PeriodicTTGramLoadSolverData5
plain-language theorem explainer
Packages a finite linear solver for the transverse-traceless Gram normal equations on the 5-cell periodic torus: a map from load vectors to coefficient vectors that inverts the Gram operator on every load induced by an edge perturbation. Track 1.D handoff endpoints cite it to close the TT split and supply projectors. Pure structure definition; no proof body.
Claim. A data package consisting of a map $S$ sending any real-valued load on the combined conformal-plus-longitudinal generator index set to a coefficient vector on the same index set, together with the identity that the finite TT Gram operator applied to $S(\ell_\varepsilon)$ recovers $\ell_\varepsilon$ for every edge-perturbation load $\ell_\varepsilon$ and every generator index.
background
Track 1.D opens the tensor/shear sector of the weak-field metric. Track 1.B's conformal ansatz puts one scalar at each vertex and varies edge lengths by averaging endpoint potentials; that slice cannot represent pure shear, so it misses transverse-traceless gravitational-wave modes. This module separates independent edge perturbations from vertex-conformal ones and records the elementary rectangle obstruction for the conformal ansatz.
The finite TT Gram operator is the normal-equation matrix of the discrete first-variation problem on the 5-periodic torus. Its combined index set joins fixed conformal vertex-delta generators with fixed longitudinal vertex-vector generators. Loads are the right-hand sides obtained by pairing an edge perturbation against those generators. A solver is a right inverse of the Gram map on the subspace of such physical loads.
Spatial dimension is the forced $D=3$ of the Recognition chain; the periodic complex is the discrete arena on which the shear sector is checked before continuum handoff.
proof idea
Definitional structure with two fields and no proof body. The first field is the solver map (load functional to coefficient functional on the combined normal-equation index). The second field is the pointwise recovery identity: Gram applied to the solved coefficients equals the load, for every edge perturbation and every index. Downstream constructors (e.g. from load-image data) inhabit the type; the structure itself only packages the finite linear-algebra obligation.
why it matters
This is the concrete finite-algebra interface that Track 1.D must discharge before the TT decomposition closes. Downstream, Track1DTTGramLoadSolverReductionEndpoint takes an inhabitant and produces nonempty Gram-system, normal-equation, and projector data. Load-image and kernel-criterion reduction endpoints construct a solver from weaker geometric hypotheses (loads lie in the Gram image; kernel criterion) and then feed the same chain. Those endpoints are consumed by Track 7 master-theorem handoff.
In the Recognition gravity program this isolates the remaining discrete linear algebra after the conformal/shear split and the rectangle obstruction: once every physical TT load is solvable, the projector onto the TT complement exists and the shear sector is no longer scaffolding. It does not itself force $D=3$ or the eight-tick octave; those enter only as ambient geometry of the periodic complex.
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