fullStar_uniformScale_decoy
plain-language theorem explainer
The full-star deficit class kernel, paired against the constant vector of ones on the 15 edge classes, evaluates to -1. Gravity/Regge analysts cite this as the uniform-scaling decoy gate for the seed hinge star. The proof reduces the Fin-15 sum to the seven supported classes via the support lemma, then evaluates the explicit kernel values.
Claim. Let $K:\{0,\ldots,14\}\to\mathbb{R}$ be the full-star deficit class kernel (nonzero only on classes $2,3,6,7,10,11,14$ with values $-1,-1,+1,-1,+1,+1,-1$). The directional pairing $\sum_{d} 1\cdot K(d)$ equals $-1$.
background
This module is the next kernel-checked increment in the QG full-theory campaign after the dihedral cosine kernel. It treats the seed triangle hinge ${0,e_0,e_0+e_1}$ and its full periodic Freudenthal star in the integer lattice: four containing unit cubes and six incident 4-simplices. The 15-class edge stencil from the imported edge layer indexes all edge orbits that can appear in that star.
The full-star class kernel $K$ is the explicit $\mathbb{R}$-valued weight on those fifteen classes, supported exactly on the seven indices $(2,3,6,7,10,11,14)$ with values $(-1,-1,+1,-1,+1,+1,-1)$. The directional functional is the ordinary pairing $\sum_d v_d K(d)$. Off-support vanishing is already proved by exhaustive fin_cases.
Uniform scaling (the constant test vector $v\equiv 1$) is one of the four gates listed in the module deliverable: nonvacuity, swap-$2\leftrightarrow 3$ symmetry, uniform-scaling decoy, and homothety stationarity. It checks that the kernel is not blind to a pure overall length rescaling.
proof idea
Unfold the directional pairing definition. Apply the private support lemma sum15_support: any summand whose class is off ${2,3,6,7,10,11,14}$ is zero because the kernel vanishes there (fullStarClassKernel_zero_off) and multiplication by zero kills the term. The Fin-15 sum therefore collapses to the seven supported classes. A final simp on the kernel definition evaluates $-1+(-1)+1+(-1)+1+1+(-1)=-1$.
why it matters
This is gate (5) of deliverable A in the Regge 4D full-star campaign: the uniform-scaling decoy for the seed hinge. Downstream copies in ReggeHinge4DStarKernel12, 13, and 22 restate the same gate for their own kernel tables (with different numerical targets), and the status-flag theorem in the 13-module records that the full-star class kernel is closed. The result does not yet assemble the flat Hessian over all hinges, nor does it prove $S_{\mathrm{RS}}$ converges to Einstein–Hilbert in 4D or flip gap_action_recovery; those remain open per the module contract. Within Recognition Science gravity it is a local algebraic sanity check on the discrete curvature weights before continuum matching.
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