periodicEdgeKernelRowVector5
plain-language theorem explainer
Extracts the e-th row of a finite edge-kernel matrix as an edge perturbation on the 5×5×5 periodic Freudenthal torus. Anyone comparing Regge TT Hessian stencils to lattice Lichnerowicz operators uses this to put matrix rows in the same type as edge strain fields. The body is pure currying: fix the kernel and source edge, return the map f ↦ K(e,f).
Claim. Given a finite edge-kernel $K:E_5\times E_5\to\mathbb{R}$ on the edges of the $5\times 5\times 5$ periodic Freudenthal torus and a fixed edge $e\in E_5$, the associated row vector is the edge perturbation $f\mapsto K(e,f)$.
background
Track 1.D opens the tensor/shear sector of the weak-field metric. The older conformal ansatz puts one scalar at each vertex and averages endpoints to get edge-length changes; that slice cannot carry pure shear, so it misses transverse-traceless gravitational-wave modes. This module therefore treats independent edge perturbations as the primary objects.
The ambient lattice is the encoded $5\times 5\times 5$ periodic Freudenthal torus. Its edges $E_5$ index both edge perturbations (maps $E_5\to\mathbb{R}$) and finite edge-kernels (maps $E_5\to E_5\to\mathbb{R}$). The kernel type is the concrete matrix surface on which the Regge TT Hessian stencil and the lattice Lichnerowicz stencil are meant to be compared.
Viewing a matrix row as an edge perturbation is the bookkeeping step that lets operator evaluation be rewritten as an edge inner product.
proof idea
One-line definitional currying. The kernel already has type $E_5\to E_5\to\mathbb{R}$; fixing the first argument $e$ yields $f\mapsto K(e,f)$, which is exactly an edge perturbation. No lemmas are applied.
why it matters
This row extraction is the hinge between matrix language and the edge-perturbation inner product. Immediately downstream, the identity that kernel-operator evaluation equals pairing against the corresponding row (up to commutativity of real multiplication) is stated in terms of it.
The same rows feed the finite residual targets for the TT Hessian-to-Lichnerowicz comparison: residual vanishing on TT perturbations, residual-row membership in the conformal-plus-longitudinal generator span, and the explicit residual-row coefficient table. Those structures are the compact calculation surface for showing the discrete Regge TT Hessian matches the lattice Lichnerowicz operator on the shear sector, which is the point of Track 1.D beyond the conformal ansatz.
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