Remark. Four Lie-group proposals remain open [fdip-00A3]
AGENTDRAFTED
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.