periodicExternalTTNormalEquationGeneratorSparseDot5
plain-language theorem explainer
Sparse row-dot of the external TT normal-equation generator on the 5×5×5 periodic Freudenthal torus: average of the two endpoint conformal coefficients plus three longitudinal difference terms weighted by edge displacement coordinates. Anyone assembling discrete transverse-traceless normal equations in Track 1.D cites this stencil. The body is a pure definitional expansion of that sparse pattern.
Claim. Given a coefficient map $c:\mathbb{N}\to\mathbb{R}$ and an edge index on the $5\times 5\times 5$ periodic Freudenthal torus, the sparse external TT generator row-dot equals $\frac{c(b)+c(h)}{2}+\sum_{i=0}^{2}\bigl(c(125+3h+i)-c(125+3b+i)\bigr)\,d_i$, where $b,h$ are the external indices of the edge's base and head vertices and $d_i$ are the three displacement coordinates of the edge.
background
Track 1.D opens the tensor/shear sector of the weak-field Regge analysis. Track 1.B's conformal ansatz puts one scalar potential at each vertex and varies edge lengths by averaging the endpoints; that scalar slice cannot carry pure shear, so it misses transverse-traceless gravitational-wave modes. This module separates independent edge perturbations from vertex-conformal ones and records the elementary rectangle obstruction for the conformal ansatz.
The ambient lattice is PeriodicTorus5, the canonical encoded $5\times 5\times 5$ periodic Freudenthal torus. Vertices are ordered externally by $v=(x\cdot 5+y)\cdot 5+z$. Edges carry a displacement code in $\mathrm{Fin},7$; periodicExternalDispCoordNat5 extracts the three Boolean bit components of that code as real $0/1$ coordinates used by the numerical generator matrix.
The coefficient vector is laid out so that conformal vertex values occupy the first block and three longitudinal components per vertex occupy the block starting at offset $125=5^3$. The present definition is the sparse stencil that dots one generator row against that layout.
proof idea
Definitional, not a proof. Decode the edge from its encoded index, read base and head external vertex indices, then form the sum of (i) the averaged conformal endpoint coefficients and (ii) three terms, one per spatial component, each the difference of the head and base longitudinal coefficients at offset $125+3\cdot\mathrm{idx}+i$, multiplied by the corresponding displacement coordinate of the edge. No lemmas are invoked; the body is the stencil written out.
why it matters
In the Recognition gravity stack, covering TT modes requires a generator matrix that mixes the conformal (scalar) sector already treated in Track 1.B with longitudinal edge degrees of freedom. This sparse row-dot is the concrete numerical interface for that combined normal equation on the $5^3$ torus. The immediately following comment in the file states that the combined generator map splits into the already-fixed conformal and longitudinal generator maps; this definition supplies the conformal-plus-longitudinal half of that split.
It sits inside the Track 1.D scaffold that aims to reach genuine shear/TT content beyond the conformal ansatz, complementary to the eight-tick and $D=3$ forcing landmarks elsewhere in the monolith. No downstream theorems yet depend on it (used_by is empty), so it is infrastructure for forthcoming normal-equation and zero-mode arguments rather than a closed claim.
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