Chromoelectric and chromomagnetic matching to scalar and spin-two nucleon structure

Compact heavy quarkonium couples through the multipole interaction to scalar and spin-two gluonic operators. At leading chromoelectric order the corresponding matching coefficients satisfy $C_2^\Phi=-C_S^\Phi$; an independent chromomagnetic polarizability lifts this relation within the general CP-even, spin-independent, local two-gluon interaction at dimension four and zero derivative order. We construct an RG-consistent realization in a fixed $MS$ convention. The QCD trace identity converts the gluon-only scalar matching condition into an invariant basis and fixes the correlated quark-mass coefficient required when the interaction is re-expressed in the scale-dependent basis away from the matching scale, whereas leading-logarithmic singlet evolution induces a quark spin-two coefficient. In threshold-aligned symmetric kinematics, the canonical-spin non-flip projection contains $A_i(t)$ and the combination $3B_i(t)-D_i(t)$. An explicit Breit-frame calculation relates this projection to an off-diagonal helicity representation for nonzero spacelike $t$; the off-diagonal form is kinematic rather than an additional dynamical spin flip. Linearity of the scalar and spin-two evolution factorizes the chromomagnetic dependence of their ratio as $R_{2/0}^{\Phi}(t;\rho_\Phi)=[(1+\rho_\Phi)/(1-\rho_\Phi)]R_{2/0}^{\Phi}(t;0)$ within the gluon-only dimension-four matching setup. The result separates state-dependent quarkonium matching from scalar and gravitational nucleon structure and states explicitly the assumptions under which this factorization holds.