periodicTTNormalEquationGenerator5
plain-language theorem explainer
On the period-5 Freudenthal torus, each combined normal-equation index is sent to an edge-length perturbation: a unit conformal vertex-delta generator or a longitudinal gauge generator. Anyone building the finite transverse-traceless normal equations cites this as the column family. The body is a two-branch match on the sum index, wiring the two already-defined generator families together.
Claim. Let $I$ be the disjoint union of conformal vertex indices $\mathrm{Fin}(n_V)$ and longitudinal gauge indices on the period-5 periodic torus. For each $i\in I$, define an edge perturbation $\delta\ell_i:\{\text{periodic edges}\}\to\mathbb{R}$ by $\delta\ell_i=$ the unit conformal generator at vertex $v$ when $i=\mathrm{inl}(v)$, and $\delta\ell_i=$ the longitudinal gauge generator at gauge index $g$ when $i=\mathrm{inr}(g)$.
background
Track 1.D isolates the tensor/shear sector of weak-field gravity on a finite periodic Freudenthal triangulation. The older conformal ansatz puts one scalar potential at each vertex and averages endpoints to get edge-length variations; that slice cannot carry pure shear, so it misses transverse-traceless gravitational-wave modes.
Here edge perturbations are maps from typed period-5 edges to real numbers. The combined index set is the sum of vertex indices (for conformal generators) and longitudinal gauge indices (for vertex-vector gauge generators). The conformal generator at a vertex is the edge log-strain induced by a unit potential spike at that vertex and zero elsewhere. The longitudinal gauge generators complete the non-TT directions that the normal equations must project out.
The module therefore treats the TT residual as whatever remains after subtracting conformal and longitudinal projections from a general edge perturbation.
proof idea
Definition by case analysis on the sum index. Left injection (vertex index $v$) returns the already-built conformal generator at $v$. Right injection (longitudinal gauge index $i$) returns the already-built longitudinal gauge generator at $i$. No further computation: it is pure wiring of the two generator families into one column family.
why it matters
This is the column family for the concrete finite TT normal equations on the period-5 torus. Downstream, the combined generator map is the linear combination of these columns; a split theorem recovers the separate conformal and longitudinal maps. Relative-frame identities identify row-translated columns with globally shifted generators. The Gram-kernel vanishing theorem uses the map to show kernel coefficient vectors produce the zero edge perturbation. Coefficient-solution data packages projectors whose residual is orthogonal to this generator family, so the TT part is defined without a separate projector.
In the Recognition gravity track this is scaffolding for the shear sector that the pure conformal ansatz cannot reach: the missing TT modes of weak-field metric perturbations on the discrete torus.
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