Remark. Four Lie-group proposals remain open [fdip-00A3]
Remark. Four Lie-group proposals remain open [fdip-00A3]
The 4 names remain open tests against the compatibility datum. The claims below were checked in arXiv:2401.13000v4, the version reached by [wilson2024discrete]. The finite maps proved for the binary tetrahedral spinor are not claimed to factor through any of the 4 names.
\(Sp(3,1)\), \(SU(3,1)\) and \(SU(3,2)\) do not occur in that version. The FDIP record lists them as candidate groups mentioned while the programme was being planned. The known algebraic datum in these notes is the name alone: no dimension, real form, matrix realization, coefficient field, or action on a finite graph is fixed here. The missing map is a stated homomorphism, kernel, cover or branching rule, together with a selector and a reference object. No finite witness is supplied. All 5 criteria are open.
\(SL(4,\mathbb R)\) is named by Wilson. In Section 6.8, page 23, three rows of products of generators drawn from two copies of \(Q_8\) extend \(SO(4)\) to \(SL(4,\mathbb R)\) [wilson2024discrete, v4, Section 6.8, p. 23]. He gives two interpretations of that group: as a double cover of \(SO(3,3)\), and so as a symmetry of a six-dimensional spacetime, or as three directions of momentum and three of angular momentum inside a four-dimensional spacetime. He writes that this breaks the identification of \(SL(2,\mathbb C)\) with \(SO(3,1)\), so that the two become disjoint subgroups of \(SL(4,\mathbb R)\), and that the gain would be a generally covariant reading of the Standard Model. Section 7.3, page 25, later says that 4 quaternionic coordinates are acted on by \(SL(4,\mathbb R)\) as spacetime symmetries. The known datum here is that pair of sentences. No matrix realization on a finite graph, selector, or reference comparison is constructed in these notes. All 5 criteria are open. The open mark does not assert that a later realization is impossible.