periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_longitudinalCoefficientData
plain-language theorem explainer
Pure coefficient-level projector data for the N=5 periodic Freudenthal torus yields the honest conformal/gauge/TT orthogonal splitting of edge perturbations, with TT defined by finite inner-product orthogonality. Gravity Track 1.D and the MasterTheorem handoff cite this as the coefficient-projector reduction step. The proof is a one-line wrapper: convert coefficient data to longitudinal projector data, then apply the existing longitudinal-data closure.
Claim. Given pure coefficient-projector data on the $N=5$ periodic torus (maps sending each edge perturbation to conformal vertex-delta coefficients, longitudinal gauge coefficients indexed by vertex$\times\{1,2,3\}$, and a TT residual, with the stated orthogonality identities), the Track 1.D decomposition target holds for the concrete longitudinal gauge: there exists a raw edge-perturbation splitting into conformal, gauge, and TT parts, with TT orthogonal to both the conformal log-strain subspace and the longitudinal gauge image.
background
Track 1.D isolates the tensor/shear sector that the Track 1.B conformal ansatz cannot reach. Vertex scalars induce only averaged endpoint log-strains on edges; pure shear and transverse-traceless wave modes need independent edge perturbations. The module therefore separates edge perturbations from vertex-conformal ones and records the elementary rectangle obstruction for the conformal slice.
The decomposition target at $N=5$ asks for a raw splitting of every periodic edge perturbation into conformal, gauge, and TT pieces, where TT means finite orthogonality (via the periodic edge inner product) to the conformal log subspace and to the image of a chosen gauge map. The remaining load is constructing the three projectors.
Coefficient-projector data strengthens ordinary longitudinal projector data: the conformal part is no longer an arbitrary map but is generated from encoded vertex-delta coefficients. The gauge index type is one spatial vector component at one periodic vertex ($D=3$ from the forcing chain). The concrete gauge map is the longitudinal gauge map on that index type.
proof idea
One-line term wrapper. Convert the coefficient-projector package to ordinary longitudinal projector data via PeriodicTTLongitudinalProjectorData5.ofCoefficientData, then apply the already-proved closure periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_longitudinalData. No new orthogonality or subspace membership is proved here; the coefficient encoding is discharged by the conversion lemma.
why it matters
Closes the coefficient-projector form of the honest $N=5$ longitudinal TT decomposition target in the tensor/shear scaffold. Downstream, the sibling residual/solution-data theorem reuses this result after converting solution data to coefficient data. The MasterTheorem handoff consumes it as the Track 1.D longitudinal coefficient-projector reduction endpoint fed into Track 7.
In the broader RS gravity program this is scaffolding for weak-field modes beyond the conformal ansatz: TT is finite orthogonality to conformal and longitudinal-gauge subspaces on the periodic Freudenthal torus, not a continuum PDE statement. Spatial dimension $D=3$ (T8/T9) enters the gauge index as three vector components per vertex. It does not yet construct continuum gravitational waves; it certifies that coefficient-level projectors suffice for the discrete orthogonal split.
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