C04.2 · Definitions/conventions

Let \(Y=\sum_a y_aE_a\) in the declared symmetric tangent basis, with the Euclidean/Frobenius metric used by the source. Let \(P_B\) be the loaded carrier projector. The joint-defect projectors are denoted \(\Pi_1,\Pi_{32},\Pi_{36}\), and \(\Xi=\Pi_{10}-\Pi_{26}\) splits the last sector. These are named inputs, not matrices reconstructed in this article.

The character and its source-reported bilinear tensor are \[ \tau(Y)=\frac{12}{54}\operatorname{tr}P_B (e^{2Y}+e^{-2Y}-2I),\qquad R_B^{\rm char}(a,b)=\operatorname{tr}P_B(E_aE_b+E_bE_a) =\tfrac12\Pi_{32}+\tfrac23\Pi_{36}. \] This is an explicit mathematical rendering of the action discussed at Appendix S p.296; the quotations below remain as extracted. The rational coefficients in this rendering are tested by the character expansion, not “recovered” by silently repairing a quote.

Use coefficient order \(c=(t,u,v,w,z)\). For \[ F_c=t\tau+\tfrac12\langle y, (u\Pi_1+v\Pi_{32}+w\Pi_{36}+z\Xi)y\rangle, \] the displayed Hessian-weight map is \[ h(c)=\bigl(u,\;4t/9+v,\;16t/27+w-z,\;16t/27+w+z\bigr). \] It is this **coefficient map**, rather than a manufactured exceptional matrix model, that the script tests.

The extended stability inventory also has a coefficient called \(t_2\). It is an independent coordinate in the source's tuple, not the square of \(t\).