Remark. The Hessian route remains an open test [fdip-00A4]
Remark. The Hessian route remains an open test [fdip-00A4]
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.