AI 中文总结
该数学论文研究Samart关于n₄(81)=40M₇的猜想，证明了L值侧成立，发现两个阻碍导致无法延拓到CM点，数值计算否定原猜想，其真值闭式仍待解决。
AI 中文摘要
本笔记记录了Samart表6中的猜想n₄(81)=40M₇，其中M₇:=L'(g₇,0)，g₇(τ)=η(τ)³η(7τ)³是S₃(Γ₀(7),χ₋₇)的新形式，n₄(s):=4m(x⁴+y⁴+z⁴+1+s¹/⁴xyz)。该猜想是Samart未解决的内点条目（判别式-7，类数1，单个L值）中最清晰的，在配套论文中被放弃作为定理目标，该论文已证明两个n₂族猜想。我们记录已证明的内容及这些方法失效的具体位置：首先，L值侧（P1）的完整证明：在CM点τ₂=(7+√-7)/4处，Samart公式基础的Eisenstein-Kronecker表达式通过Q(√-7)整数环和主理想(ϖ̄)上的格求和，精确计算得EK₄(τ₂)=40M₇，寄生的ζ_K(2)项完全抵消。其次，剩余一半n₄(81)=EK₄(τ₂)的两个定量阻碍：n₄族的临界图像是包含参数c=3在内部的二维星形线圆盘（与n₂族的一维狭缝[0,64]形成对比），因此没有延拓路径能接近CM点；Samart的U级数在上半平面处处收敛，但在Imτ<1/√2时处处离开全纯Mahler测度的几何叶，导致微分比较延拓的前提不成立。随后的20位直接环面积分从数值上判定该猜想：n₄(81)-40M₇=+0.0586706795972872…，比积分误差下限高五个数量级，因此字面表述的等式被否定；n₄(81)真值的闭式仍未解决，似乎需要调节器/单值群机制。
英文摘要
This note archives the status of Samart's Table-6 conjecture $n_4(81)=40M_7$, $M_7:=L'(g_7,0)$, where $g_7(τ)=η(τ)^3η(7τ)^3$ is the newform of $S_3(Γ_0(7),χ_{-7})$ and $n_4(s):=4m(x^4+y^4+z^4+1+s^{1/4}xyz)$. The conjecture is the cleanest of Samart's open interior-point entries (discriminant $-7$, class number $1$, a single $L$-value), and it was dropped as a theorem target in the companion paper, where the two $n_2$-family conjectures were proved. We record what is proved and precisely where those methods fail. First, a complete proof of the $L$-value side (P1): at the CM point $τ_2=(7+\sqrt{-7})/4$ the Eisenstein--Kronecker expression underlying Samart's formula evaluates exactly to $\mathrm{EK}_4(τ_2)=40M_7$, via lattice sums over the ring of integers of $\mathbb{Q}(\sqrt{-7})$ and the principal ideal $(\bar\varpi)$, with an exact cancellation of the parasitic $ζ_K(2)$-terms. Second, two quantitative obstructions to the remaining half $n_4(81)=\mathrm{EK}_4(τ_2)$: the critical image of the $n_4$-family is a two-dimensional astroid disc containing the parameter $c=3$ in its interior (in contrast to the one-dimensional slit $[0,64]$ of the $n_2$-family), so no continuation path can approach the CM point; and Samart's $U$-series converges on all of the upper half-plane but leaves the geometric sheet of the holomorphic Mahler measure everywhere below $\mathrm{Im}\,τ=1/\sqrt{2}$, so the premise of the differential-comparison continuation fails. A 20-digit direct torus integration then decides the conjecture numerically: $n_4(81)-40M_7=+0.0586706795972872...$, five orders of magnitude above the integration error floor, so the identity as literally stated is refuted; a closed form for the true value $n_4(81)$ remains open and appears to require regulator/monodromy machinery.
Comments12 pages. Status report; companion paper proves the two n2-family conjectures (submitted simultaneously). Code: https://doi.org/10.5281/zenodo.21711884
