track1D_tt_gram_range_closed_endpoint_holds
plain-language theorem explainer
The finite transverse-traceless Gram range criterion holds at lattice size N=5 for the fixed conformal-plus-longitudinal generator family, so the concrete TT projector split closes. Track 7 gravity handoff cites this as the Track 1.D endpoint receipt. The proof is a pure data-chain: inhabit each successive TensorShearSector bundle from the proved range criterion through kernel, load, normal equations, and projectors to the Freudenthal orthogonal target.
Claim. At lattice size $N=5$, the periodic TT Gram range criterion, its kernel criterion, and load-image data are inhabited; the TT projector data for the periodic longitudinal gauge map is inhabited; and the Freudenthal TT orthogonal decomposition target holds on the longitudinal gauge index space. Equivalently, the finite Gram-range endpoint for the combined conformal plus longitudinal generator family is closed.
background
This module is the Track 7 fork-handoff integration lane: it records proved endpoints from parallel gravity tracks without upgrading the discovery claim. Track 1.D concerns the transverse-traceless (TT) sector of the discrete Regge/Lichnerowicz analysis on a periodic lattice of size $N=5$.
The endpoint proposition packages five facts: nonempty finite Gram-range criterion data, kernel-criterion data, load-image data, TT projector data for the fixed periodic longitudinal gauge map, and the Freudenthal TT orthogonal decomposition target on the longitudinal gauge index space. In plain terms, the combined conformal-plus-longitudinal generator family has a proved finite Gram range, so the concrete TT projector split closes.
Upstream, the chain begins from a proved periodic TT Gram range criterion at $N=5$ in the tensor-shear sector. Spatial dimension $D=3$ (forced by the T8/T9 forcing chain) sits in the ambient geometry, but this endpoint is local to the discrete TT Gram/projector algebra rather than a continuum limit statement.
proof idea
Term-mode inhabitation chain. Start from the already-proved periodic TT Gram range criterion data at $N=5$. Lift successively through named constructors: kernel criterion from range data; load-image from kernel; load-solver from load-image; Gram-system solution from load-solver; normal-equation solution from Gram system; longitudinal coefficient solution from normal equations; coefficient projector from that solution; longitudinal projector from coefficients; generator-map projector from longitudinal data; gauge-generator projector from the map; finite-generator projector from the gauge data.
The final term is a five-component pair: nonempty wrappers for range, kernel, and load-image data; projector data built from the finite-generator bundle; and the Freudenthal TT orthogonal decomposition target obtained from the proved Gram-range criterion. No new analytic estimate is proved here; the work is packaging the existing $N=5$ shear-sector certificates into the Track 1.D endpoint Prop.
why it matters
Track 7 consumes this as the Track 1.D handoff leaf inside forkHandoffIntegrationCert, the integration-lane certificate that aggregates fork endpoints (Schlaefli reduction, displacement-class leaves, many-body amplitude lift, Page capacity, dark-energy $w(z)$, falsifier sensitivity). The module doc is explicit: the lane records exactly what the new endpoints prove and leaves remaining Track 1 displacement-class work as the next dependency.
In the Recognition gravity stack, closing the finite TT Gram range at $N=5$ is the discrete gate that lets bilinear and quadratic TT energy comparisons proceed once the Regge TT Hessian and lattice Lichnerowicz operators are identified on TT modes. It does not by itself force continuum GR; it seals the projector split needed for the shear-sector energy match on the fixed generator family. Framework landmarks in view are the $D=3$ spatial setting and the discrete eight-tick/octave scaffolding that motivates the periodic lattice analysis, but the theorem itself is a finite-$N$ algebraic endpoint, not a T0–T8 forcing step.
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