发表机构
College of Information and Network Engineering, Anhui Science and Technology University（安徽科技学院信息工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文将微分比较延拓方法公理化为一台认证机器，用于在CM点证明Mahler测度恒等式，为Samart猜想提供了十二个新证明，并发现其适用性边界并非拓扑边界。
AI 中文摘要
我们将[Zf]中的微分比较延拓方法公理化，发展为一个在CM点处证明Mahler测度恒等式的一般机器：一个带有严格尾部界的模参数化、在临界像补集上的单值全纯Mahler微分、一个取代所有单值性和万有覆盖论证的传播引理、区间算术路径认证，以及一个精确的三轨CM求值。将该机器应用于Samart族$n_4(s)=4\mathrm m(x^4+y^4+z^4+1+s^{1/4}xyz)$，该机器证明了他2015年表格[Sa15, 表6]中的另外十二个条目，包括该族中首次在类数二（对应于$\mathbb Q(\sqrt{-6})$和$\mathbb Q(\sqrt{-15})$）及类数四（对应于$\mathbb Q(\sqrt{-21})$，具有四个新形式和亏格群$(\mathbb Z/2)^2$）的非退化CM点处证明的恒等式。新要素包括通过grossencharacter导子（Hecke定理）得到的精确扭曲级，其中区间锁定的Fricke比值作为精确的根数符号锁定；一个扭曲级陷阱（$g_{16}\otimes\chi_8$的级为64，而非32）；类数二中格点和的亏格特征分解；以及发现Samart的适用性边界$\{\mathrm{Im}\\,\tau=1/\sqrt2\}$并非延拓区域的拓扑边界。每个恒等式都由精确的代数轨道、严格的最终恒等式区间锁定（半宽度从$4.6\times10^{-53}$到$5.1\times10^{-52}$）以及独立的50至60位交叉验证所认证；所有脚本均为公开。求值轨道还产生了在Heegner点（$D=11,19,27,43,67,163$）处的六个新恒等式，作为带有60位证据的猜想陈述；我们以一个伞形猜想结束，该猜想组织了已证明和猜想的恒等式，并对剩余五个未解决条目进行了精确分析。
英文摘要
We axiomatize the differential-comparison continuation method of [Zf] into a general machine for proving Mahler measure identities at CM points: a modular parametrization with rigorous tail bounds, a single-valued holomorphic Mahler differential on the complement of the critical image, a propagation lemma replacing all monodromy and universal-cover arguments, interval-arithmetic path certification, and an exact three-track CM evaluation. Applied to Samart's family $n_4(s)=4\mathrm m(x^4+y^4+z^4+1+s^{1/4}xyz)$, the machine proves twelve further entries of his 2015 table [Sa15, Table 6], including the first identities in this family proved at non-degenerate CM points of class number two (attached to $\mathbb Q(\sqrt{-6})$ and $\mathbb Q(\sqrt{-15})$) and of class number four (attached to $\mathbb Q(\sqrt{-21})$, with four newforms and the genus group $(\mathbb Z/2)^2$). New ingredients include exact twist levels via grossencharacter conductors (Hecke's theorem), with the interval-locked Fricke ratio as an exact root-number sign lock; a twist-level trap ($g_{16}\otimesχ_8$ has level $64$, not $32$); a genus-character decomposition of the lattice sums in class number two; and the discovery that Samart's applicability boundary $\{\mathrm{Im}\,τ=1/\sqrt2\}$ is not the topological boundary of the continuation region. Every identity is certified by an exact algebraic track, rigorous final-identity interval locks (half-widths $4.6\times10^{-53}$ to $5.1\times10^{-52}$), and independent 50--60-digit cross-checks; all scripts are public. The evaluation track also yields six new identities at Heegner points ($D=11,19,27,43,67,163$), stated as conjectures with 60-digit evidence; we close with an umbrella conjecture organizing the proved and conjectured identities, and a precise analysis of the five remaining open entries.
Comments38 pages, 5 tables. Certification scripts available at https://github.com/huiminZheng-collab/samart-mahler and archived at https://doi.org/10.5281/zenodo.21711884 . Research carried out with the assistance of the AI system Kimi (Moonshot AI); see the declaration in the article
