Intrinsic Charge Transport in Lieb--Schultz--Mattis Gap Bounds

Geometric interaction range can overestimate the energy cost of a Lieb–Schultz–Mattis twist. We derive finite-ring upper bounds on fractional-sector gaps from the twist phases connecting an interaction’s joint charge sectors. The shortest circular phase-arc radius is the exact minimum norm of a charge-algebra logarithm implementing the local twist, and its decomposition envelope is independent of a chosen interaction presentation. For a specified local term, a finite-dimensional transport quotient bounds this phase cost and has a closed wrapped-polarization formula for arbitrary-order correlated hopping. For a spin-1/2 chain, we exactly evaluate the quadratic and finite-angle envelopes. On graph-fiber cylinders, the bound is direction resolved: charge-conserving dynamics within a transverse slice has exactly zero longitudinal cost. Power-law moving tails give an explicit width–decay crossover. Even when every pair of transverse channels is coupled by a Euclidean dipolar tail, the bound vanishes if W_L^2[1 + log(L/W_L)] = o(L); a charge-moving van der Waals tail remains O(W_L/L). For even longitudinal size, an odd-leg ladder corollary is independent of arbitrary translated SU(2)-invariant rung dynamics. These are upper bounds on the gap, not evaluations of the physical gap itself.