vector \(x\in A_{2k-1}\), thejoint coefficients with fixed alternating moment and fixed Jacobian classfactor through two fixed-sum square-counting functions equipped with theadditional congruence\[ J(a)-J(b)=\frac{c+M}{2}\pmod k, \qquad J(a)=\sum_{j=0}^{k-1}ja_j.\] After subtracting the floor,\[ H=\frac{E-2\mu_k(M)}2,\]all layers \(H\ge0\) occur for every \(k\ge3\). For \(k=2\), the remainingsupport is governed by the classical two-square theorem. We solve thelow-rank cases \(k=2,3,4,5\) explicitly. The \(k=3\) formulas identify thealternating-slice coefficients with classical representation numbers of theEisenstein quaternary form, as recorded by Huard--Ou--Spearman--Williams andattributed there to Lomadze. The \(k=4\) components are cusp-free, while at\(k=5\) the cusp component is exactly a rational multiple of the eta product\[ \eta(\tau)^4\eta(5\tau)^4.\] Finally, the interior coefficients are coefficients of a coset thetacomponent for\[ A_{k-1}\oplus A_{k-1}.\]We prove a summatory asymptotic\[ \sum_{0\le h\le X}B_{k,r}(h) = \frac{2^{k-1}\pi^{k-1}}{k\Gamma(k)} X^{k-1} + O_{k,r}(X^{k-3/2}),\]and, for \(k\ge4\), a pointwise local-density expansion\[ B_{k,r}(H) = \frac{2^{k-1}\pi^{k-1}}{k\Gamma(k-1)} \Sing_{k,r}(H) \left(H+\frac{r(k-r)}{k}\right)^{k-2} + O_{k,r,\varepsilon}(H^{(k-2)/2+\varepsilon}).\]The local densities are normalized for the height variable\[ Q_r(a,b)=\frac{\sum a_i^2+\sum b_i^2-2r}{2},\]which is essential at \(p=2\). The good-prime local factors\(p\nmid 2k\) are computed in closed form. A stabilization lemma controlsthe bad primes, yielding uniform upper and lower bounds for the singularseries for fixed \(k\ge4\), and therefore\[ B_{k,r}(H)\asymp_k (H+1)^{k-2}.\]For \(k=4\), the cusp-free formulas determine the remaining \(2\)-adicdensity in closed form.