PeriodicTTLongitudinalCoefficientProjectorData5
plain-language theorem explainer
Packages the pure coefficient-projector data for the longitudinal split of edge perturbations on the 5×5×5 periodic Freudenthal torus: conformal coefficients from vertex deltas, longitudinal gauge coefficients, and a residual TT projector. Anyone citing the Track 1.D Gram/normal-equation handoff needs this interface. It is a structure of maps plus three axioms (TT orthogonal to conformal and gauge generators; exact reconstruction).
Claim. A data package on the $5\times 5\times 5$ periodic Freudenthal torus consisting of: (i) a map sending each edge perturbation $\varepsilon$ to conformal coefficients on vertices; (ii) a map sending $\varepsilon$ to longitudinal gauge coefficients indexed by (vertex, spatial direction); (iii) a residual projector $P_{\mathrm{TT}}(\varepsilon)$. Required: $P_{\mathrm{TT}}(\varepsilon)$ is orthogonal (edge inner product) to every conformal generator and every longitudinal gauge generator, and $\varepsilon$ equals the sum of the conformal image, the gauge image, and $P_{\mathrm{TT}}(\varepsilon)$ on every edge.
background
Track 1.D opens the tensor/shear sector that Track 1.B's conformal ansatz cannot reach. The conformal ansatz puts one scalar at each vertex and averages endpoints to get edge-length variations; that slice misses pure shear and thus transverse-traceless weak-field modes.
The ambient geometry is the canonical encoded $5\times 5\times 5$ periodic Freudenthal torus. Edge perturbations are real functions on the typed periodic edges. Longitudinal gauge indices are pairs (periodic vertex, direction in $\mathrm{Fin},3$), matching $D=3$ spatial dimensions.
This structure upgrades an earlier abstract longitudinal split: the conformal part is no longer an arbitrary map, but is generated from encoded vertex-delta coefficients. The residual after subtracting conformal and longitudinal projections is the TT piece used downstream in Gram and normal-equation data.
proof idea
No proof body: this is a structure definition packing three maps and three propositional fields. Inhabitants must supply conformal and gauge coefficient extractors, a residual TT map, orthogonality of that residual to every conformal generator and every longitudinal gauge generator under the periodic edge inner product, and pointwise reconstruction of any edge perturbation as conformal image plus gauge image plus residual. Downstream constructors discharge these fields from Gram/load/normal-equation data.
why it matters
This is the coefficient-level interface for the concrete longitudinal split in Track 1.D. Downstream MasterTheoremHandoffIntegration endpoints (Gram kernel criterion, generator-map-zero, load image/solver, range closed/criterion, and the longitudinal coefficient projector reduction endpoint) thread instances of this data into Track 7 handoffs.
In the Recognition gravity program it separates the pure shear/TT residual from the vertex-conformal and longitudinal gauge sectors on a finite periodic complex, which is the discrete stand-in for TT gravitational-wave content missing from the conformal ansatz. It sits under the forced $D=3$ spatial setting and the eight-tick/Clifford scaffolding only indirectly (via the ambient torus and constants), not as a T0–T8 step itself.
It does not close the full continuum TT projection; it fixes the finite $5^3$ coefficient-projector contract that later Gram solvability and range results must inhabit.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.