periodicTTOrthogonal5_zero
plain-language theorem explainer
The zero edge-length perturbation on the period-5 Freudenthal torus is transverse-traceless in the finite sense: it is orthogonal to every conformal-log mode and every gauge mode under the periodic edge inner product. Anyone wiring Track 1.D TT projectors or the master handoff cites this as the trivial basepoint of the TT cone. The proof is a two-branch constructor that applies left-vanishing of the edge inner product.
Claim. For any gauge-potential type $G$ and any map $\gamma: G \to (\text{periodic edges}_5 \to \mathbb{R})$, the zero edge perturbation $\varepsilon \equiv 0$ is TT-orthogonal: $\langle 0, c \rangle_5 = 0$ for every $c$ in the periodic conformal-log subspace, and $\langle 0, g \rangle_5 = 0$ for every $g$ in the image gauge subspace of $\gamma$.
background
Track 1.D opens the tensor/shear sector that Track 1.B's vertex-conformal ansatz cannot reach. The conformal slice assigns one scalar per vertex and averages endpoints onto edges; pure shear and transverse-traceless gravitational-wave modes live outside that slice. This module therefore treats independent edge perturbations on the typed periodic Freudenthal complex and isolates them from vertex-conformal modes.
A periodic edge perturbation is simply a real function on the period-5 edge set. TT is defined finitely: an edge field $\varepsilon$ is TT-orthogonal when it is orthogonal, under the periodic edge inner product, both to the conformal-log subspace and to a supplied gauge subspace (the image of an arbitrary gauge map). The zero field is the canonical basepoint of that cone.
The ambient geometry is the $D=3$ recognition setting forced by the T8/T9 chain; the period-5 torus is the finite triangulation carrier used for the Track 1.D surface endpoints.
proof idea
Unfold the conjunction in the TT-orthogonality predicate. The first branch takes an arbitrary conformal-log edge field $c$ and applies left-vanishing of the periodic edge inner product at zero. The second branch does the same for an arbitrary gauge-subspace field $g$. No conformal or gauge structure is used beyond the type of the witnesses; bilinearity (or the dedicated zero-left lemma) finishes both goals.
why it matters
This is the trivial endpoint of the honest Track 1.D TT decomposition target: TT means finite orthogonality to conformal and gauge subspaces, with the remaining load being construction of the three projectors. Downstream, track1D_tt_orthogonal_surface_endpoint_holds packages this zero witness with the N=5 Freudenthal decomposition target and hands the surface endpoint to Track 7.
In the Recognition gravity program the conformal ansatz alone cannot carry pure shear, so the TT cone must be opened separately before weak-field gravitational-wave modes can sit on the ledger. The result is elementary but load-bearing: every nontrivial TT projector and every master-theorem handoff that quotes Track 1.D orthogonality needs a proved basepoint. It does not itself force $D=3$ or the eight-tick octave; those enter through the ambient geometry and the period-5 carrier.
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