PeriodicTTGeneratorMapProjectorData5
plain-language theorem explainer
Packages finite-dimensional projector data that splits any periodic edge perturbation into conformal, longitudinal-gauge, and transverse-traceless pieces, with the gauge piece spanned by an explicit finite generator family. Gravity Track 1.D cites it as the exact decomposition input for TT/shear modes on the periodic Freudenthal torus. As a structure, the content is the field bundle and the three algebraic side-conditions (conformal membership, TT orthogonality, reconstruction).
Claim. For a finite index type $G$, projector data consist of maps $P_c$, $P_{\mathrm{tt}}$ on periodic edge perturbations, coefficient maps $c_g$ into $\mathbb{R}^G$, and generators $g_i$, such that: $P_c(\varepsilon)$ always lies in the vertex-conformal log-strain subspace; $P_{\mathrm{tt}}(\varepsilon)$ is orthogonal (under the periodic edge inner product) to every conformal generator and every $g_i$; and pointwise $\varepsilon = P_c(\varepsilon) + \mathrm{GaugeMap}(g,c_g(\varepsilon)) + P_{\mathrm{tt}}(\varepsilon)$.
background
Track 1.D opens the tensor/shear sector of the weak-field metric on the periodic Freudenthal torus. Track 1.B's conformal ansatz assigns one scalar potential per vertex and induces edge-length variations by averaging endpoints; that scalar slice cannot represent pure shear, so it misses transverse-traceless gravitational-wave modes.
A periodic edge perturbation is a real function on the typed periodic edges. The conformal log-strain subspace consists of those perturbations that arise as encoded conformal edge log-strains of some vertex potential. The structure packages a three-way split of an arbitrary edge perturbation relative to that subspace and a finite longitudinal gauge basis.
Spatial dimension $D=3$ (forced by T8/T9) fixes the underlying hypercube/torus combinatorics; the present object is the finite-dimensional linear-algebra interface on that fixed complex, not a continuum continuum limit.
proof idea
Definitional structure, not a proved theorem. The four data fields name the conformal projector, gauge-coefficient projector, TT projector, and finite gauge generators. Three Prop fields enforce: image of the conformal projector lies in the periodic conformal log-strain subspace; the TT image is orthogonal to every conformal generator and every gauge generator under the periodic edge inner product; and the three pieces reconstruct the original perturbation pointwise on edges (gauge piece via the generator map applied to the coefficient projector). Instantiating the structure means exhibiting concrete maps and discharging those three algebraic obligations.
why it matters
This is the exact finite-dimensional decomposition input still owed by Track 1.D: once the gauge map is defined from finite generators, the separate gauge-span obligation collapses (the gauge map is the span). Downstream, MasterTheoremHandoffIntegration consumes it as Track1DTTGeneratorMapProjectorReductionEndpoint, which states that any such generator-map projector data yields nonempty ordinary TT gauge-generator projector data.
From there a chain of reduction endpoints (Gram kernel criterion, generator-map-zero, load image/solver, range-closed, system solution) feeds Track 7's master handoff. In the RS gravity program this is the discrete stand-in for the TT projection that isolates shear/GW content from conformal and longitudinal gauge junk on the eight-tick, $D=3$ recognition complex.
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