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We present the first discovery from the COol Companions ON Ultrawide orbiTS (COCONUTS) program, a large-scale survey for wide-orbit planetary and substellar companions. We have discovered a co-moving system COCONUTS-1, composed of a hydrogen-dominated white dwarf (PSO J058.9855+45.4184; $d=31.5$ pc) and a T4 companion (PSO J058.9869+45.4296) at a $40.6''$ (1280 au) projected separation. We derive physical properties for COCONUTS-1B from (1) its near-infrared spectrum using cloudless Sonora atmospheric models, and (2) its luminosity and the white dwarf's age ($7.3_{-1.6}^{+2.8}$ Gyr) using Sonora evolutionary models. The two methods give consistent temperatures and radii, but atmospheric models infer a lower surface gravity and therefore an unphysically young age. Assuming evolutionary model parameters ($T_{\rm eff}=1255^{+6}_{-8}$ K, $\log{g}=5.44^{+0.02}_{-0.03}$ dex, $R=0.789^{+0.011}_{-0.005}$ R$_{\rm Jup}$), we find cloudless model atmospheres have brighter Y- and J-band fluxes than the data, suggesting condensate clouds have not fully dispersed around 1300 K. The W2 flux (4.6 $μ$m) of COCONUTS-1B is fainter than models, suggesting non-equilibrium mixing of CO. To investigate the gravity dependence of the L/T transition, we compile all 60 known L6-T6 benchmarks and derive a homogeneous set of temperatures, surface gravities, and masses. As is well-known, young, low-gravity late-L dwarfs have significantly fainter, redder near-infrared photometry and $\approx200-300$ K cooler temperatures than old, high-gravity objects. Our sample now reveals such gravity dependence becomes weaker for T dwarfs, with young objects having comparable near-infrared photometry and $\approx100$ K cooler temperatures compared to old objects. Finally, we find that young objects have a larger amplitude J-band brightening than old objects, and also brighten at H band as they cross the L/T transition.