紧窗口中的Weil正性：有限约化、认证的双侧界及Landau-Widom衰减定律
Weil positivity in compact windows: a finite reduction, certified two-sided bounds, and a Landau-Widom decay law
- School of Mechanics and Aerospace Engineering（力学与航空航天工程学院）
- Dalian University of Technology（大连理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对紧窗口的Weil正性，通过认证计算得到支撑[-0.8,0.8]的二次型下界，经区间算术得到L=2时的上界，验证了Landau-Widom衰减定律，还明确了正性路径无法单独推导RH的原因。
AI中文摘要：
Weil准则指出，黎曼假设（Riemann Hypothesis, RH）等价于测试函数上显式二次型Q(f)的非负性。对于支撑在[-L, L]内的f，我们从两侧研究归一化下确界λ_min(L) = inf Q(f)/||f||²。下界（无条件）：支撑在[-(log 2)/2, (log 2)/2]内的f，其有界支撑上的正性是经典结果（Yoshida；Connes-Consani）。我们通过认证计算证明，对于所有支撑在[-0.8, 0.8]内的f，即自相关支撑为1.6（是经典范围的2.3倍），Q(f) ≥ 8.9×10⁻¹⁸ ||f||²。该证明基于一次式约化：Weil符号的逐点上界，结合Weyl均匀分布理论证明梳状常数最优，将整个窗口上的正性转化为单个有限矩阵的半正定性。奇宇称部分表明，认证正性对所有支撑为1.6的复测试函数成立，且窗口基态是简单偶宇称，这是Connes、Consani、Moscovici及van Suijlekom算子理论方案所需的谱假设。上界（无条件）：变分界在区间算术的几何侧计算，未依赖零点或RH，当L=2时低至3.2×10⁻²⁸³。这些界遵循经验定律 -ln λ_min(L) ~ 2π²N(T*)/ln N(T*)，其中T* = 2πe^(2L)，N为零点计数函数；2π²与Landau-Widom特征值骤降率匹配。在RH下，对所有足够大的L，λ_min(L) ≤ exp(-L e^L)。综合结果：在L=0.8时，两侧界围出了λ_min(0.8)的范围，即8.9×10⁻¹⁸ ≤ λ_min(0.8) ≤ 2.27×10⁻¹⁷，二者均已认证。我们还解释了为何正性路径无法单独推导RH：任何一次式认证必须分辨频率高达2πe^(A_L)，其中A_L ~ 4e^L，这是一个双指数阈值，素数梳状的任何逐点界都无法降低该阈值，同时谱余量以Landau-Widom速率坍缩。
英文摘要：
Weil's criterion states that the Riemann Hypothesis (RH) is equivalent to the non-negativity of an explicit quadratic form Q(f) on test functions. For f supported in [-L,L] we study the normalized infimum lambda_min(L) = inf Q(f)/||f||^2 from both sides. Lower bounds (unconditional): positivity on bounded support is classical for supp f in [-(log 2)/2,(log 2)/2] (Yoshida; Connes-Consani). We prove by certified computation that Q(f) >= 8.9e-18 ||f||^2 for all supp f in [-0.8,0.8], i.e. autocorrelation support 1.6, 2.3 times the classical range. The proof rests on a one-stroke reduction: a pointwise envelope for the Weil symbol, with a comb constant shown optimal by Weyl equidistribution, converts positivity on the whole window into positive semidefiniteness of a single finite matrix. The odd parity sector shows that the certified positivity holds for arbitrary complex test functions of support 1.6, and that the window ground state is simple and even, the spectral hypothesis required by the operator-theoretic program of Connes, Consani, Moscovici and van Suijlekom. Upper bounds (unconditional): variational bounds are evaluated on the geometric side in interval arithmetic, with no appeal to zeros or RH, down to 3.2e-283 at L=2. They follow the empirical law -ln lambda_min(L) ~ 2 pi^2 N(T*)/ln N(T*), T* = 2 pi e^{2L}, with N the zero-counting function; 2 pi^2 matches the Landau-Widom eigenvalue-plunge rate. Under RH, lambda_min(L) <= exp(-L e^L) for all large L. Synthesis: at L=0.8 the two halves enclose the profile, 8.9e-18 <= lambda_min(0.8) <= 2.27e-17, both certified. We show why the positivity route cannot reach RH unassisted: any one-stroke certificate must resolve frequencies up to 2 pi e^{A_L}, A_L ~ 4e^L, a doubly exponential threshold that no pointwise bound on the prime comb lowers, while the spectral margin collapses at the Landau-Widom rate.
