periodicTTNormalEquationConformalCoeff5
plain-language theorem explainer
Extracts the conformal (vertex-scalar) half of a combined normal-equation coefficient vector on the 5×5×5 periodic Freudenthal torus. Anyone assembling the TT residual or splitting the combined generator map cites this projection. The body is the left-sum inclusion: evaluate the coefficient only on conformal vertex indices.
Claim. Given a real coefficient assignment on the disjoint union of conformal vertex indices and longitudinal gauge indices for the $5\times 5\times 5$ periodic torus, return its restriction to the conformal vertices: a map $V\to\mathbb{R}$ on the torus vertex set.
background
Track 1.D separates pure shear (tensor) edge perturbations from the older Track 1.B conformal ansatz, which assigns one scalar per vertex and cannot represent transverse-traceless gravitational-wave modes. The module works on the canonical encoded $5\times 5\times 5$ periodic Freudenthal torus.
The combined normal-equation index is the sum type of conformal vertex labels and fixed longitudinal vertex-vector gauge labels. A single coefficient vector on that sum packages both sectors for the finite normal equation that projects edge perturbations onto conformal plus longitudinal generators.
This definition is the left-hand extractor: it reads only the conformal block. The companion longitudinal extractor and the split of the combined generator map sit beside it in the same file.
proof idea
Definitional one-liner. The combined index is a sum type; the conformal block is the left injection. The map sends each torus vertex $v$ to the coefficient at $\mathrm{inl}(v)$. No lemmas are invoked.
why it matters
The combined generator map splits as conformal map of these extracted coefficients plus the longitudinal gauge map. That identity feeds the residual after conformal-plus-longitudinal projection, which is how the file states the TT part without a separate projector.
Downstream, the residual and the split appear in orthogonality of longitudinal TT perturbations to every combined generator, in relative generator-space closure under row-frame translation, and in the coefficient-solution data structure. Within Recognition gravity scaffolding this is bookkeeping for the tensor/shear sector on the discrete torus, not a forcing-chain step (T5–T8) or an RCL identity.
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