Single-scale magnetoelastic Landau quantization: Thermodynamics, quantum oscillations, and metrology

We develop a unified single-scale framework for the thermodynamics and quantum oscillations of a non-relativistic two-dimensional electron gas with a uniform areal density of screw dislocations under a magnetic field. Dislocation-induced torsion acts as a geometric gauge potential, preserving a Landau-like ladder while renormalizing its spacing into a single effective energy scale that governs equilibrium observables. From a compact harmonic-oscillator partition function, we derive closed forms for the free energy, internal energy, entropy, heat capacity, magnetization, susceptibility, and magnetocaloric response, all collapsing onto universal temperature kernels. A compensated-field regime closes the transverse gap, producing a thermodynamic crossover and an equipartition-like heat-capacity plateau. The same effective scale shifts the Hall fan and modifies de Haas--van Alphen and Shubnikov--de Haas periods, enabling extraction of screw-dislocation density from field sweeps. In mesoscopic samples, boundary corrections generate calorimetric oscillations, while moderate disorder and weak interactions mainly smooth amplitudes without altering the underlying single-scale structure.