PeriodicTTProjectorData5
plain-language theorem explainer
Packages the three linear maps that split a finite N=5 periodic edge perturbation into conformal, gauge, and transverse-traceless pieces, with membership and pointwise reconstruction. Gravity Track 1.D cites it as the concrete projector interface the tensor lane must inhabit. As a structure, there is no proof body: fields are the maps plus the four Prop obligations.
Claim. For a gauge-potential type $G$ and a gauge map $g: G\to\{\text{periodic edge perturbations on the $N=5$ torus}\}$, a projector datum consists of three maps $P_c,P_g,P_{\mathrm{TT}}$ on edge perturbations such that: $P_c(\varepsilon)$ lies in the vertex-conformal log-strain subspace; $P_g(\varepsilon)$ lies in the image of $g$; $P_{\mathrm{TT}}(\varepsilon)$ is orthogonal (under the finite periodic-edge inner product) to both the conformal and gauge subspaces; and $P_c(\varepsilon)+P_g(\varepsilon)+P_{\mathrm{TT}}(\varepsilon)=\varepsilon$ pointwise on every edge.
background
Track 1.D opens the tensor/shear sector of the weak-field metric. Track 1.B's conformal ansatz puts one scalar potential on each vertex and induces edge-length changes by averaging endpoints; that scalar slice cannot carry pure shear, so it misses transverse-traceless gravitational-wave modes. This module therefore treats independent edge perturbations separately from vertex-conformal ones.
On the finite periodic Freudenthal torus with $N=5$, an edge perturbation is simply a real function on the typed periodic edges. The conformal log-strain subspace consists of those perturbations that arise as conformal edge log-strains of some vertex potential. A gauge subspace is the image of a supplied forward gauge operator (left parametric so the longitudinal/diffeomorphism discretization need not be fixed yet). TT means orthogonality, under the finite periodic-edge inner product, to both the conformal and gauge slices.
Spatial dimension $D=3$ is the forced value from the forcing chain (T8/T9), which fixes the underlying hypercube and torus combinatorics used by the periodic edge indexing.
proof idea
No proof: this is a structure (definitional interface). The four field obligations are the mathematical content. Downstream constructors (e.g. ofFiniteGeneratorData) inhabit the structure by building the three maps from finite spanning generators and discharging membership plus reconstruction. The handoff theorems then wrap those constructors in one-line exact ⟨⟨...⟩, ...⟩ style reductions.
why it matters
This is the concrete projector interface that closes the finite $N=5$ conformal/gauge/TT decomposition on the periodic edge space. Without inhabited projector data, the tensor lane cannot claim an orthogonal split of edge perturbations into pure shear (TT) versus conformal and gauge junk.
Downstream, MasterTheoremHandoffIntegration consumes it heavily: the Track 1.D finite-generator, gauge-generator, and generator-map projector-reduction endpoints all reduce to producing (or deriving) an instance of this structure, then packaging it for Track 7. Doc-comments there state that finite spanning-generator projector data is sufficient to build the projector data and close the orthogonal decomposition target.
In the broader RS gravity program this is the discrete stand-in for the continuum TT projection that isolates gravitational-wave degrees of freedom once $D=3$ and the eight-tick combinatorics are fixed. It remains theorem-shaped: the tensor lane must still construct the maps for a chosen gauge operator.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.