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 $n_4(81)=40M_7$, $M_7:=L'(g_7,0)$, where $g_7(\tau)=\eta(\tau)^3\eta(7\tau)^3$ is the newform of $S_3(\Gamma_0(7),\chi_{-7})$ 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 $\tau_2=(7+\sqrt{-7})/4$ the Eisenstein--Kronecker expression underlying Samart's formula evaluates exactly to $\mathrm{EK}_4(\tau_2)=40M_7$, 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 $\zeta_K(2)$-terms. Second, two quantitative obstructions to the remaining half $n_4(81)=\mathrm{EK}_4(\tau_2)$: 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 $\mathrm{Im}\,\tau=1/\sqrt{2}$, 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.