Large-$N$ scaling of Tan's contact for the harmonically trapped Tonks--Girardeau gas at finite temperature

We derive the canonical-ensemble scaling of Tan’s contact for $N$ harmonically trapped Tonks–Girardeau bosons at finite temperature in the large-$N$ limit. The leading scaling coefficient reproduces the local-density-approximation result and is obtained from a contour-integral representation of the canonical partition function followed by a saddle-point reduction to a phase-space integral with a self-consistent scaled chemical potential. The subleading coefficient is the central new object of this work: it admits an explicit representation in terms of universal phase-space integrals of the Fermi factor, has closed-form Sommerfeld and virial limits, and is identified with the canonical-versus-grand-canonical ensemble difference at fixed mean particle number. In the high-temperature Boltzmann regime the ratio of subleading to leading coefficients collapses to a universal value, traceable to the Poissonian particle-number statistics of the dilute grand-canonical gas. We construct Padé approximants for both scaling functions that interpolate uniformly between the low-temperature Sommerfeld and high-temperature virial regimes; for the subleading coefficient we report a form that is uniformly accurate on our working range of temperatures and asymptotically correct beyond. The scaling law is verified against canonical contour-integration data across the full temperature range.