Noncommutative wormholes with generalized uncertainty principle in f(R, T) gravity: Energy conditions and gravitational wave echo time delay

A central difficulty in the physics of traversable wormholes is the requirement of exotic matter, which arises from the violation of the null energy condition within classical general relativity. This motivates the study of modified gravitational frameworks and quantum-gravitational corrections that collectively reshape the effective stress-energy sector. In this work, we develop a unified construction combining three physically well-motivated ingredients: noncommutative geometry (NCG), the generalized uncertainty principle (GUP), and f ( R, T ) modified gravity with f ( R , T ) = R + 2 λ T . We adopt the matter Lagrangian L m = − ρ and derive the corresponding modified field equations carefully, specifying the explicit form of the effective stress-energy tensor. The NCG Gaussian matter distribution regularises the source profile over a minimal length scale θ and may reduce the amount of exotic matter required near the throat, while the GUP introduces quantum-gravitational corrections to the energy density and pressure at sub-Planckian scales. We perform a complete derivation of the wormhole shape function, writing it as b ( r ) = b N C G ( r ) + b G U P ( r ) , where each contribution is obtained through an explicit integration of the corresponding source term. The throat conditions, flare-out criterion, and asymptotic flatness are verified analytically, and the key limiting cases β → 0, λ → 0, and θ → 0 are checked for consistency. An analysis of the null energy condition shows that the NEC violation is spatially localised near the throat owing to the Gaussian suppression provided by the NCG matter distribution, and the volume integral quantifier is evaluated with the f ( R, T ) coupling included explicitly. The effective potential for axial perturbations is derived and used to compute the gravitational wave echo time delay, which provides potential phenomenological signatures relevant for future gravitational-wave echo searches.