periodicEdgeKernelOperator5
plain-language theorem explainer
Defines the action of a finite edge-to-edge kernel matrix on an edge-length perturbation over the 5×5×5 periodic Freudenthal torus: each output edge value is the sum of kernel entries times input perturbations. Gravity and lattice-GR workers cite it when comparing Regge TT Hessians to lattice Lichnerowicz stencils. The body is a one-line matrix–vector product.
Claim. Given a real kernel $K$ on pairs of edges of the $5\times5\times5$ periodic Freudenthal torus and an edge perturbation $\varepsilon$, define the operator output by $(K\varepsilon)(e)=\sum_f K(e,f)\,\varepsilon(f)$ for every edge $e$.
background
Track 1.D isolates the tensor/shear sector of weak-field gravity on a discrete triangulation. The conformal (vertex-scalar) ansatz cannot carry pure shear or transverse-traceless modes, so the module works with independent edge-length perturbations rather than endpoint-averaged potentials.
On the canonical $5\times5\times5$ periodic Freudenthal torus, an edge perturbation is a real function of the typed periodic edges. A finite edge-kernel is a real matrix on those edges: the concrete surface on which a Regge TT Hessian stencil and a lattice Lichnerowicz stencil are meant to be compared.
This definition turns such a matrix into a linear map on edge perturbations by the usual row-sum product. Downstream lemmas rephrase that map as an inner product against a row vector and compare two kernels by residual vanishing on the TT subspace.
proof idea
Definitional one-liner: the operator is the function sending each edge $e$ to the finite sum over edges $f$ of $K(e,f)\varepsilon(f)$. No lemmas or tactics; pure matrix–vector multiplication on the finite edge set.
why it matters
This is the shared operator surface for the TT Hessian-to-Lichnerowicz match in Track 1.D. Downstream, evaluation is identified with pairing against the kernel row; entrywise or rowwise kernel equality lifts to operator equality; the residual kernel acts as the difference of the two operators.
Those facts feed the data structures that package the physical closure target: instantiate a Regge second-variation edge Hessian kernel and a spin-2 lattice Lichnerowicz kernel, then prove their rows (or residual) agree on longitudinal-TT edge perturbations. The residual-vanishing package is described as the most compact finite calculation target for that comparison.
In the broader Recognition gravity program this is scaffolding for discrete weak-field tensor modes beyond the conformal slice, not a continuum GR theorem by itself.
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