Problem. Let \(\Phi\) be the classical even Riemann theta kernel and define \[ C_\Phi(a,b)=2\int_{|a+b|/2}^{\infty}q\,\Phi\!\left(\frac{a-b}{2}+q\right)\Phi\!\left(\frac{a-b}{2}-q\right)\,dq. \] Fix the interval \([-2,2]\). For each even integer \(m\ge 2\), set \(h=4/m\) and \(t_j=-2+jh\) for \(0\le j\le m\), and form the anchor Gram matrix \(G_m=[C_\Phi(t_i,t_j)]_{0\le i,j\le m}\). Whenever \(G_m\) is invertible, define \[ c_m(a)=\bigl(C_\Phi(t_0,a),\ldots,C_\Phi(t_m,a)\bigr)^T \] and the intrinsic Schur residual \[ S_m(a,b)=C_\Phi(a,b)-c_m(a)^*G_m^{-1}c_m(b). \] For the cell \(I_j=[t_j,t_{j+1}]\), define the normalized Dirichlet modes \[ \phi_{j,k}(x)=\sqrt{\frac2h}\sin\!\left(k\pi\frac{x-t_j}{h}\right),\qquad k\ge1. \] For \(k,\ell\ge1\), let \(H_{k\ell}^{(m)}\) be the \(m\times m\) block whose \((i,j)\) entry is \[ (H_{k\ell}^{(m)})_{ij}=\int_{I_i}\int_{I_j}S_m(x,y)\phi_{i,k}(x)\phi_{j,\ell}(y)\,dx\,dy. \] Whenever \(H_{11}^{(m)}\) is invertible, define \[ P_1^{(m)}=H_{11}^{(m)},\qquad P_2^{(m)}=H_{22}^{(m)}-H_{21}^{(m)}(H_{11}^{(m)})^{-1}H_{12}^{(m)}. \] Determine the exact small-cell asymptotic of the first even modal pivot \(P_2^{(m)}\) as \(m\to\infty\), equivalently \(h\downarrow0\). In particular, decide whether there exist constants \(m_0\) and \(0<c_-\le c_+<\infty\) such that for every even \(m\ge m_0\), the matrices \(G_m\) and \(H_{11}^{(m)}\) are invertible, the pivots \(P_1^{(m)}\) and \(P_2^{(m)}\) are positive semidefinite, and \[ c_-h^6\le \frac{\operatorname{Tr}P_2^{(m)}}{\operatorname{Tr}P_1^{(m)}}\le c_+h^6. \] Preferably determine whether the finite positive limit \[ \lim_{m\to\infty}h^{-6}\frac{\operatorname{Tr}P_2^{(m)}}{\operatorname{Tr}P_1^{(m)}} \] exists and identify it. A negative resolution may instead exhibit a rigorous infinite sequence of even \(m\to\infty\) for which the required invertibility, positivity, or either side of the \(h^6\) comparison fails.