The modular torus maximizes the Mahler product

The stable norm of a complete finite-area hyperbolic once-punctured torus encodes its simple closed geodesic lengths in a planar convex body. We prove that the modular torus uniquely maximizes the Mahler product of this body and enclose the maximum between 9.62109317748 and 9.62109317752. A simultaneous enclosure theorem constructs four bounding polygons from the primal points and polar faces of selected Farey curves. The face calculation also yields a convergent simple-length series for the polar area. A symmetric Taylor expansion with certified coefficients, together with second-derivative bounds for the infinite tail, gives a quantitative strict local maximum. A twist-uniform cusp estimate and certified interval comparisons complete the global proof. We also show that the product tends uniformly to the planar Mahler minimum $8$ under degeneration. Combined with the stable-norm area-minimization theorem of Dang-Dao-Doan, the result implies that the modular torus uniquely maximizes the polar area.