AI 中文总结
本文证明Mellin周期泛函行列式严格正性，得出满支撑周期定理，进而证明连续Rankin-Cohen括号构成模形式空间基，并给出除数-tau卷积的精确公式及模素数约化结果。
AI 中文摘要
对于$r\ge1$，令$K=2r+14$且$d=\dim S_K$。我们证明了Mellin周期泛函行列式的严格正性，从而得到了基本半区间上周期的满支撑定理，特别是Xue的混合奇偶周期独立性猜想。作为推论，连续的首次Rankin-Cohen括号$[E_{2r+10-2j},E_{2j+2}]_1$（$1\le j\le d$）构成$S_K$的一组基，并且我们确定了所有可容许的首个Eisenstein括号有序行列式的符号。这给出了除数-tau卷积$C_r(n)=\sum_{m=1}^{n-1}\sigma_{2r+1}(m)\tau(n-m)$的一个统一的精确全权重公式。将相同的坐标恒等式模素数约化，我们在每个权重中获得了无穷多个素数的典范逐素数约化，并通过消失的Cramer坐标刻画了稀疏Ramanujan型特化。最后，对于齐次的$f,g\in\mathbf Q[E_4,E_6]$，我们证明了$[f,g]_1/\Delta=-3456\det((f_{E_4},f_{E_6}),(g_{E_4},g_{E_6}))$，将Cramer系统归结为$T=E_6^2/E_4^3$中的单变量坐标问题。
英文摘要
For $r\ge1$, let $K=2r+14$ and $d=\dim S_K$. We prove strict positivity for determinants of Mellin period functionals, yielding a full-spark theorem for periods in a fundamental half-range and, in particular, the mixed odd--even period independence conjecture of Xue. As a consequence, the consecutive first Rankin--Cohen brackets $[E_{2r+10-2j},E_{2j+2}]_1$, $1\le j\le d$, form a basis of $S_K$, and we determine the signs of all admissible ordered determinants of first Eisenstein brackets. This gives a uniform exact all-weight formula for the divisor--tau convolution $C_r(n)=\sum_{m=1}^{n-1}σ_{2r+1}(m)τ(n-m)$. Reducing the same coordinate identity modulo primes, we obtain canonical prime-wise reductions for infinitely many primes in every weight and characterize sparse Ramanujan-type specializations through vanishing Cramer coordinates. Finally, for homogeneous $f,g\in\mathbf Q[E_4,E_6]$, we prove $[f,g]_1/Δ=-3456\det((f_{E_4},f_{E_6}),(g_{E_4},g_{E_6}))$, reducing the Cramer system to a one-variable coordinate problem in $T=E_6^2/E_4^3$.
