periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_projectorData
plain-language theorem explainer
Concrete conformal/gauge/TT projector data on the N=5 periodic Freudenthal edge space is enough to close the finite orthogonal decomposition target. Track 1.D gravity and the master handoff cite this reduction when they already hold the three projectors with subspace membership. The proof packages those projectors into a raw splitting and reuses the three membership fields.
Claim. Fix a gauge-potential type $G$ and a linear gauge map $g\colon G\to\{\text{edge perturbations on the $N=5$ periodic Freudenthal complex}\}$. If three endomorphisms of the edge-perturbation space are given that land, respectively, in the conformal-log subspace, the gauge subspace of $g$, and the finite TT-orthogonal complement of those two subspaces, then every edge perturbation admits an orthogonal conformal/gauge/TT splitting with those membership properties.
background
Track 1.D isolates the tensor/shear sector of weak-field gravity on the periodic Freudenthal complex. Track 1.B's conformal ansatz assigns one scalar per vertex and averages to edges; that slice cannot represent pure shear, so it misses transverse-traceless gravitational-wave modes. This module therefore treats independent edge perturbations and separates them from vertex-conformal ones.
Edge perturbations at $N=5$ are real functions on the typed periodic Freudenthal edges. The honest decomposition target asks for a raw splitting of every such perturbation into conformal, gauge, and TT parts, with TT defined by finite orthogonality to the conformal-log and gauge subspaces. Projector data is the concrete package: three endomorphisms together with proofs that each image lands in the corresponding subspace (and, in the structure, reconstruction). Spatial dimension $D=3$ is the forced RS value from the T8/T9 chain, fixing the underlying complex.
proof idea
Term-mode packaging, not a calculation. Form the existential witness by converting the three projectors into a raw edge-perturbation splitting (toRawSplitting), then discharge the three membership conjuncts by the corresponding fields of the projector-data structure: conformal membership, gauge membership, and TT-orthogonal membership. No new geometric identity is proved here.
why it matters
This is the intended consumption theorem for any concrete periodic Freudenthal projector construction: once the three maps and memberships exist, the Track 1.D orthogonal-decomposition target is closed. Downstream, the master handoff records the projector-data reduction endpoint used by Track 7, and the finite-generator variant reduces to this theorem after building projectors from spanning generators. In the RS gravity program it sits on the path from the conformal obstruction (rectangle shear not vertex-conformal) toward genuine TT modes on the eight-tick, $D=3$ complex. The remaining tensor-lane work is constructive: produce generators, solve the projector system, prove reconstruction.
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