areaGradA_t12

This module assembles the zero-momentum per-cell Hessian of the 4D Regge action from committed star-deficit kernels and Heron area gradients, replacing the provisional weight-1 aggregate. Scope is constant edge-class perturbations only; finite-momentum Bloch folding remains open.

The area of a triangle with sides $a,b,c$ is written via Heron as $A=\sqrt{(2ab+2bc+2ca-a^2-b^2-c^2)/16}$. Differentiating under the square root yields the closed first-edge gradient $$\partial A/\partial a=(b+c-a)/(16A),$$ recorded as areaGradA. The four committed flat representatives are $(1,1,2)$, $(1,2,3)$, $(1,3,4)$, $(2,2,4)$.

Upstream, hingeArea_t12 already fixes $A(1,2,3)=\sqrt{2}/2$ by evaluating the Heron square and taking the positive root.

One-line algebraic evaluation. Unfold the gradient definition to $(2+3-1)/(16\cdot A(1,2,3))$, rewrite the denominator via hingeArea_t12 so $A=\sqrt{2}/2$, clear the nonzero factor $\sqrt{2}$ by field_simp, replace $(\sqrt{2})^2$ by $2$, and finish with norm_num to obtain $\sqrt{2}/4$.

Delivers one of the four explicit flat area-gradient values required by deliverable A of the Regge 4D Hessian campaign. Downstream, areaCov12_eq_grads identifies the type-(1,2) area-covariance components with these closed gradients, and hasDerivAt_area_t12_a packages the same number as the derivative of $t\mapsto A(t,2,3)$ at $t=1$.

Those identities feed the orbit-count-weighted sum $(dA\cdot c)(d\delta\cdot c)$ over the six $S_4$ types (counts $72/48/48/24/24/24$) that defines the true-weight zero-momentum Hessian. With true weights the Hessian kills pure gauge on the decoy directions (provisional weight-1 did not). The result does not touch continuum EH recovery or the gap-action flag; it only supplies a numerical brick inside the flat second-variation class.