worstRadical_flatAngleJacobian_value

This module is the panel-locked C11 Regge TT Bloch interface audit. It keeps the first gate deliberately non-tautological: the raw cell stencil is a literal $6\times 6\times 6$ triple sum over tetrahedra and ordered slot pairs, with the inner flat-slot angle derivative expanded and the square-root velocity term written as plane-wave tet velocity over $2\sqrt{a^*}$. Full rational bucket aggregation and zero-mode cancellation are not claimed here; Gate A2-full and Gate A3 remain open.

The flat-angle Jacobian is the edge-pair derivative of the dihedral cosine built from Cayley–Menger cofactors on squared edge lengths. Upstream, dihedralCos3SqClosedFormDeriv_eq_poly equates the closed-form cofactor-ratio derivative to its polynomial-cofactor form $(C_k D - C D_k)/D^2$, where $C$ is the off-diagonal CM cofactor, $D$ is the product-of-diagonals square-root denominator, and subscripts denote partials in a squared-edge coordinate. The Freudenthal tetrahedron supplies the concrete squared-edge vector at which the entry is evaluated.

Rewrite the flat-angle Jacobian into cofactor form, then replace the closed-form dihedral-cosine derivative by the polynomial-cofactor expression via dihedralCos3SqClosedFormDeriv_eq_poly. Unfold the polynomial numerator, denominator, and their closed derivatives (including cmCofactor3Poly, opposite CM vertices, and the Freudenthal squared edges) and discharge the resulting rational expression by norm_num. The only remaining radical identity is $\sqrt{32}=4\sqrt{2}$, proved by factoring $32=16\cdot 2$ and using multiplicativity of square root; finish with ring.

This is the selected genuine radical-row entry for the C11 audit. Downstream, worstRadical_rawJacobianCoefficient_closedForm normalizes the same Jacobian by $2\sqrt{a^*}$ (Freudenthal edge length) and obtains the rational value $-1/8$, so the $\sqrt{2}$ in the numerator cancels against the normalization denominator. The doc-comment is explicit: individual raw coefficients are irrational at the row-Jacobian layer; the panel rationality claim lives only at bucket-fiber aggregation and remains open. The result therefore pins the radical witness without closing Gate A2-full or the hinge-aware zero-mode (Gate A3). It sits inside the gravity second-variation pipeline that feeds continuum-limit and Bloch-mode analysis of the Regge TT operator, not the T0–T8 forcing chain itself.