旧结构，一个消失的公式
Old structures, a vanishing formula
AI总结：
本文针对四阶Hahn差分方程的行列式构造，发现其存在归一化缺陷，通过与Foupouagnigni的方法对比，明确了其在特定正交族上的失效问题及修正方式。
摘要：
我们在四阶Hahn差分方程的行列式构造中识别出一个归一化缺陷，并将其标量方程与Foupouagnigni 1998年的消元法及2001年发表的对应方法进行比较。在一个明确的正正交族上，所有五个规定系数均消失：该公式给出0=0且不施加任何条件。沿一个明确的正则扰动，在特化前除以ε²可恢复该族上的非零方程，与正确执行的旧移位-消元程序得到的标量方程相同。在下文指定的公共非退化域上，两种构造给出的标量算子相差一个非零有理因子。一阶次数界直接由1998年的有效恒等式得出；保留抵消可在明确的容许输入上得到任意更紧的界，连同精确的最高系数和抵消准则。明确的系数公式本身并未在该域上产生新的标量方程。我们的比较不排除校正性或说明性贡献；额外的计算增益或特殊覆盖必须得到证明。
英文摘要：
We identify a normalization defect in a determinantal construction of fourth-order Hahn difference equations and compare its scalar equation with Foupouagnigni's 1998 elimination and its published 2001 counterpart. On an explicit positive orthogonal family, all five prescribed coefficients vanish: the formula gives \(0=0\) and imposes no condition. Along an explicit regular perturbation, division by \(\varepsilon^2\) before specialization recovers a nonzero equation on that family, the same scalar equation as the correctly executed old shift-and-eliminate procedure. On the common nondegenerate domain specified below, both constructions give the same scalar operator up to a nonzero rational factor. The first-order degree bounds follow directly from valid identities of 1998; preserving cancellations gives a bound that can be arbitrarily sharper on explicit admissible inputs, together with exact top coefficients and cancellation criteria. Explicit coefficient formulas do not by themselves yield a new scalar equation on this domain. Our comparison does not exclude corrective or expository contributions; additional computational gain or exceptional coverage must be demonstrated.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
