Collective Protection and Exchange-Summed No-Jump Survival in Dipositronium

Decoherence-free subspaces are defined relative to a specified noise algebra. Dipositronium, \( Ps_2=e^-e^-e^+e^+ \), is the minimal positronium molecule in which the fixed-pair picture becomes inadequate: it contains two electrons and two positrons, but no permanent electron–positron pairs. A fixed-pair picture of two positronium atoms can suggest a collective-spin protected sector. In the true molecule, however, annihilation must be summed over all electron–positron contacts. This paper compares two scalarity tests on the same finite spin space: protection under modeled collective generators and undistorted conditional survival under the exchange-summed no-jump sink. In an equal-contact spin approximation, the operator governing conditional annihilation loss takes the form \( \begin{equation} K_{\rm ann}=C\left[\Gamma_2+3\Gamma_3+(\Gamma_3-\Gamma_2)\mathbf S_e\cdot\mathbf S_p\right]\end{equation} \) (1). It is therefore scalar on fixed \( (S_e,S_p;F) \) sectors. The two spin sectors (0,0;0) and (1,1;0) are both annihilated by the modeled collective generators, but the same two sectors have no-jump sink eigenvalues \( (C(\Gamma_2+3\Gamma_3) \) and \( C(3\Gamma_2+\Gamma_3) \), respectively. Thus the collective protected \( F=0 \) spin space is generally not scalar for conditional survival. The calculation is a channel-level comparison, not a molecular rate calculation. It shows that collective protection and conditional no-jump survival define different scalar decompositions.