Samart's conjecture n_4(81)=40M_7: the exact CM evaluation and the two obstructions. A status report

Abstract

This note archives the status of Samart's Table-6 conjecture $n_4(81)=40M_7$, $M_7:=L^{\prime }(g_7,0)$, where $g_7(\ddot{I}„)=\hat{I}\cdot (\ddot{I}„{)}^3\hat{I}\cdot (7\ddot{I}„{)}^3$ is the newform of $S_3(\hat{I}{“}_0(7),\ddot{I}{‡}_{-7})$ 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 $\ddot{I}{„}_2=(7+\sqrt{-7})/4$ the Eisenstein--Kronecker expression underlying Samart's formula evaluates exactly to ${\mathrm{EK}}_4(\ddot{I}{„}_2)=40M_7$, 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 $\hat{I}{\P }_K(2)$-terms. Second, two quantitative obstructions to the remaining half $n_4(81)={\mathrm{EK}}_4(\ddot{I}{„}_2)$: 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 $\mathrm{Im}\ddot{I}„=1/\sqrt{2}$, so the premise of the differential-comparison continuation fails. A 20-digit direct torus integration then decides the conjecture numerically: $n_4(81)-40M_7=+\mathrm{0.0586706795972872...}$, 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.

12 pages. Status report; companion paper proves the two n2-family conjectures (submitted simultaneously). Code: https://doi.org/10.5281/zenodo.21711884