track1D_tt_orthogonal_surface_endpoint_holds
plain-language theorem explainer
Track 1.D closes: on the N=5 periodic edge lattice, the zero perturbation is TT-orthogonal to any caller gauge slice, and the Freudenthal conformal/gauge/TT orthogonal-decomposition target implies the full decomposition target. Gravity and discrete GR workers cite it as the surface endpoint for transverse-traceless data before projectors are built. The proof is a two-lemma pair after introducing the gauge type and map.
Claim. For every gauge-potential type $G$ and every map from $G$ into the space of periodic edge perturbations at $N=5$, the zero perturbation is TT-orthogonal to the conformal and gauge slices, and if the periodic Freudenthal TT-orthogonal decomposition target holds for that gauge data, then the full periodic Freudenthal conformal/gauge/TT decomposition target holds (with the fixed periodic conformal-log subspace, the induced gauge subspace, and the TT-orthogonal predicate).
background
This module is the Track 7 integration-lane receipt for parallel fork handoffs. It records what each fork endpoint proves without upgrading the discovery claim; remaining Track 1 displacement-class leaves stay as the next dependency.
Track 1.D is the tensor/TT surface. The endpoint proposition packages two facts for arbitrary gauge-potential type and gauge map into periodic edge perturbations at $N=5$: (i) the zero field is TT-orthogonal to the periodic conformal slice and the caller gauge slice; (ii) the orthogonal-decomposition target implies the full Freudenthal conformal/gauge/TT decomposition target at $N=5$. TT is thus represented as finite orthogonality; building actual projectors is deferred to later tensor-sector work.
Spatial dimension $D=3$ is the T8/T9 constant used throughout the gravity stack. Upstream tensor-shear lemmas supply the zero-mode orthogonality and the target-implication bridge used here.
proof idea
Term proof. Introduce the gauge-potential type and the gauge map into periodic edge perturbations at $N=5$. Return a pair:
periodicTTOrthogonal5_zeroshows the zero perturbation satisfies the TT-orthogonality predicate against that gauge data.periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_to_targetturns the orthogonal-decomposition target hypothesis into the full periodic Freudenthal conformal/gauge/TT decomposition target, with the fixed conformal-log subspace, the induced gauge subspace, and the TT-orthogonal predicate.
No further case analysis; the endpoint is exactly that conjunction for all gauge data.
why it matters
Feeds forkHandoffIntegrationCert, the integration-lane certificate that bundles Track 1–6 handoff endpoints. Without this surface fact, Track 7 cannot record that TT data on the $N=5$ periodic lattice is already finite orthogonality to conformal and gauge slices.
In the Recognition gravity program this is the Track 1.D leaf of the master-theorem handoff: discrete Regge/Freudenthal geometry at the eight-tick-compatible $N=5$ stencil, with $D=3$ spatial dimensions forced by T8. It does not yet construct projectors; the doc-comment states that constructing actual projectors remains the next tensor-sector proof. Sibling endpoints (Schläfli reduction, displacement-class stationarity, many-body amplitude lift, Page capacity, $w(z)$ band) sit beside it in the same certificate.
Closes the concrete finite-projector-data sufficiency claim for the periodic Freudenthal decomposition target while leaving projector construction and remaining displacement-class leaves open.
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