periodicTTNormalEquationGenerator5_relativeColumn_eq_shift
plain-language theorem explainer
On the 5×5×5 periodic Freudenthal torus, evaluating any combined TT normal-equation generator (conformal vertex-delta or longitudinal gauge) on a row-relative edge column equals evaluating the globally index-shifted generator on the absolute column. Discrete GR / Regge weak-field workers cite this when moving between local edge frames and global stencil maps. The proof is a two-branch case split that reuses the conformal and longitudinal shift lemmas.
Claim. Let $E$ be the edge set of the encoded $5\times5\times5$ periodic Freudenthal torus, and let $I$ be the combined index set of conformal vertex-delta generators and longitudinal vertex-vector generators. For any row edge $r\in E$, column edge $c\in E$, and generator index $i\in I$, the combined TT normal-equation generator at $i$ evaluated on the $r$-relative column of $c$ equals the same generator at the $r$-base translate of $i$ evaluated on $c$.
background
Track 1.D isolates the tensor/shear sector of weak-field gravity on a discrete lattice. The conformal ansatz of Track 1.B assigns one scalar potential per vertex and averages endpoints to get edge strains; that slice cannot carry pure shear or transverse-traceless modes. This module therefore treats independent edge perturbations separately from vertex-conformal ones, on the canonical encoded $5\times5\times5$ periodic Freudenthal torus (PeriodicEdge5, PeriodicTorus5).
The combined normal-equation index $I$ is a disjoint sum: conformal generators labelled by vertices, plus fixed longitudinal vertex-vector (gauge) generators. The relative-column map re-expresses a global edge column in the local frame of a chosen row edge. Upstream, the conformal branch already has the matching shift identity (row-frame translate of a conformal generator column equals the globally shifted conformal column), and likewise for the longitudinal gauge generators.
proof idea
Term-mode proof by cases on the combined index, which is a Sum of vertex labels and longitudinal-gauge labels.
- Left injection (conformal vertex index $v$): apply
periodicConformalGenerator5_relativeColumn_eq_shiftat the same row, column, and $v$. - Right injection (longitudinal gauge index $i$): apply
periodicLongitudinalGaugeGenerator5_relativeColumn_eq_shiftat the same row, column, and $i$.
No further algebra: the combined generator is defined by cases on that sum, so the two sector lemmas glue directly.
why it matters
This is the generator-level commutation of relative framing with global index translation for the full TT normal-equation package (conformal plus longitudinal gauge). Downstream, periodicRelativeTTNormalEquationGeneratorMap5_eq_shiftedMap lifts the identity to coefficient maps: a relative-frame combined generator map equals a global map whose coefficient vector is reindexed by the row-base translation equivalence.
In the Recognition gravity track this is scaffolding for a discrete weak-field shear/TT sector beyond the pure conformal ansatz, on the same periodic Freudenthal geometry used for stencil and continuum-bridge work. It does not yet close continuum GR recovery or the full forcing chain (T0–T8), but it is a necessary algebraic step before TT mode equations can be written invariantly under edge-frame choice.
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