标题： 关于保守矩阵场：连续渐近和算术 显示英文标题

标题： On Conservative Matrix Fields: Continuous Asymptotics and Arithmetic

摘要： D-有限序列的比值及其极限——称为阿佩里极限——自阿佩里1979年突破性地证明了$\zeta(3)$的无理性以来，推动了许多无理性证明的工作。 我们通过引入保守矩阵场（CMF）将D-有限序列的比值扩展到高维设置。 我们展示了如何通过该对象作为特殊情况包含经典的阿佩里极限。 提供了一种CMF的有用构造，建立了与规范变换以及有限维Ore代数模块中移位算子表示之间的联系。 最后，对这些对象的数值实验揭示了令人惊讶的算术和动力学现象，这些现象被 formulated 成猜想。 如果得到证实，这些猜想将把庞加莱-佩朗渐近扩展到更高维度，可能会为基于优化的寻找新的无理性证明打开大门。 显示英文摘要

摘要： Ratios of D-finite sequences and their limits -- known as Ap\'ery limits -- have driven much of the work on irrationality proofs since Ap\'ery's 1979 breakthrough proof of the irrationality of $\zeta(3)$. We extend ratios of D-finite sequences to a high-dimensional setting by introducing the Conservative Matrix Field (CMF). We demonstrate how classical Ap\'ery limits are included by this object as special cases. A useful construction of CMFs is provided, drawing a connection to gauge transformations and to representations of shift operators in finite dimensional modules of Ore algebras. Finally, numerical experiments on these objects reveal surprising arithmetic and dynamical phenomena, which are formulated into conjectures. If established, these conjectures would extend Poincar\'e--Perron asymptotics to higher dimensions, potentially opening the door to optimization-based searches for new irrationality proofs.