fullStarDirectional
plain-language theorem explainer
Pairs any real assignment on the fifteen edge-length stencil classes with the type-(2,2) full-star deficit class kernel by a weighted sum. Gravity analysts cite it when checking homothety stationarity and decoy-trace orthogonality on the (2,2) hinge orbit. The body is the one-line stencil dot product against the local class kernel.
Claim. For a real-valued assignment $v$ on the fifteen edge-length stencil classes, the full-star directional response on the type-$(2,2)$ hinge orbit is $\sum_{d=0}^{14} v(d)\,K_{22}(d)$, where $K_{22}$ is the closed-form full-star deficit class kernel of that orbit.
background
This module treats the type-(2,2) triangle hinge ${0, e_0+e_1, e_0+e_1+e_2+e_3}$ (masks ${0,3,15}$, difference masks $(3,12)$) and its full periodic Freudenthal star in the 4D Regge lattice. The campaign imports the Freudenthal incidence layer, the fifteen-class edge stencil, and the Gram-projection cosine calculus; it does not redefine those APIs.
The sibling class kernel $K_{22}$ records the closed-form deficit coefficients on the fifteen stencil classes for this orbit (four incident Kuhn simplices, each with flat cosine $0$, so the star angle sum is exactly $2\pi$). Parallel directional pairings already exist for the (1,1), (1,2), and (1,3) orbits; each is the same stencil sum against its own class kernel.
The directional form is the linear functional used to test first-order stationarity of the star deficit under class-weighted edge variations (homotheties, uniform scalings, decoy traces).
proof idea
Pure definition: expand as the finite sum over Fin 15 of $v(d)$ times the local type-(2,2) full-star class kernel at class $d$. No lemmas, no tactics; the mathematical content is entirely in the sibling kernel table and in downstream rewrites that restrict the sum to the kernel support.
why it matters
Supplies the linear pairing that the flat 4D Hessian assembly uses for the (2,2) orbit. Downstream, deficitKernel22_dot_decoyTrace shows the class-dot of this kernel against the decoy trace vanishes, matching the parallel (1,1)/(1,2)/(1,3) decoy-orthogonality theorems. Sibling modules reuse the same directional shape for homothety stationarity and uniform-scale decoy gates on their orbits.
In the QG full-theory campaign this is deliverable A item 4 for the (2,2) star: the kernel is in place so first-order deficit tests can run without re-deriving incidence. It does not finish the full multi-hinge Hessian, does not prove $S_{\mathrm{RS}}\to\mathrm{EH}_{4\mathrm{d}}$, and does not flip gap-action recovery. It is the next kernel-checked increment after the (1,1) seed orbit.
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