Symmetry-selected extreme shear compliance in a stable carbon monolayer

Solids that are stiff to most loads and extremely compliant to one are normally built, not found. That internal atomic relaxation lowers elastic constants is textbook; what has remained untested is whether it can be driven far enough to separate the elastic scales of a crystal by an order of magnitude while leaving it mechanically stable. Density functional theory calculations of the complete in-plane elastic tensor of four non-hexagonal carbon monolayers sharing a square lattice metric show that it can: their longitudinal stiffnesses agree to within a factor of two while their shear constants span more than a factor of fifty. Twin T-graphene, previously reported as elastically isotropic from normal strains alone, is the extreme case, with $C_{66}=3.03$~N,m${}^{-1}$, nearly fifty times below graphene, at an axial stiffness typical of semiconducting monolayers, and a soft Kelvin eigenvalue a factor of $29$ below the others while the Born criteria hold. Symmetry confines the strain-induced relaxation to the $B_{2g}$ sector, a counter-torsion of the carbon sublayers about the short mid-plane bonds that removes $96\mathrm{\%}$ of the clamped-ion shear stiffness with almost no bond stretching and leaves the semiconducting gap intact. Transverse acoustic phonons, finite-temperature dynamics, and large-deformation simulations show that the channel survives beyond the harmonic, zero-temperature limit. Because the selection rule is a point-group statement rather than a chemical one, it yields a screening criterion for other crystals and other strain components.