Primitive Bernoulli Seam Matrices, Class Numbers, and Cyclotomic q-Sweep Spectra

We introduce and study finite \emph{Bernoulli seam matrices} and \emph{cyclotomic \(q\)-sweep matrices}. These are explicit finite matrices on the unit group \[ G_N=(\mathbb Z/N\mathbb Z)^\times \] whose character spectra recover special values of Dirichlet \(L\)-functions, imaginary and real quadratic class-number invariants, Gauss-twisted \(q\)-series, Ramanujan sums, and Dedekind correlations. The normalized Bernoulli seam kernel is \[ b_N(u)=\frac12-\frac{\widetilde u}{N}, \qquad u\in G_N, \] where \(\widetilde u\in\{1,\dots,N-1\}\) is the standard representative. The primitive Bernoulli seam matrix is \[ \mathsf A_N=\left(b_N(ab^{-1})\right)_{a,b\in G_N}. \] We prove that \(\mathsf A_N\) is diagonalized by the Dirichlet characters modulo \(N\), and that \[ \boxed{ \mathsf A_Nv_\chi=L_N(0,\overline\chi)v_\chi, } \] where \[ L_N(0,\chi)=\sum_{a=1}^{N}\chi(a)\left(\frac12-\frac{a}{N}\right) \] is the value at \(0\) of the Dirichlet \(L\)-function associated with the character modulo \(N\), including the modulus-\(N\) Euler-factor convention. Equivalently, for the unnormalized seam-defect matrix \[ \mathsf B_N=2\pi\mathsf A_N, \] one has \[ \boxed{ \mathsf B_Nv_\chi=2\pi L_N(0,\overline\chi)v_\chi. } \] Quadratic eigenspaces give class numbers. If \(D3\), then \[ \boxed{ \mathsf A_pv_{\chi_{-p}}=h(-p)v_{\chi_{-p}}, \qquad \mathsf B_pv_{\chi_{-p}}=2\pi h(-p)v_{\chi_{-p}}. } \] Thus imaginary quadratic class numbers occur as explicit eigenvalues of finite rational matrices. We derive exact seam Plancherel identities. For \(N>1\), \[ \boxed{ \sum_{\chi\bmod N}|L_N(0,\chi)|^2 = \frac{\varphi(N)}{12} \left( \varphi(N)+\frac{2}{N}\prod_{p\mid N}(1-p) \right). } \] For an odd prime \(p\), this becomes \[ \boxed{ \sum_{\substack{\chi\bmod p\\ \chi(-1)=-1}} |L(0,\chi)|^2 = \frac{(p-1)^2(p-2)}{12p}. } \] We also prove seam-dilation floor-sum identities. If \(\chi\) is nonprincipal modulo \(N\) and \((m,N)=1\), then \[ \boxed{ \sum_{a=1}^{N}\left\lfloor\frac{ma}{N}\right\rfloor\chi(a) = \left(\overline{\chi(m)}-m\right)L_N(0,\chi). } \] For \(p\equiv3\pmod4\), \(p>3\), this gives the class-number formula \[ \boxed{ \sum_{a=1}^{p-1} \left\lfloor\frac{ma}{p}\right\rfloor \left(\frac ap\right) = \left(\left(\frac mp\right)-m\right)h(-p). } \] In particular, the quadratic half-interval residue excess \[ E_p=\sum_{1\le a<p/2}\left(\frac ap\right) \] satisfies \[ \boxed{ E_p= \left(2-\left(\frac2p\right)\right)h(-p) = \begin{cases} h(-p), & p\equiv7\pmod8,\\[0.3em] 3h(-p), & p\equiv3\pmod8. \end{cases} } \] We also intr