Hyperon-pair spin tomography beyond scalar spin correlations

Spin correlations measured after hadronization probe how quantum information is transported through confinement. We formulate a process-independent spin-tomography framework for $\Lambda\bar{\Lambda}$ pairs in which the weak-decay angles determine the full two-spin tensor rather than only its scalar trace. Applied to the short-range STAR correlation in unpolarized $pp\to\Lambda\bar{\Lambda}X$, the framework shows that the published scalar $P_{\Lambda\bar{\Lambda}}$ leaves a continuous density-matrix degeneracy: separable tensor completions and completions that violate the positive-partial-transpose (PPT) criterion can have the same trace. The missing observables are the transverse and longitudinal tensor components and their anisotropy $A_C=C_\perp-C_\parallel$. A local ${}^{3}P_0$ string-breaking benchmark turns the STAR trace into the falsifiable pattern $C_\perp>0$, $C_\parallel<0$, and $A_C>0$. We further show that feed-down must be treated as a tensor response rather than as a universal scalar dilution. In the complementary $e^+e^-$ calibration channel, the same benchmark predicts a centrally enhanced $A_C$ of order $0.3$ in Belle II kinematics, measurable with about $10^4$ selected pairs for nominal spin-transfer parameters. A claim of entanglement requires the reconstructed two-spin density matrix, not the scalar trace alone, to violate the PPT bound.