periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_gaugeGeneratorData
plain-language theorem explainer
Given gauge-generator projector data on the N=5 periodic Freudenthal torus (conformal generators already fixed to the spanning vertex-delta family), the Track 1.D orthogonal decomposition target holds: every edge perturbation splits into conformal, gauge, and TT-orthogonal parts. Gravity handoff and generator-map reduction cite this. One-line term proof: coerce to finite-generator data, then apply the finite-generator decomposition lemma.
Claim. Let $\mathrm{gaugeMap}$ send gauge potentials to real edge perturbations on the $N=5$ periodic Freudenthal edges. If one is given projector data consisting of conformal, gauge, and TT projectors together with a finite family of gauge generators whose conformal slice is the already-spanning vertex-delta family, then there exists a raw edge-perturbation splitting such that every perturbation decomposes into a conformal-log part, a gauge part in the image of $\mathrm{gaugeMap}$, and a part orthogonal to both the conformal and gauge subspaces.
background
Track 1.D opens the tensor/shear sector of weak-field gravity on the Recognition lattice. Track 1.B's conformal ansatz assigns one scalar potential per vertex and averages endpoints to get edge-length variations; that scalar slice cannot represent pure shear, so it misses transverse-traceless gravitational-wave modes. This module separates independent edge perturbations from vertex-conformal ones and records the elementary rectangle obstruction for the conformal ansatz.
Edge perturbations here are maps PeriodicEdge5 → ℝ. The decomposition target asserts existence of a raw splitting into three parts: a conformal-log subspace member, a gauge-subspace member for the chosen gauge map, and a TT part defined by finite orthogonality to both subspaces. The remaining load is constructing the three projectors.
Gauge-generator projector data is the sharper package after the conformal half is discharged: conformal generators are fixed to the encoded vertex-delta family already proved to span the conformal slice, and a finite index type supplies the gauge generators. Spatial dimension $D=3$ (T8/T9) underlies the Freudenthal torus geometry, but enters only as ambient lattice data.
proof idea
Pure term-mode one-liner. First apply PeriodicTTFiniteGeneratorProjectorData5.ofGaugeGeneratorData to the given gauge-generator projector data, producing the weaker finite-generator projector package. Then feed that package into periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_finiteGeneratorData, which already closes the orthogonal-decomposition target once finite generator data is in hand. No new algebraic work: the conformal span hypothesis is inherited from the gauge-generator structure, and the finite-generator lemma does the rest.
why it matters
Closes the gauge-generator form of the Track 1.D finite orthogonal decomposition target on the N=5 periodic Freudenthal torus. Downstream, periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_generatorMapData specializes further to the gauge map generated by its own finite basis, and track1D_tt_gauge_generator_projector_reduction_endpoint_holds packages the reduction as the Track 7 handoff endpoint ("Gauge-generator projector-data reduction endpoint consumed by Track 7").
In the broader RS gravity program this is the step that lets TT modes sit honestly orthogonal to conformal and gauge slices once projector data is supplied, rather than being smuggled in by the scalar conformal ansatz. It does not yet construct the projectors from first principles; it converts an already-assembled gauge-generator data bundle into the Prop-shaped decomposition target. Framework landmarks: D=3 spatial lattice (T8) and the eight-tick/Freudenthal discrete geometry that fixes N=5 as the working triangulation size.
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