PeriodicTTLongitudinalProjectorData5
plain-language theorem explainer
Packages the finite-dimensional split of periodic edge perturbations into conformal, longitudinal-gauge, and transverse-traceless (TT) parts on the typed Freudenthal torus. Track 1.D shear/tensor work and the Gram-system handoff to Track 7 cite this as the exact projector input. It is a pure structure: three maps plus membership, orthogonality, and reconstruction axioms; no proof body.
Claim. A data bundle of maps on periodic edge perturbations $\varepsilon$: a conformal projector $P_c$, a longitudinal gauge coefficient map $c_g$ (indexed by vertex $\times$ spatial direction), and a TT projector $P_{TT}$, such that $P_c(\varepsilon)$ lies in the vertex-conformal log-strain subspace, $P_{TT}(\varepsilon)$ is orthogonal under the periodic edge inner product to every conformal generator and every longitudinal gauge generator, and pointwise $\varepsilon = P_c(\varepsilon) + G(c_g(\varepsilon)) + P_{TT}(\varepsilon)$ reconstructs the original perturbation.
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.
Here edge perturbations are real functions on the typed periodic Freudenthal edges. The conformal log-strain subspace consists of those edge maps that arise from a vertex potential via the encoded conformal edge log-strain. Longitudinal gauge indices are pairs (periodic vertex, spatial direction in $\mathrm{Fin},3$), matching the forced spatial dimension $D=3$. The periodic edge inner product is the finite Gram form used throughout the shear track.
This structure is the exact finite-dimensional decomposition input still owed by Track 1.D: projector data for the concrete periodic longitudinal gauge basis, with the conformal part generated from encoded vertex-delta coefficients rather than left arbitrary.
proof idea
No proof: the declaration is a structure (definitional interface). It bundles three maps on periodic edge perturbations together with four propositional fields: conformal image lands in the conformal log-strain subspace; the TT image is orthogonal to every conformal generator; the TT image is orthogonal to every longitudinal gauge generator; and the three pieces sum to the original edge perturbation pointwise via the longitudinal gauge map applied to the coefficient projector. Downstream constructions inhabit this interface; the structure itself only states the axioms.
why it matters
This is the concrete projector interface that closes the Track 1.D longitudinal split and feeds the Gram-system reduction chain into Track 7. Downstream handoff theorems (Gram kernel criterion, kernel generator-map-zero, load image, load solver, range-closed, range criterion, and the longitudinal coefficient projector reduction endpoint) all consume data built from this decomposition.
In the Recognition gravity program it separates the pure shear/TT sector from vertex-conformal and longitudinal gauge junk, which is required before weak-field tensor modes can be treated on the same footing as the conformal ansatz. It sits on the $D=3$ spatial forcing (T8) via the three-component gauge index. It does not by itself prove existence of the projectors; it fixes the contract that existence and Gram solvability proofs must discharge.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.