PeriodicTTLongitudinalCoefficientSolutionData5
plain-language theorem explainer
Packages coefficient projectors that extract the transverse-traceless edge residual on the 5×5×5 periodic Freudenthal torus by subtracting conformal and longitudinal-gauge components. Anyone citing Track 1.D TT Gram handoffs or the residual-as-TT formulation needs this bundle. It is a pure data structure: two projector maps plus the two residual-orthogonality axioms that define a valid coefficient solve.
Claim. A coefficient-solution datum on the canonical $5\times5\times5$ periodic Freudenthal torus consists of a conformal coefficient map $P_{\mathrm{conf}}:\{\text{edge perturbations}\}\times V\to\mathbb{R}$ and a longitudinal-gauge coefficient map $P_{\mathrm{g}}:\{\text{edge perturbations}\}\times(V\times\{1,2,3\})\to\mathbb{R}$ such that, for every edge perturbation $\varepsilon$, the residual $\varepsilon-P_{\mathrm{conf}}(\varepsilon)-P_{\mathrm{g}}(\varepsilon)$ is orthogonal (in the periodic edge inner product) to every conformal vertex generator and every longitudinal gauge generator.
background
Track 1.D opens the tensor/shear sector that Track 1.B's conformal ansatz cannot reach. The conformal slice assigns one scalar per vertex and stretches edges by averaging endpoint potentials; pure shear and transverse-traceless gravitational-wave modes lie outside that slice. This module therefore treats independent edge perturbations separately from vertex-conformal ones, on the fixed encoded $5\times5\times5$ periodic Freudenthal torus.
Edge perturbations are real functions on the typed periodic edges. Longitudinal gauge indices are pairs (vertex, spatial component in $\mathrm{Fin},3$), matching the forced spatial dimension $D=3$. The residual after conformal and gauge projection is the candidate TT component: no separate TT projector is required.
The structure records exactly the data of that coefficient solve: the two projector maps and the two families of residual-orthogonality identities against the fixed conformal generators and the fixed longitudinal gauge generators.
proof idea
No proof body: this is a structure definition. Inhabitants must supply the two projector maps and discharge the two residual-orthogonality fields (inner product of the residual against every conformal generator vanishes, and likewise against every longitudinal gauge generator). Downstream constructors such as ofNormalEquationData build instances from Gram/normal-equation solution data rather than proving the axioms in place.
why it matters
This is the coefficient-solve interface that lets Track 1.D treat the TT sector as the residual after conformal and longitudinal projections, matching the module's residual-as-TT design. Downstream, MasterTheoremHandoffIntegration consumes it in the Track 1.D TT Gram reduction chain: load-solver, load-image, kernel-criterion, generator-map-zero, range-criterion, and range-closed endpoints all thread through PeriodicTTLongitudinalCoefficientSolutionData5 (via ofNormalEquationData and the Gram-system stack) before handing off to Track 7.
In the broader Recognition gravity program this closes the algebraic bookkeeping needed to move beyond the pure conformal weak-field slice toward genuine shear/TT modes on a finite periodic lattice with $D=3$. It does not by itself prove existence of the projectors; it only names the data a solution must carry.
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