track1D_conformal_generator_span_endpoint_holds
plain-language theorem explainer
On the N=5 periodic torus, every edge perturbation in the conformal log-subspace is a finite linear combination of encoded-vertex conformal generators. Gravity Track 7 cites this as the Track 1.D handoff endpoint before gauge-projector data is supplied. The proof is a one-line term wrapper of the TensorShearSector spanning theorem.
Claim. For every periodic edge perturbation $c$ on the $N=5$ torus, if $c$ lies in the periodic conformal log-subspace, then there exist real coefficients indexed by the torus vertices such that $c(e)=\sum_v a_v\,G_v(e)$ for every edge $e$, where $G_v$ is the encoded-vertex conformal generator at vertex $v$.
background
Track 7 is the fork-handoff integration lane for gravity: it records what parallel endpoints already prove without upgrading the discovery claim. Track 1.D is the conformal-generator span leaf on the $N=5$ periodic torus.
A periodic edge perturbation assigns a real weight to each edge of the $N=5$ periodic complex. The periodic conformal log-subspace is the linear slice of those perturbations that arise as conformal (log-scale) variations. The encoded-vertex conformal generators $G_v$ are an explicit finite family, one per torus vertex, living in that slice.
The endpoint proposition asserts exact span: membership in the conformal log-subspace is equivalent to existence of vertex coefficients realizing $c$ as $\sum_v a_v G_v$. Spatial dimension $D=3$ (forced by T8) fixes the underlying hypercube combinatorics that feed the torus construction, but the statement itself is purely about the $N=5$ conformal slice.
proof idea
One-line term proof: the claim is definitionally identical to TensorShearSector.periodicConformalLogSubspace5_spanned_by_encodedVertexGenerators, so the theorem is that lemma under the Track 1.D endpoint name. No extra algebra or case split is performed here.
why it matters
This is the Track 1.D receipt consumed by Track 7. Downstream, forkHandoffIntegrationCert packages the parallel fork endpoints; the module doc states that after the conformal span is fixed by vertex generators, it is enough to supply gauge-generator projector data. Closing the span leaf therefore clears the linear-algebra obstruction on the conformal slice and leaves projector/gauge data as the remaining handoff input.
In the broader RS gravity stack this sits under the master-theorem structural lane rather than under T5–T8 forcing: it does not re-derive $J$, $\varphi$, the eight-tick octave, or $D=3$. It records a finite spanning family on the $N=5$ conformal sector so later residual and stationarity reductions can quote an explicit basis instead of an abstract subspace membership hypothesis.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.