A certified continuation machine for Mahler measure identities at CM points, with twelve new proofs of conjectures of Samart

Abstract

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 ($g_{16}\otimes {\chi }_8$ 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 $\{\mathrm{Im}\tau =1/\sqrt{2}\}$ 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 $4.6\times 10^{-53}$ to $5.1\times 10^{-52}$), 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.

38 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