We show that the discrete set of paired amplitudes $A_m$ introduced by Haldane is an angular-momentum-resolved generalization of the Tan two-body contact. They provide a complete description of translation- and rotation-invariant states at the lowest Landau levels (LLLs), both compressible and incompressible. To guide the important order beyond the non-interacting high temperature limit, they are analytically determined with respect to the Haldane pseudopotential parameter Vm, and are qualitatively determined for the crossover to the incompressible ground state with various filling fractions. provide an explanation.Furthermore, we show that for the contact interaction $\sim g_2 \delta^{(2)}(\mathbf{x})$ that is scale-invariant at the classical level, the non-commutativity of the guide center coordinates arises. Quantum anomaly of commutator $i [\hat{H}_{\rm LLL}, \hat{D}_R] = (2 + \ell \partial_\ell) \hat{H}_{\rm LLL}$ scale anomalies due to interaction plus LLL dilation operator $\hat{D}_R$. The interaction-induced scale invariance violation causes a finite frequency shift of the breathing mode in the harmonic trap. It describes a transition between different Landau levels, whose strength is the associated dimensionless coupling constant $\tilde{ g}_2$.
