periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_generatorMapData
plain-language theorem explainer
Generator-map projector data on the N=5 periodic Freudenthal edge space closes the finite conformal/gauge/TT orthogonal decomposition for the gauge map spanned by its own generators. Track 1.D gravity cites this when the gauge operator is presented as a finite generator family rather than an abstract subspace. The proof is a one-line reduction through the gauge-generator projector-data constructor.
Claim. Let $D$ be projector data for a gauge map on the $N=5$ periodic edge space that is itself given by a finite generator family (conformal, gauge-coefficient, and TT projectors together with those generators). Then the finite orthogonal decomposition target holds: every edge perturbation splits into conformal, gauge, and TT parts, with the TT part orthogonal (in the periodic edge inner product) to both the conformal log-subspace and the image of the gauge map generated by $D$'s generators.
background
Track 1.D opens the tensor/shear sector of weak-field gravity on the periodic Freudenthal geometry. The older Track 1.B conformal ansatz assigns one scalar potential per vertex and induces edge-length changes by averaging endpoints; that scalar slice cannot represent pure shear, so it cannot cover transverse-traceless gravitational-wave modes. This module separates independent edge perturbations from vertex-conformal ones and records elementary rectangle obstructions for the conformal ansatz.
The decomposition target is an existence statement: a raw splitting of periodic edge perturbations into conformal, gauge, and TT parts, where TT means finite orthogonality to the conformal log-subspace and to the image of a chosen gauge map. At $N=5$ one works on the 5-periodic Freudenthal torus (tied to the discrete geometry forced alongside $D=3$ spatial dimensions in the T0–T8 chain).
Generator-map projector data packages the three projectors plus a finite generator family for the gauge map itself. As the structure doc states, this "removes a separate gauge-span obligation: the gauge map is the span."
proof idea
One-line term proof. Convert the supplied generator-map projector data into gauge-generator projector data via the structure map ofGeneratorMapData, then apply the already-proved lemma that gauge-generator projector data closes the finite $N=5$ orthogonal decomposition target. No new algebraic identities are proved at this layer; the work is packaging and reduction.
why it matters
This is the generator-map entry point for the honest Track 1.D finite TT decomposition. Downstream, the MasterTheorem handoff consumes it as the generator-map projector-data reduction endpoint for Track 7 (track1D_tt_generator_map_projector_reduction_endpoint_holds). The longitudinal specialization also routes through it: concrete longitudinal projector data is re-packaged as generator-map data, yielding the same target for the vertex-vector longitudinal gauge map.
In the Recognition gravity program the point is discreteness and honesty: TT is defined by finite orthogonality to conformal and gauge slices on the periodic edge space, not by continuum PDE language. Closing this target is a necessary scaffold step toward representing pure shear and gravitational-wave modes that the pure conformal ansatz cannot reach.
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