PeriodicTTGaugeGeneratorProjectorData5
plain-language theorem explainer
Bundles finite-generator projector data that splits periodic edge perturbations into conformal, gauge, and transverse-traceless (TT) parts on the Freudenthal torus. Gravity Track 1.D and the master-theorem handoff cite it as the sharper TT target once the conformal slice is already spanned by vertex generators. It is a pure data structure: projectors, generators, membership, orthogonality, and reconstruction axioms, with no proof body.
Claim. A package of maps and finite generators on periodic edge perturbations $\varepsilon: E_5\to\mathbb{R}$ consisting of projectors $P_c$, $P_g$, $P_{TT}$, and generators $g_i$ ($i\in G$, $G$ finite) such that: $P_c\varepsilon$ lies in the vertex-conformal log-strain subspace; $P_g\varepsilon$ lies in the image of a supplied gauge map; every gauge-slice element is a real linear combination of the $g_i$; $P_{TT}\varepsilon$ is orthogonal (under the periodic edge inner product) to every conformal generator and every $g_i$; and $P_c\varepsilon+P_g\varepsilon+P_{TT}\varepsilon=\varepsilon$ pointwise on edges.
background
Track 1.D opens the tensor/shear sector that Track 1.B's conformal ansatz cannot reach. The conformal ansatz assigns one scalar potential per vertex and induces edge-length variations by averaging endpoints; that scalar slice misses pure shear and therefore cannot cover transverse-traceless gravitational-wave modes. This module separates independent edge perturbations from vertex-conformal ones on the periodic Freudenthal torus.
A periodic edge perturbation is simply a real function on the typed periodic edges. The conformal log-strain subspace consists of those perturbations equal to the encoded conformal edge log-strain of some vertex potential. The gauge subspace is the image of a still-parametric forward gauge map (the longitudinal/diffeomorphism discretization is not fixed here). TT is defined by orthogonality to both slices under the finite periodic-edge inner product.
Upstream dimension landmarks fix $D=3$ (T8/T9) and the eight-tick period, which set the discrete geometry on which these edge spaces live. The conformal half of the decomposition is already discharged by the encoded vertex-delta family; this structure is the residual gauge-generator target.
proof idea
No proof: this is a structure (definitional data package). The fields are the three projectors, the finite gauge-generator family, membership of the conformal and gauge images, the spanning property of the generators for the gauge slice, TT orthogonality to conformal generators and to each gauge generator, and pointwise reconstruction of every edge perturbation as the sum of the three projected pieces. Downstream code instantiates or converts into this package (for example via ofGeneratorMapData) rather than proving a theorem here.
why it matters
This is the sharper Track 1.D handoff object after the conformal span is fixed: master-theorem integration treats "supply gauge-generator projector data" as the remaining obligation. It is the hypothesis of Track1DTTGaugeGeneratorProjectorReductionEndpoint and the codomain of Track1DTTGeneratorMapProjectorReductionEndpoint, which observes that when the gauge map is itself the span of finite generators the separate gauge-span proof is automatic.
Those endpoints feed the proved reduction theorems consumed by Track 7 (track1D_tt_generator_map_projector_reduction_endpoint_holds and the Gram-kernel / load-image / solver chain). In the broader RS gravity program this is the discrete stand-in for isolating TT shear modes once conformal and gauge (diffeomorphism) directions are projected out, on the $D=3$ eight-tick geometry forced by the unified chain. It does not yet construct the projectors; it names the interface the construction must satisfy.
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