中文:本存档包含非交换离散傅里叶变换(NCDFT)框架的完整数学推导。该框架通过参数 $\alpha\in[0,1]$ 定义一族酉矩阵,揭示了在 $\alpha=1/2$ 处的严格自对偶性。谱分析表明,当且仅当 $\alpha=1/2$ 时,尺度化生成元谱退化为点谱;否则谱无界扩张。通过 Weil 公式,证明 $\alpha=1/2$ 时的谱测度收敛于黎曼 $\zeta$ 函数非平凡零点处的 Dirac 测度之和,从而推出黎曼猜想等价于 $\alpha=1/2$ 的自对偶性。所有步骤均给出严格证明,适用于理论验证与数值实验。PS:中文md文档是最新的 English:This deposit contains the complete mathematical derivation of the Noncommutative Discrete Fourier Transform (NCDFT) framework. The family of unitary matrices, parametrized by $\alpha\in[0,1]$, exhibits a critical self-duality at $\alpha=1/2$. Spectral analysis shows that the scaled generator has a degenerate point spectrum iff $\alpha=1/2$; otherwise its spectrum expands unboundedly. Via Weil’s formula, the spectral measure at $\alpha=1/2$ is proved to converge to the sum of Dirac measures at the non‑trivial zeros of the Riemann zeta function, establishing that the Riemann Hypothesis is equivalent to the self-duality at $\alpha=1/2$. All steps are rigorously proved, making the framework suitable for both theoretical study and numerical experiments.Note:English md format no update.plese use chinese md format.