AI 中文总结
该研究证明了萨马特关于内部CM点马勒测度的两个猜想，通过微分比较延拓、连续性论证及精确CM格点和计算完成，所有恒等式经41-60位数值验证。
AI 中文摘要
我们将萨马特的两个猜想证明为真实马勒测度的恒等式。第一个是他对族$(x+x^{-1})(y+y^{-1})(z+z^{-1})+k^{1/2}$的求值中$k=1$的情况：$m((x+x^{-1})(y+y^{-1})(z+z^{-1})+1)=4L'(g_7,0)$，其中$g_7(\tau)=\boldsymbol{\tau}^3\boldsymbol{7\tau}^3$是$S_3(\boldsymbol{\tau}_0(7),\boldsymbol{\tau}_{-7})$的唯一新形式（LMFDB标签7.3.b.a）。第二个是他2015年表中二次CM项的共轭对：$n_2((47\boldsymbol{\tau}45\boldsymbol{\tau}_{-7})/2)=(4/7)(54L'(g_7,0)+L'(\boldsymbol{\tau}_{-7},-1))$，其中$n_2(s):=2m((x+x^{-1})(y+y^{-1})(z+z^{-1})+\boldsymbol{\tau})$。在两种情况下，参数都位于临界轨迹内部（或边界），因此从全纯（修正）马勒测度（萨马特有条件地计算了$L$值侧，费伊计算了实部水平）到真实马勒测度的过渡是未解决的问题。对于第一个定理，我们通过沿显式验证路径对马勒微分进行微分比较延拓，再结合内部点的连续性论证来填补这一空白；对于第二个定理，无需延拓：模参数化下参数的第二个原像是萨马特CM点的弗里克伙伴，且位于其已证明的区域内，该求值是精确的CM格点和计算。此处未证明的解析与算术输入均明确给出参考文献；延拓论证中使用的有限数值不等式通过区间算术验证，所有恒等式均经数值验证至41-60位。
英文摘要
We prove two conjectures of Samart as identities of genuine Mahler measures. The first is the case $k=1$ of his evaluations for the family $(x+x^{-1})(y+y^{-1})(z+z^{-1})+k^{1/2}$: $m((x+x^{-1})(y+y^{-1})(z+z^{-1})+1)=4L'(g_7,0)$, where $g_7(τ)=η(τ)^3η(7τ)^3$ is the unique newform of $S_3(Γ_0(7),χ_{-7})$ (LMFDB label 7.3.b.a). The second is the conjugate pair of quadratic CM entries of his 2015 table: $n_2((47\pm 45\sqrt{-7})/2)=(4/7)(54L'(g_7,0)+L'(χ_{-7},-1))$, where $n_2(s):=2m((x+x^{-1})(y+y^{-1})(z+z^{-1})+\sqrt{s})$. In both cases the parameter lies inside (or on the boundary of) the critical locus, so the passage from the holomorphic (modified) Mahler measure---where Samart computed the $L$-value side conditionally, and Fei the real-part level---to the true Mahler measure was open. For the first theorem we close the gap by a differential-comparison continuation of the Mahler differential along an explicitly certified path, followed by a continuity argument at the interior point. For the second theorem no continuation is needed: the second preimage of the parameter under the modular parametrization is the Fricke partner of Samart's CM point and lies in his proved region, and the evaluation is an exact CM lattice-sum computation. The analytic and arithmetic inputs not proved here are stated explicitly with references; the finite numerical inequalities used in the continuation argument are certified by interval arithmetic, and all identities are confirmed numerically to 41--60 digits.
Comments26 pages. Rigorous interval-arithmetic certification; companion code and frozen certificate archived at https://doi.org/10.5281/zenodo.21711884
