Traversable Wormholes in f(T,𝒯 ) Gravity: A Matter-Trace Threshold for the Null Energy Condition at the Throat

We study static spherically symmetric traversable wormholes in [Formula: see text] gravity, the teleparallel theory whose Lagrangian couples the torsion scalar [Formula: see text] to the trace [Formula: see text] of the matter energy–momentum tensor. Working with the rotated tetrad, for which the off-diagonal field equation is met identically by the linear model [Formula: see text], we obtain the field equations for an anisotropic source and solve them on the Morris–Thorne geometry. For the polynomial extension [Formula: see text] the off-diagonal sector fixes a single constraint that, in the zero-tidal-force case, reduces the shape-function problem to a first-order Abel equation of the first kind. The radial null-energy quantity at the throat reduces to [Formula: see text], which is negative in general relativity but changes sign once the matter-trace coupling exceeds the gravitational coupling, [Formula: see text], with [Formula: see text]; the full set of null conditions is then met provided the additive constant satisfies [Formula: see text], with the throat slope locked to [Formula: see text] on the boundary of that region. A direct integration shows that small nonlinear couplings keep the matter ordinary across a finite neighborhood of the throat, while strong nonlinear couplings restrict the favorable behavior to the throat alone; none of the favorable configurations is asymptotically flat. A Bayesian scan over [Formula: see text] maps the admissible region, and an explicit zero-tidal-force model with [Formula: see text] realises the saturating slope [Formula: see text]. The construction shows how the trace coupling relaxes the matter requirement that ordinarily forces exotic content at a wormhole throat.