periodicRelativeColumnOfRow5
plain-language theorem explainer
Rewrites a column edge in the coordinate frame fixed by a row edge on the 5×5×5 periodic Freudenthal torus: relative base via torus subtraction, displacement unchanged. Cited by TT Hessian residual diagnostics and by conformal and longitudinal gauge generator shift lemmas that need translation-covariant columns. Pure structure constructor, no proof content.
Claim. On the $5\times 5\times 5$ periodic Freudenthal torus, given edges $e_{\mathrm{row}}=(b_r,d_r)$ and $e_{\mathrm{col}}=(b_c,d_c)$, the row-relative column is the edge $(b_r^{-1}\cdot b_c,\, d_c)$: base equal to the torus-difference of the two bases, displacement equal to that of the column.
background
Track 1.D opens the tensor/shear sector beyond the Track 1.B conformal ansatz. The conformal slice assigns one scalar per vertex and averages endpoints, so it cannot carry pure shear or transverse-traceless gravitational-wave modes. This module separates independent edge perturbations from vertex-conformal ones and records elementary rectangle obstructions.
Edges live on the canonical encoded $5\times 5\times 5$ periodic Freudenthal torus (PeriodicEdge5). Each edge is a base vertex plus a displacement in a finite stencil. The sibling periodicRelativeVertex5 subtracts bases componentwise on the torus (periodic finite differences), giving vertex coordinates of $v$ in the frame whose origin is a chosen base.
The present definition lifts that vertex re-basing to edges: keep the column displacement, replace the column base by its position relative to the row base. Downstream residual and generator lemmas then treat each matrix row in its own edge-centered frame.
proof idea
Definition only: pack a new PeriodicEdge5 whose base is periodicRelativeVertex5 row.base col.base and whose disp is exactly col.disp. No tactics, no lemmas beyond the relative-vertex constructor.
why it matters
Translation covariance of physical stencils is the bookkeeping backbone of the TT/Lichnerowicz residual track. After each Hessian or generator row is re-based at its own edge base, coefficients must match globally shifted columns. This relative-column map is the common re-basing primitive.
It feeds EncodedTTHessianLichnerowiczCoeffRelativeTranslatedFormulaData5 (relative-frame coefficient residual data), the conformal and longitudinal gauge identities periodicConformalGenerator5_relativeColumn_eq_shift and periodicLongitudinalGaugeGenerator5_relativeColumn_eq_shift, origin-edge simplification, and several endpoint-equality iff lemmas that convert row-frame endpoint tests into globally shifted ones. Without a uniform relative column, those shift and residual certificates cannot even be stated.
In the broader RS gravity program this is scaffolding for the shear sector that must eventually cover TT modes the conformal ansatz misses; it is geometry bookkeeping, not a forcing-chain step (T5–T8).
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