We present a large-scale computational study combining arbitrary-precision arithmetic, sequence acceleration, and the PSLQ integer relation algorithm to discover exact closed-form expressions for fundamental constants arising in asymptotic analysis. We compute the Stokes multipliers C_M of the one-dimensional anharmonic oscillators H = p^2/2 + x^2/2 + g x^{2M} for M = 2, 3, ..., 11, extracting 17-30 significant digits from up to 1200 perturbation coefficients computed at 300-digit working precision. The computational pipeline consists of three stages: (i) Rayleigh-Schrodinger recursion in the harmonic oscillator basis, (ii) Richardson extrapolation of order 40-100 to accelerate convergence of ratio sequences, and (iii) PSLQ searches over bases of Gamma-function values and algebraic numbers. This pipeline discovers three new exact identities: C_3^2 pi^4 = 32, C_5^4 Gamma(1/4)^4 pi^5 = 2^{12} 3^2, and C_7^6 Gamma(1/3)^9 pi^6 = 2^{20} 3^3, in addition to confirming the known C_2^2 pi^3 = 6. Equally significant is a negative result: exhaustive PSLQ searches at 30-digit precision with coefficient bounds up to 2000 find no closed form for C_4, strongly suggesting the x^8 case introduces a genuinely new transcendental number. A number-theoretic pattern emerges: closed-form existence correlates with Euler's totient function phi(M-1)/2, which counts algebraically independent Gamma-function transcendentals at denominator M-1. We formulate conjectures connecting computational constant recognition to classical number theory, and provide all code and data for full reproducibility.

翻译：我们开展了一项大规模计算研究，结合任意精度算术、序列加速和PSLQ整数关系算法，以发现渐近分析中基本常数的精确闭式表达式。我们计算了一维非谐振子H = p^2/2 + x^2/2 + g x^{2M}（M = 2, 3, ..., 11）的Stokes乘子C_M，从高达1200个扰动系数（工作精度为300位）中提取了17-30位有效数字。计算流程包含三个阶段：(i) 谐振子基下的Rayleigh-Schrodinger递归，(ii) 阶数为40-100的Richardson外推以加速比率序列收敛，以及(iii) 基于Gamma函数值与代数数基底的PSLQ搜索。该流程发现了三个新的精确恒等式：C_3^2 π^4 = 32、C_5^4 Γ(1/4)^4 π^5 = 2^{12} 3^2 和 C_7^6 Γ(1/3)^9 π^6 = 2^{20} 3^3，同时验证了已知的 C_2^2 π^3 = 6。同样重要的是一个负面结果：在30位精度下对系数上限达2000的穷尽PSLQ搜索未能发现C_4的闭式表达式，这强烈表明x^8情形引入了一个真正全新的超越数。一个数论模式浮现：闭式存在性与欧拉函数φ(M-1)/2相关，该函数计数了分母为M-1的代数无关Gamma函数超越数。我们提出了将计算常数识别与经典数论关联的猜想，并提供全部代码与数据以确保完全可复现性。