Scope of the identical-particles examples page

What the computations establish. Both items are conditional statements of declared models. The two spin-half fermion sector: the spin-singlet and spin-triplet parts and their orbital symmetry decomposes the antisymmetric two-particle sector of a declared spin-half species with a finite orbital factor into its spin-singlet and spin-triplet parts, exhibits the opposite orbital exchange symmetry of the two parts, and computes the components of an antisymmetrised product of two declared one-particle states. Exchange versus direct Coulomb terms: the pair-energy split of the two-orbital spin-half model specialises a two-orbital model with a declared Coulomb-form interaction and shows, for real normalized orthogonal $W^{1,2}$ orbitals with a nonzero product, that the declared parallel-spin (triplet) preparation has strictly smaller pair-interaction expectation than the singlet preparation, the difference being twice the exchange integral.

What they rest on. Both rest on the adopted antisymmetric sector of P6 (Identical-particle and spin assignments P6: permutation sectors and finite-spin representations) in the reading of Declared species sectors and their preparations: the framework's reading of P6: the assignment of the species to the antisymmetric sector and the choice of orbital factor are model declarations, and in the second item the two orbitals and the interaction form are declared idealizations. The spin-statistics association is not derived, no interaction is derived from the sector and no dynamics is introduced.

What is not claimed. The page records no measured energy, no count and no apparatus response; the terms "singlet", "triplet" and "exchange" name components and differences of the declared models. Whether a laboratory pair of particles is described by one of the declared preparations is a tested identification with uncertainty under the preparation and apparatus postulate Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions, and agreement of these conditional numbers with data would test the combined model and apparatus assumptions rather than prove the model.