fullStarDirectional
plain-language theorem explainer
Directional pairing of a 15-class edge displacement against the full-star deficit class kernel of the seed triangle hinge in the 4D Freudenthal lattice. Anyone checking stationarity gates or assembling the flat Regge Hessian cites this scalar. The body is the plain finite sum of class values times kernel weights on Fin 15.
Claim. For a real assignment $v$ on the fifteen edge-orbit classes, the full-star directional is $\sum_{d=0}^{14} v(d)\,K(d)$, where $K$ is the full-star deficit class kernel (supported on classes $2,3,6,7,10,11,14$ with values $-1,-1,+1,-1,+1,+1,-1$).
background
This module is the QG full-theory increment after the two-simplex dihedral cosine calculus. It treats the seed triangle hinge ${0,e_0,e_0+e_1}$ and its full periodic Freudenthal star: four containing unit cubes and six incident 4-simplices in the integer lattice. Edge lengths are reduced to a 15-class stencil; the star is never redefined from the incidence layer.
The companion kernel $K$ (full-star class kernel) is the sparse map on those fifteen classes with values $(-1,-1,+1,-1,+1,+1,-1)$ on indices $(2,3,6,7,10,11,14)$ and zero elsewhere. Flat cosine data on the six simplices (four at $1/\sqrt{2}$, two at $0$) and the exact $2\pi$ angle sum already fix that support. Parallel kernels exist for other hinge orbits (12, 13, 22) with different numerical weights; each exposes the same directional pairing shape.
The local deliverable is the class-level linear form used by the stationarity and decoy gates, not a continuum Einstein–Hilbert limit.
proof idea
Pure definition: the real number is the sum over $d:\mathrm{Fin},15$ of $v(d)$ times the full-star class kernel at $d$. No tactics, no lemmas inside the body. Downstream proofs unfold this sum, restrict to the seven-class support via the off-support vanishing lemma, and evaluate the remaining finite arithmetic.
why it matters
Supplies the linear form for the two local gates proved in this file: uniform-scale decoy (directional of the constant-1 vector equals $-1$) and homothety stationarity (directional of the natural class-weight vector equals $0$). Those gates feed the flat 4D Hessian assembly, where the same pairing appears as class-dot of each orbit kernel against the decoy trace, all forced to vanish.
In the Recognition gravity campaign this is deliverable A item 4–5 of the full-star kernel: the discrete deficit direction on the seed hinge orbit. It does not yet close $S_{\mathrm{RS}}\to\mathrm{EH}_{4d}$ or flip gap-action recovery; other hinge orbits remain open. The construction stays inside the eight-tick / $D=3$ lattice geometry already forced upstream, without reverse-engineering continuum weights.
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