Track1DTTOrthogonalSurfaceEndpoint
plain-language theorem explainer
Track 1.D surface endpoint: on the N=5 periodic Freudenthal complex, TT modes are finite orthogonality to the conformal/log-strain slice and a caller-supplied gauge slice. Gravity Track 7 handoff and tensor-shear decomposition work cite this package. It records zero-perturbation orthogonality plus the implication from the orthogonal split target to the classical three-way split; actual projector construction stays open.
Claim. For every gauge-potential type $G$ and every map $g: G \to$ (real periodic edge perturbations at $N=5$), the zero perturbation is transverse-traceless in the finite-orthogonality sense (orthogonal to every conformal/log-strain mode and every image of $g$), and if the orthogonal Freudenthal decomposition target holds for $(G,g)$, then the classical three-way decomposition target holds with conformal part in the periodic conformal/log subspace, gauge part in the supplied gauge subspace, and TT part in the finite-orthogonality TT class.
background
This module is the Track 7 integration-lane receipt for parallel fork handoffs (A–F). It records what the new endpoints prove without upgrading the discovery claim, and leaves remaining Track 1 displacement-class leaves as open dependencies.
In the tensor-shear sector, edge data live on typed periodic Freudenthal edges at $N=5$: a periodic edge perturbation is a real function on those edges. The conformal/log-strain subspace consists of perturbations that arise as encoded conformal edge log-strains of a vertex potential. A gauge subspace is the image of a caller-supplied forward map from an abstract gauge-potential type (the longitudinal/diffeomorphism discretization is not fixed here).
TT is defined by finite orthogonality: a perturbation is TT when its periodic-edge inner product vanishes against every conformal/log mode and every gauge mode. The classical decomposition target asks for a raw splitting into conformal, gauge, and TT parts under three predicates. The orthogonal variant replaces the TT predicate by this finite-orthogonality class; its doc states the remaining load is construction of the three projectors.
proof idea
Definitional packaging, not a derived proof. The Prop is a universal quantification over gauge-potential type and gauge map, conjoined of two facts: (i) the zero edge perturbation satisfies the finite TT-orthogonality predicate for that gauge data; (ii) the orthogonal Freudenthal decomposition target implies the classical three-way decomposition target, instantiated with the periodic conformal/log subspace, the supplied gauge subspace, and the finite-orthogonality TT predicate. No tactics; the body is the claim surface consumed downstream.
why it matters
Fills the Track 1.D handoff surface for tensor/TT orthogonality that Track 7 consumes. The companion theorem track1D_tt_orthogonal_surface_endpoint_holds discharges the Prop by zero-orthogonality and the orthogonal-to-classical target implication; the integration certificate ForkHandoffIntegrationCert records fork handoffs including Track 1 reduction/interface packages without closing open Schläfli leaves.
In the Recognition gravity stack this is the honest interface between discrete shear data and a continuum-style TT split: TT is no longer a named residual class but finite orthogonality to conformal and gauge slices on the periodic complex. Projector construction remains the next tensor-sector proof, so the master theorem still uses structural witnesses where the plan requires them. No direct T0–T8 forcing step; the link is the discrete gravity/shear sector feeding the structural master theorem path.
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