PeriodicTTNormalEquationSolutionData5
plain-language theorem explainer
Packages finite TT normal-equation solution data on the concrete periodic triangulation: a coefficient map from edge perturbations to generator indices whose residual is orthogonal to every conformal and longitudinal generator. Track 1.D handoff endpoints cite it to close the Gram-system reduction toward the TT projector. Pure structure definition; no proof content.
Claim. A record of a coefficient map $c$ sending each edge perturbation $\varepsilon$ and each normal-equation index $i$ (conformal vertex-delta or longitudinal gauge) to a real, such that for all $\varepsilon$ and $i$ the residual of $\varepsilon$ after the linear combination prescribed by $c$ is orthogonal, under the periodic edge inner product, to the $i$-th normal-equation generator.
background
Track 1.D isolates the tensor/shear sector that the Track 1.B conformal ansatz cannot reach. Vertex-scalar potentials induce only averaged endpoint edge strains; pure shear and transverse-traceless gravitational-wave modes need independent edge perturbations. The module separates those from vertex-conformal variations and records the elementary rectangle obstruction for the conformal slice.
On the concrete finite periodic Freudenthal torus, edge perturbations are real functions on typed edges. The normal-equation index set is the disjoint sum of conformal vertex-delta generators and longitudinal vertex-vector generators. The residual of a trial coefficient map is the edge perturbation minus the corresponding linear combination of those generators; the normal equations demand that this residual be orthogonal to every generator under the periodic edge inner product.
That finite Gram system is exactly the linear-algebra problem left by the TT decomposition track once the continuous continuum limit has been replaced by the discrete periodic complex.
proof idea
Definitional structure, not a proved theorem. It declares two fields: a coefficient projector from edge perturbations and normal-equation indices into $\mathbb{R}$, and the universal normal-equation identity that the residual built from that projector is orthogonal to every generator. No tactics or lemmas are invoked; inhabitants are supplied later by constructors such as ofGramSystemData in the handoff chain.
why it matters
This structure is the normal-equation endpoint of the Track 1.D finite TT split. Downstream, MasterTheoremHandoffIntegration threads it through the Gram-load solver, Gram-system solution, and projector data: track1D_tt_gram_load_solver_reduction_endpoint_holds, track1D_tt_gram_load_image_reduction_endpoint_holds, and the kernel/range criterion endpoints all require a nonempty instance. Those endpoints are consumed by Track 7 of the master gravity handoff.
In the broader Recognition framework the object sits inside the discrete weak-field gravity program on the forced $D=3$ lattice (T8), where the eight-tick octave and phi-ladder supply the background units. It does not itself force dimensions or constants; it packages the linear algebra that lets shear modes be projected off the conformal-plus-longitudinal span so that a genuine TT remainder can be identified.
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