Boyd's conductor-11 Mahler measure conjecture: proof of the split-integral identity (C3), with an exact structural analysis of the family S_k

Abstract

Boyd's 1998 tables of conjectural identities $m(P)=r|L^{\prime }(E,0)|$ between Mahler measures of two-variable polynomials and $L$-values of elliptic curves begin with the smallest possible conductor, $N=11$. The third conductor-$11$ identity, (C3), concerns the polynomial $S_0=y^2+(x^2+1)y+x^3$, which vanishes on the unit torus, and asserts that a signed split integral $I_{\mathrm{split}}$ of $\log |y|$ around the branch cut equals $b_{11}=L^{\prime }(E_{11},0)$. We prove (C3). The proof identifies the split integral with a regulator integral along Samart's signed open chain $\tilde{\gamma }$, closed by a small-branch arc ${\beta }_0$ into a closed anti-invariant cycle $C^{\prime }$, and proves its homology class is $2{\gamma }^{-}$ with ${\gamma }^{-}$ generating $H_1(E,\mathbb{Z}{)}^{-}$: the period ratio is a-priori integral, and ball arithmetic (Arb) pins it to $2$. A direct regulator computation via Brunault's proved Siegel-unit formula---the symbol $\{x,y\}$ being a pair of modular units on $X_1(11)$, so Bloch's diamond theorem is not needed---then yields $I_{\mathrm{split}}=b_{11}$; the sign is certified in interval arithmetic, and the identity agrees with the numerical value to $366$ digits. For the family $S_k=y^2+(x^2+kx+1)y+x^3$ we determine exactly the torus intersections and the modular-unit/tempered cases; further Boyd-type evaluations are recorded as conjectures. At $k=-1$ (conductor $53$) the mechanism provably fails; an appendix treats Samart's conductor-$17$ analogue conditionally. All computations are reproducible from the accompanying code; every certification step is carried out within interval arithmetic.

34 pages. Certification scripts available at https://github.com/huiminZheng-collab/boyd-conductor11 and archived at https://doi.org/10.5281/zenodo.21820650 . Research carried out with the assistance of the AI system Kimi (Moonshot AI); see the declaration in the article