seed_summand_mul_angle

This module lifts the 3D Gate-A2 Schläfli closed form from the tetrahedron ($n_H=n_E=6$) to the Freudenthal/Kuhn 4-simplex ($n_H=n_E=10$). The flat Schläfli summand table $S(h,e)$ records each hinge-edge contribution at the flat seed; the seed row is hinge $h=0$.

The angle kernel $\theta'(k)= -\sqrt{2},(\cos\theta)'(k)$ is the directional derivative of the dihedral angle along squared-edge coordinates through the flat seed (flat $1/\sin\theta=\sqrt{2}$). It is the per-simplex contribution to the deficit gradient $\delta=2\pi-\sum\theta$.

The local goal is to identify the seed row with hinge area times this kernel, as input to the flat directional Schläfli kill and the non-vacuous SchlaefliIdentityN witness.

One-line algebraic wrapper. Rewrite the target equality by the hypothesis $\theta'(e)=2,S(0,e)$, then close with ring to obtain $S(0,e)=\tfrac12,\theta'(e)$. No geometric lemmas are invoked.

Feeds flatSchlaefliSummand_seed_eq_area_angleKernel, which states $S(0,e)=A_{\mathrm{flat}}(0),\theta'(e)$ after inserting the unit seed hinge area. That identity is the THEOREM-tagged match between the seed-hinge row of the flat summand table and hingeArea · angleKernel from the dihedral kernel module.

It is a tiny but necessary bridge in the Gate A2-style flat input at $n_H=n_E=10$: column sums vanish, seed row matches area times angle kernel, and flat directional Schläfli kill holds along every affine velocity through the seed. Downstream open items (full pathwise identity off the flat seed, elevation to the RS candidate, $S_{RS}\to$ EH in 4D) sit above this layer; the lemma itself only rearranges the seed-row scalar relation.