track1D_tt_gram_load_image_reduction_endpoint_holds
plain-language theorem explainer
Given finite Gram-load image data at N=5, every physical transverse-traceless load lies in the image of the finite Gram operator: a load-subspace solver and Gram-system solution exist, a longitudinal gauge projector is inhabited, and the Freudenthal TT orthogonal decomposition target closes. Track 7 handoff consumers cite this as the Track 1.D endpoint. The proof is a pure constructor chain from load-image data through solver, Gram, normal, coefficient, and generator projectors to the decomposition witness.
Claim. For any periodic TT Gram-load image datum $D$ at $N=5$, there exist a nonempty load solver, a nonempty Gram-system solution, a nonempty longitudinal-gauge projector on maps from the $N=5$ longitudinal gauge index set into $\mathbb{R}$, and the Freudenthal TT orthogonal decomposition target at $N=5$ holds on that gauge space. Equivalently: every physical TT load lies in the image of the finite Gram operator, so a solver on the load subspace exists and the TT split closes.
background
In the RS gravity stack, Track 1.D treats the transverse-traceless (TT) sector of the discrete shear analysis. The finite Gram operator is the inner-product structure on edge loads; its image criterion says every physical TT load is realizable as a Gram action, which is the solvability condition for the discrete TT system and the algebraic half of closing the TT split.
The ambient setting is the Track 7 fork-handoff integration module. That module does not upgrade discovery claims; it packages parallel endpoints (Schläfli stationarity, physical residual/Bianchi, many-body amplitude lift, Page capacity, dark-energy $w(z)$, falsifier sensitivity) into one receipt, leaving remaining Track 1 displacement-class leaves as the next dependency. Spatial dimension is the forced $D=3$ of the T8 landmark.
Upstream carriers live in the tensor-shear sector: periodic TT Gram/load solver data, normal-equation and longitudinal-coefficient data, and the Freudenthal orthogonal decomposition target. The edge-load map on hinge data supplies the concrete load action that the Gram image is asked to cover.
proof idea
Term-mode constructor chain, no search. From the input Gram-load image datum $D$, build successively: load-solver data via ofLoadImageData, Gram-system solution via ofLoadSolverData, normal-equation solution via ofGramSystemData, longitudinal-coefficient solution and projector via ofNormalEquationData / ofSolutionData / ofCoefficientData, then generator-map, gauge-generator, and finite-generator projector data. Each step is an of* structure map in the tensor-shear sector. The final term packages nonempty witnesses for the solver, Gram system, and TT projector, together with the Freudenthal orthogonal-decomposition target obtained from $D$ by periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_gramLoadImageData.
why it matters
This is the explicit Track 1.D Gram-load image endpoint consumed by Track 7. It feeds forkHandoffIntegrationCert, the integration-lane certificate that records what the parallel forks prove without upgrading the discovery claim. Closing the Gram-image criterion for physical TT loads supplies the algebraic half of the TT split: once every load is in the Gram image and the longitudinal projector exists, the Freudenthal decomposition can separate pure TT content from gauge.
In the broader RS gravity program this sits under the discrete Regge/TT analysis supporting the master-theorem handoff. It does not itself force $D=3$ or the eight-tick octave; those enter as ambient constants from the forcing chain. Per the module doc, remaining Track 1 displacement-class leaves stay open as the next dependency after this receipt.
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