track1D_tt_gram_kernel_criterion_reduction_endpoint_holds
plain-language theorem explainer
Given finite Gram-kernel criterion data on the periodic transverse-traceless sector at N=5, every physical TT load lies in the Gram image, a load solver exists, longitudinal projector data is inhabited, and the Freudenthal orthogonal TT split closes. Track 7 gravity handoff cites this as the Track 1.D endpoint. The proof is a pure constructor pipeline through TensorShearSector intermediate data, then a package of nonempty witnesses plus the Freudenthal target lemma.
Claim. If $D$ is finite Gram-kernel criterion data for the periodic transverse-traceless (TT) sector at $N=5$, then the physical TT load space is nonempty in the Gram image, a load solver exists, longitudinal gauge projector data for the periodic longitudinal gauge map is inhabited, and the Freudenthal orthogonal TT decomposition target at $N=5$ holds for real-valued maps on the periodic longitudinal gauge index set.
background
This module is the Track 7 integration-lane receipt for parallel gravity fork handoffs. It records what each new endpoint proves without upgrading the discovery claim; remaining Track 1 displacement-class leaves stay open. Track 1.D is the Gram-kernel criterion reduction on the periodic TT (tensor/shear) sector.
The endpoint proposition says: from finite Gram-kernel criterion data at $N=5$, one obtains nonempty Gram-load-image data, nonempty load-solver data, nonempty longitudinal projector data for the periodic longitudinal gauge map, and the Freudenthal orthogonal TT decomposition target on real maps from the longitudinal gauge index. Informally, if Gram-kernel coefficient vectors generate the zero edge perturbation, the load-annihilates-kernel half of the finite criterion is automatic and the TT split closes.
Upstream geometry sits in the tensor-shear analysis stack (edge TT load maps, Regge hinge-aware zero modes, periodic Freudenthal stencils). Spatial dimension is the forced $D=3$ from the foundation chain (T8).
proof idea
Term-mode pipeline on input criterion data $D$. Build successive TensorShearSector witnesses: load-image data from the kernel criterion; load-solver from the image; Gram-system solution from the solver; normal-equation solution from the Gram system; longitudinal coefficient solution from the normal equation; coefficient projector, then longitudinal projector, generator-map projector, gauge-generator projector, and finite-generator projector, each by the corresponding of* constructor.
The final witness is a four-part package: nonempty load-image, nonempty load-solver, nonempty projector data obtained from the finite-generator stage, and the Freudenthal orthogonal decomposition target lemma applied directly to $D$. No extra analytic work; the criterion data already carries the hypotheses each constructor needs.
why it matters
Supplies the Track 1.D field of the fork handoff integration certificate, the module-level receipt that Track 7 consumes when assembling parallel gravity endpoints (Schläfli/stationarity reductions, physical residual, many-body amplitude lift, Page capacity, dark-energy $w(z)$, falsifier sensitivity). Downstream packaging treats this as the explicit Gram-kernel criterion endpoint for the TT split at $N=5$.
In the Recognition gravity program this closes the finite-criterion half of the periodic TT shear analysis: Gram image membership for physical loads plus Freudenthal orthogonal decomposition, so the discrete tensor sector can hand off a clean projector/decomposition interface. It does not finish the full master theorem; displacement-class leaves remain the next dependency, consistent with the module stance that integration records endpoints rather than upgrading discovery.
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