BBN to late-time acceleration in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi mathvariant="script">L</mml:mi> <mm

We present, to our knowledge, the first systematic study of early-late cosmic evolution and acceleration in the framework of f ( T , L m ) gravity, an extension of teleparallel theories coupling torsion with the matter Lagrangian. By incorporating the Big-Bang Nucleosynthesis (BBN) bound on the freeze-out temperature, we obtain a tight constraint on the inverse-torsion parameter, ensuring consistency with early-time physics. Employing Markov Chain Monte Carlo analyses with progressively richer observational datasets, CC, Union3, and SN22 supernovae, we constrain a well-motivated model and reconstruct key cosmological functions. The reconstructed Hubble and distance modulus functions show excellent agreement with the observations, confirming the observational viability of the model. The model successfully reproduces the observed late-time expansion history, yielding a transition from deceleration to acceleration through the deceleration parameter. The effective equation of state is found to remain negative throughout, with present values w 0 > − 1 , indicating a quintessence-like behavior rather than a cosmological constant or phantom regime. These results highlight the ability of f ( T , L m ) gravity to mimic the concordance scenario while allowing controlled deviations in the expansion history.