Track1DTTGeneratorMapProjectorReductionEndpoint
plain-language theorem explainer
Track 1.D endpoint: whenever the gauge map is built from finite generators, the gauge-span, finite-generator, and full TT projector packages are inhabited and the Freudenthal TT orthogonal decomposition target at N=5 holds. Gravity Track 7 cites it as a handoff receipt, not a discovery claim. The body is a pure Prop packaging four conjuncts over an arbitrary finite index type.
Claim. For every finite index type $G$ and every $N=5$ generator-map projector datum $D$, writing $g$ for the periodic gauge-generator map induced by $D$, the following all hold: the gauge-generator projector data for $g$ is inhabited; the finite-generator projector data for $g$ is inhabited; the full TT projector data for $g$ is inhabited; and the Freudenthal TT orthogonal decomposition target at $N=5$ holds for $g$ (a raw edge-perturbation splitting into conformal, gauge, and TT-orthogonal parts exists).
background
Module setting is Gravity Track 7 fork-handoff integration: a receipt lane that records what parallel forks prove without upgrading the discovery claim. Track 1.D sits in the tensor-shear / TT sector on the periodic $N=5$ torus.
The key upstream target is the honest Track 1.D decomposition: TT means finite orthogonality to the conformal and gauge subspaces, and "the remaining mathematical load is the construction of the three projectors." The gauge map here is the periodic generator map induced from finite generators, so the separate gauge-span argument becomes automatic once projector data is assembled from that map.
Spatial dimension $D=3$ (T8) and the bridge ratio $K=\varphi^{1/2}$ appear only through the ambient torus and constant infrastructure; they are not free parameters of this endpoint.
proof idea
Definitional Prop, not a proved theorem. It quantifies over a finite index type and a PeriodicTTGeneratorMapProjectorData5 datum, then conjoins four obligations on the induced periodicGaugeGeneratorMap5: nonempty gauge-generator projector data, nonempty finite-generator projector data (indexed by the torus vertex set), nonempty full TT projector data, and the Freudenthal TT orthogonal decomposition target at $N=5$.
The companion theorem discharges it by constructing the gauge-generator package from the generator-map datum, then the finite-generator package from that, and chaining the resulting nonempty witnesses into the four conjuncts.
why it matters
Supplies the Track 1.D generator-map projector-reduction leaf consumed by Track 7. Downstream, the holds theorem asserts the Prop, and ForkHandoffIntegrationCert folds Track 1 reduction/interface packages into the multi-fork receipt (alongside many-body, Page-capacity, $w(z)$, and falsifier-sensitivity handoffs).
Per the module doc, this does not upgrade the discovery claim and does not close open Schläfli or displacement-class leaves; it records that the finite-generator route makes the gauge-span projector path automatic. Framework context is the gravity master-theorem structural lane on the $D=3$ recognition geometry, not a new forcing-chain step (T0–T8).
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