periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_longitudinalCoefficientSolutionData
plain-language theorem explainer
Residual-defined longitudinal coefficient-solution data on the N=5 periodic Freudenthal torus yields the concrete conformal/gauge/TT orthogonal decomposition target for the longitudinal gauge. Gravity Track 1.D and the Track 7 handoff cite it as the reduction step from coefficient solutions to the honest TT splitting. The proof is a one-line term wrapper: convert solution data to projector data, then apply the existing coefficient-data closure.
Claim. Given residual-defined coefficient-solution data $D$ for the $N=5$ periodic torus (conformal and longitudinal gauge coefficient projectors whose residual is orthogonal to both subspaces), the concrete Track 1.D target holds: every edge perturbation splits into conformal, longitudinal-gauge, and TT parts, with TT meaning finite orthogonality to the conformal and longitudinal gauge subspaces under the longitudinal gauge map.
background
Track 1.D opens the tensor/shear sector of the weak-field metric. Track 1.B's conformal ansatz puts one scalar at each vertex and averages to edge lengths; that slice cannot carry pure shear, so it misses transverse-traceless gravitational-wave modes. This module separates independent edge perturbations from vertex-conformal ones and targets a finite conformal/gauge/TT split on the periodic Freudenthal torus at $N=5$.
The decomposition target asserts existence of a raw edge-perturbation splitting whose three parts land in the conformal log-strain subspace, the longitudinal gauge subspace (indexed by vertex times three spatial directions, matching $D=3$), and the TT residual defined by orthogonality to both. Coefficient-solution data packages the two projectors and the residual-orthogonality identities without a separate TT projector: TT is whatever remains after subtracting conformal and longitudinal projections.
Spatial dimension $D=3$ is the T8 landmark; the gauge index type is built from that three-dimensional fiber.
proof idea
One-line term proof. Convert the residual-defined solution data $D$ into longitudinal coefficient projector data via PeriodicTTLongitudinalCoefficientProjectorData5.ofSolutionData, then apply the already-proved closure periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_longitudinalCoefficientData, which builds the three-way splitting and checks subspace membership and TT orthogonality for the longitudinal gauge map.
why it matters
Closes the residual-defined coefficient-solution route into the honest $N=5$ longitudinal TT decomposition target, the finite stand-in for a conformal/gauge/TT split in the tensor/shear lane. Downstream, periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_normalEquationData reduces normal-equation solutions through this theorem, and the master handoff track1D_tt_longitudinal_coefficient_solution_reduction_endpoint_holds consumes the same reduction chain as a Track 7 endpoint ("Residual-defined coefficient-solution reduction endpoint consumed by Track 7").
In the framework this is scaffolding closure for Track 1.D: once coefficient (or normal-equation) data exist, the TT part is the orthogonal residual, not an extra projector. It sits under the gravity tensor sector needed beyond the conformal ansatz, with $D=3$ from T8 fixing the longitudinal gauge fiber. It does not yet construct the projectors from first principles; it only discharges the target once solution data are supplied.
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