PeriodicRelativeTTGeneratorClosure5
plain-language theorem explainer
Defines the relative-frame generator-closure property on the encoded 5×5×5 periodic Freudenthal torus: every row-translated combined TT normal-equation generator must split as a fixed conformal image plus a longitudinal-gauge image. Gravity Track 1.D cites it as the bridge from shifted generators to TT orthogonality and the Lichnerowicz Hessian package. Pure Prop packaging; a companion theorem discharges it.
Claim. For every edge $r$ of the $5\times5\times5$ periodic torus and every coefficient map $c$ on the TT normal-equation index set, there exist conformal vertex coefficients $a$ and longitudinal-gauge coefficients $b$ such that, for every column edge, the row-frame translated combined generator of $(r,c)$ equals the conformal generator of $a$ plus the longitudinal-gauge generator of $b$.
background
Track 1.D opens the tensor/shear sector beyond the Track 1.B conformal ansatz. Vertex-scalar potentials induce edge-length changes by endpoint averaging; that slice cannot represent pure shear, hence cannot cover transverse-traceless weak-field modes. The module therefore treats independent edge perturbations and isolates conformal versus longitudinal-gauge subspaces on the encoded $5\times5\times5$ periodic Freudenthal torus.
The conformal generator map sends a coefficient vector on vertices to an edge perturbation (two-point support at base and head). The longitudinal-gauge map does the same for pure gauge edge variations. The relative combined generator is the normal-equation generator translated into the frame of a chosen edge row.
Closure asks that this row-frame translate never leave the sum of the fixed conformal and gauge images: after the shift, the generator still factors through those two unshifted maps.
proof idea
Definitional Prop only: no proof body beyond the quantified equality. It packages (i) universal quantification over row edges and normal-equation coefficient vectors, (ii) existence of conformal vertex coefficients and longitudinal-gauge coefficients, and (iii) pointwise equality, on every column, of the row-translated combined generator with the sum of the fixed conformal and longitudinal-gauge generator maps.
The companion theorem proves the Prop by pushing coefficients through the row-base translation equivalence on the normal-equation index set, then invoking unshifted generator closure in the translated frame.
why it matters
This is the preferred mathematical target for the relative-frame TT-zero route: prove generator closure once, then orthogonality follows from the existing TT definition. Downstream, it is a field of the relative-translated Lichnerowicz Hessian closure data structure, the statement discharged by the holds theorem, and the missing bridge behind shifted-generator orthogonality on the TT subspace (every TT edge perturbation orthogonal to every row-translate of the combined conformal/gauge generator).
In the broader scaffold it advances discrete Regge TT/shear analysis toward a Lichnerowicz-type Hessian on the periodic Freudenthal geometry, separating pure shear from the conformal slice that cannot carry gravitational-wave polarizations. It is local to Gravity Track 1.D, not a T0–T8 forcing step.
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