Track1DTTProjectorDataReductionEndpoint
plain-language theorem explainer
Concrete finite projector data (conformal, gauge, and TT maps on periodic edge perturbations at N=5) is enough to obtain the periodic Freudenthal orthogonal decomposition target. Track 1.D and Track 7 handoff authors cite this as the reduction interface between projector construction and the decomposition claim. It is a Prop packaging: an implication from projector data to the decomposition target, not a constructive proof.
Claim. For every gauge-potential type $G$ and every gauge map $g:G\to\mathbb{R}^{E_5}$ (real edge perturbations on the typed periodic Freudenthal edges at $N=5$), the existence of concrete conformal, gauge, and TT projectors satisfying the membership and pointwise-reconstruction axioms implies the periodic Freudenthal conformal/gauge/TT orthogonal decomposition target at $N=5$.
background
Module context is Gravity Track 7 fork-handoff integration: a receipt lane that records what parallel forks prove without upgrading the discovery claim. Fork A covers Track 1.B Schläfli stationarity reduction at $N=5$; Track 1.D sits in the tensor-shear sector as the conformal/gauge/TT split.
PeriodicEdgePerturbation5 is the space of real-valued perturbations on typed periodic Freudenthal edges. The decomposition target asserts a raw splitting of every edge perturbation into conformal, gauge, and TT parts, with TT read as finite orthogonality to the conformal and gauge subspaces. The remaining load is constructing three projectors.
PeriodicTTProjectorData5 packages that load: three maps (conformal, gauge, TT projectors) plus membership and reconstruction obligations for a chosen gauge operator. The quantum Projector structure (idempotent observable) is the abstract ancestor; here the data are finite combinatorial maps on edge perturbations, not Hilbert-space operators.
proof idea
Definition-only packaging, not a tactic proof. The Prop is the universal implication: for any gauge-potential type and gauge map, PeriodicTTProjectorData5 yields PeriodicFreudenthalTTOrthogonalDecompositionTargetAtN5. The sibling theorem that discharges the endpoint applies periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_projectorData after introducing the gauge data and the projector-data witness.
why it matters
Closes the Track 1.D reduction interface that Track 7 consumes. Downstream, track1D_tt_projector_data_reduction_endpoint_holds proves the endpoint, and ForkHandoffIntegrationCert records fork handoffs (A–F) including Track 1 reduction/interface packages. The cert doc is explicit: the Track 1 result is a reduction package, not closure of open Schläfli leaves.
In the gravity lane this separates projector construction from the orthogonal-decomposition claim at finite $N=5$, so later work can supply the three maps without reopening the target statement. It does not touch T0–T8 forcing, RCL, or the mass ladder; it is local to the tensor-shear / Freudenthal stencil side of the master-theorem handoff.
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