Remark. The Hessian route remains an open test [fdip-00A4]
AGENTDRAFTED
The Hessian proposal is the route recorded in
the Hessian-group outlook. That remark begins
with
\[Q_8\rtimes C_3\cong 2T\]
and then names the iterated semidirect product
\((G_{27}\rtimes Q_8)\rtimes C_3\). The actions in those
products are part of the data. The name of the product does
not choose them. The same remark states that a central scalar
subgroup \(C_3\leq G_{27}\) has quotient the Hessian group of
order 216, and that the larger group is a triple cover of that
quotient. This chapter does not reconstruct \(G_{27}\), the
semidirect actions, or that quotient.
The source sentence was rechecked in arXiv:2401.13000v4.
Section 2.2 is page 5 of that file
[wilson2024discrete, v4, Section 2.2, p. 5].
The locator pages 4 to 5 in Remark [fgap-001X] do not match
this version. On page 5 Wilson states that the matrices for
\(G_{27}\) and \(Q_8\) generate a group of order 216, the
semidirect product \(G_{27}\rtimes Q_8\), and that the full
group has order 648 and is the triple cover of the Hessian
group. He writes that it can be interpreted as a finite
analogue of \(U(1)\times SU(2)\times SU(3)\), and that the
finite group is the iterated semidirect product
\(G_{27}\rtimes Q_8\rtimes Z_3\). The cyclic factor of order
3 is the group written \(C_3\) in the display above. These
notes record the interpretation and do not adopt it.
The known algebraic datum is the binary tetrahedral group
already used in the finite spinor identities.
The displays \(Q_8\rtimes C_3\cong 2T\) and
\((G_{27}\rtimes Q_8)\rtimes C_3\) are not presentations and
are not further data: their actions are not chosen, and
neither \(G_{27}\) nor the order-216 quotient is constructed
here. The missing map is a typed homomorphism from the
iterated product, or from its order-216 quotient, into
\(U(1)\times SU(2)\times SU(3)\) or into a group already
constructed in these notes.
The comparison with \(U(1)\times SU(2)\times SU(3)\) stays an
attributed proposal in Remark [fgap-001X]. It is not an
isomorphism, and the comparison motivation recorded beside
that outlook is not a construction of the finite group.
The finite witness available now is \(2T\) alone. It does
not determine \(G_{27}\) or the order-216 quotient. A witness
that would meet the compatibility datum would
specify one of those groups by a concrete model, a
coefficient field, a representation on a named fiber, an
action on a named finite graph, a selector, and one
reference comparison. None of those slots is filled, and a
matrix model would not by itself fill the missing action.
Every criterion is open.
That absence is not a no-go theorem and not a physical
identification.