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Mixing of the Eta octet state and eta singlet state to produce physical neutral pseudoscalar mass eigenstates. It limits mass identities derived by treating the physical eta meson as a pure octet state.

Use natural units and write the common light constituent quark mass as $m$, and the strange one as $M$. For two spin one-half constituents,The relative hyperfine contributions are $-3$ and $+1$. Absorb their normalization and spatial overlap into pair-dependent parameters $h_{\ell \ell },h_{\ell s},h_{ss}$. The meson hyperfine mass splitting model givesA common color-magnetic coefficient gives $h_{ij}\propto 1/(m_i{m}_j)$, but its numerical value is not needed.

In the specified Eta octet state used for the eta meson the light-flavor probability is $1/3$ and the strange-flavor probability $2/3$. Assuming a flavor-diagonal mass operator,Inserting the five mass expressions into the proposed combination makes the coefficient of $h_{\ell s}$ equal to $4-4=0$. Those of $m,M,h_{\ell \ell },h_{ss}$ are $2/3,4/3,-1,-2$. ConsequentlyThis is an identity within the assumed mass operator and octet flavor assignment. It is not an exact relation among measured masses: eta mixing and additional spin-independent binding effects are outside those assumptions.