AI 中文总结
该研究为 Salikhov-Zeilberger-Zudilin-Bai 族给出闭式界公式，通过二维搜索和计算证据表明 Bai 点 (1857,1857,2785) 是全局最优，并给出猜想性下取整值。
AI 中文摘要
Salikhov（2008）、Zeilberger-Zudilin（2020）和 Bai（2026）都以相同形状的复围道积分来界定 π 的无理性度量，仅在三个指数上有所不同。我们给出了一个闭式表达式，将所得界表示为这些指数的函数。鞍点数据来自一个显式三次式；算术因子——2 的幂、最小公倍数的范围，以及推广 Zeilberger-Zudilin 引理 2 的素数消去窗口——对任意指数均已写出，因此对于新的选择无需重做任何算术引理。该定律在 (2,2,3) 处返回 7.1032053341370017，与 Zeilberger-Zudilin 的结果在 17 位数字上一致；在 Bai 点 (1857,1857,2785) 处返回 7.1018628323563507，与其发表的 7.101862832357 一致；它还能从 Salikhov 自己的双分支窗口重现其 7.606308。我们证明窗口度量是精确齐次的，W = this http URL (a0/a1, b/a1)，因此该界仅依赖于两个比值，三参数格塌缩为二维对象；证明 π 系数是单次系数提取；并且证明在 j = 0 时整除性判据是精确的，出现一个超卡特兰数，随后 Kummer 定理给出 ord_p 的等式而非不等式。在该二维搜索上，我们给出计算证据表明 Bai 点是该族的全局最小值：在切片 a0 = a1 上，该界仅是 Q = 2a0/(3a0-2b) 的函数，在 Q = 3714 处有一个尖锐顶点，仅由 (1857,1857,2785) 本原实现。这依赖于三个有限计算，标记为已验证而非已证明；由此得到的下取整值 7.101862832356 被表述为一个猜想。一个附带的 Python 脚本（仅用标准库）在大约半分钟内重新推导出每一项声明。
英文摘要
Salikhov (2008), Zeilberger-Zudilin (2020) and Bai (2026) bound the irrationality measure of pi with the same shape of complex integral. This note gives one closed-form law mu(a0,a1,b) for the whole family, shows that it reproduces all three published records, proves that it is exactly scale-invariant, and gives evidence that Bai's parameter point is the global optimum of the family - a statement Bai explicitly declines to make about her own result. The admissible configurations are forced by a Machin condition: zeros at 0, +-w, +-wbar and poles at +-P give a linear form in 1 and pi if and only if P^2 - N(w) = 2P Im(w). The law depends only on the two ratios a0/a1 and b/a1, so the infinite three-parameter lattice is exactly two-dimensional. The 2-adic saving is exactly 5(2a1-b)/2 per n, in two regimes, and no further saving exists. The decay rate is governed by the min-max value of the steepest-descent method, the lowest level at which the two endpoints of the integral become connected within a sublevel set of the integrand; the value obtained from the least critical level is a lower bound for mu and serves as a screening. Among the sixteen admissible configurations with P < 100, exactly one carries a triple that yields a bound, namely Salikhov's (P = 5, w = 1+2i), and the least mu over all sixteen and over all primitive triples with a0, a1 <= 10 and b <= 22 is 7.103205334137 at (2,2,3). Within that configuration the minimum is 7.101862832356 at (1857,1857,2785), which is Bai's point, the vertex sitting exactly at Q = 2a/(3a-2b) = 3714. Each statement carries an explicit label, PROVED or VERIFIED or CONJECTURE, and the reservations of scope are stated in the note. An ancillary file re-derives every numerical claim from scratch, with no external dependency.
Comments11 pages. Ancillary file: verify_szzb.py (standalone verifier, standard library only). v3: Theorem 2 restated as a min-max value; arithmetic factor completed and checked at a second configuration; Section 9 redone: exactly one configuration yields a bound, least mu 7.103205334137
