periodicTTNormalEquationGeneratorMap5
plain-language theorem explainer
Defines the linear map sending coefficient vectors on the combined conformal-plus-longitudinal generator index to edge-length perturbations on the period-5 Freudenthal torus. Anyone assembling the finite TT normal equations or Hessian/Lichnerowicz residual certificates cites it. The body is a one-line specialization of the generic finite gauge map to the fixed TT generator family.
Claim. The combined TT normal-equation generator map is the linear map $(c_i) \mapsto \sum_i c_i\, g_i$ from real coefficient vectors indexed by the sum of conformal vertex indices and longitudinal gauge indices into edge perturbations on the period-5 periodic Freudenthal edges, where $g_i$ runs over the fixed conformal vertex-delta and longitudinal vertex-vector generator family.
background
Track 1.D separates independent edge perturbations from the Track 1.B vertex-conformal ansatz, which cannot represent pure shear or transverse-traceless gravitational-wave modes. On the period-5 Freudenthal torus, edge perturbations are real functions of typed periodic edges.
The combined index is the disjoint sum of conformal vertex indices (one per torus vertex) and longitudinal gauge indices (vertex-vector generators). The generator family assigns to each such index either a conformal edge perturbation from a vertex delta or a longitudinal gauge edge perturbation.
The generic gauge map turns any finite family of edge generators into the linear combination map: coefficients times generators, summed edgewise. Specializing that map to the combined TT family yields the concrete normal-equation generator map used throughout the residual certificates.
proof idea
One-line definitional wrapper: apply the generic finite gauge generator map to the fixed combined TT normal-equation generator family. No further proof obligations; the resulting map is the edgewise sum of coefficients times the matched conformal or longitudinal generators.
why it matters
This map is the generator-facing surface for the finite TT normal equations in the tensor/shear sector. Downstream certificate structures (origin-column, translated, relative-frame, and raw residual formula data for the Regge Hessian versus lattice Lichnerowicz residual) all take residual-dispersion coefficients against this same index and prove scalar formulas directly against the generator map.
It sits inside the gravity track that aims to isolate pure shear from conformal modes, complementary to the conformal ansatz obstruction already proved for rectangles. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is discrete geometric scaffolding for weak-field TT modes on the periodic triangulation.
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