periodicTTNormalEquationResidual5_eq_sub_generatorMap
plain-language theorem explainer
On the 5×5×5 periodic Freudenthal torus, the combined TT normal-equation residual at each edge equals the edge perturbation minus its reconstruction by the combined conformal-plus-longitudinal generator map. Track 1.D workers cite this when reducing residual pairings to load-minus-Gram form. The proof unfolds the residual definitions, rewrites by the generator-map split, and finishes by ring.
Claim. For any coefficient projector from edge perturbations to combined normal-equation indices, any edge perturbation $\varepsilon$, and any edge $e$ of the $5\times 5\times 5$ periodic Freudenthal torus, the combined TT normal-equation residual of $\varepsilon$ at $e$ equals $\varepsilon(e)$ minus the value at $e$ of the combined generator map applied to the projected coefficients of $\varepsilon$.
background
Track 1.D builds the tensor/shear sector that the conformal (vertex-scalar) ansatz cannot reach. Conformal edge-length variations average endpoint potentials and therefore miss pure shear; transverse-traceless gravitational-wave modes live outside that slice. The module therefore treats independent edge perturbations on the typed periodic Freudenthal edges as the primary unknowns.
The ambient geometry is the canonical encoded $5\times 5\times 5$ periodic Freudenthal torus. An edge perturbation is simply a real function on those edges. The combined normal-equation index is a sum type: conformal vertex-delta generators on one side and fixed longitudinal vertex-vector generators on the other. A coefficient projector maps an edge perturbation to coefficients on that combined index set.
The residual under study is the pointwise difference between the input perturbation and the reconstruction produced by feeding those coefficients through the combined generator map. That residual is the object later paired against generators to obtain the discrete load-minus-Gram identity.
proof idea
Short tactic proof. Unfold the combined residual and the longitudinal coefficient residual so both sides are expressed in elementary arithmetic of the edge values and the generator-map pieces. Rewrite with the already-proved split identity for the combined generator map (conformal part plus longitudinal part). The two sides then match by ring.
why it matters
This identity is the edgewise algebraic step that the next theorem consumes: pairing the combined residual against a generator equals load minus Gram. That parent result is the discrete normal-equation statement for the mixed conformal-plus-longitudinal gauge on the periodic torus.
In the broader Recognition gravity track, the conformal ansatz alone cannot represent pure shear or TT wave modes. Establishing a clean residual-versus-generator-map relation on independent edge perturbations is the first concrete bookkeeping needed before a hinge-aware Regge TT analysis can be closed on the Freudenthal lattice. The result sits inside the Track 1.D scaffold that separates shear from vertex-conformal strain and prepares the ground for a full weak-field metric sector beyond scalar potentials.
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