periodicTTNormalEquationGaugeCoeff5
plain-language theorem explainer
Extracts the longitudinal-gauge half of a combined normal-equation coefficient vector on the five-cell periodic torus. Anyone assembling the finite TT residual, the conformal-plus-gauge generator split, or the coefficient solution data cites this projection. The body is the right Sum inclusion: evaluate the combined map on gauge indices only.
Claim. Given coefficients $c$ on the combined index set (conformal vertex indices $\oplus$ longitudinal gauge indices), the longitudinal gauge coefficients are the restriction $i\mapsto c(\iota_R(i))$, where $\iota_R$ is the right inclusion of the gauge index type (periodic vertex $\times\{0,1,2\}$) into the sum.
background
Track 1.D builds the tensor/shear sector that the Track 1.B conformal ansatz cannot reach: pure shear and transverse-traceless edge modes on a periodic Regge complex. The conformal slice assigns one scalar per vertex and averages endpoints; it is orthogonal to genuine TT content.
On the concrete five-cell periodic torus, normal equations are indexed by a disjoint sum: left summands are fixed conformal vertex-delta generators, right summands are longitudinal gauge generators. The gauge index type is one vector component at one periodic vertex (vertex $\times$ three spatial directions).
A combined coefficient vector therefore packs both conformal and gauge unknowns. This definition is the typed projection onto the gauge block, used whenever the residual or generator map must be split.
proof idea
Pure definitional projection. The combined index is a Sum type; the gauge half is the right injection. The map sends each longitudinal gauge index $i$ to the combined coefficient at $\mathrm{inr}(i)$. No lemmas, no tactics.
why it matters
Local plumbing for the finite TT normal-equation layer in the tensor/shear scaffold. Downstream, the combined generator map splits as conformal map of the left coefficients plus longitudinal gauge map of these right coefficients; the residual after both projections is the TT part used by the coefficient-solution structure (no separate TT projector). Orthogonality of longitudinal TT perturbations to every combined generator, and closure of the row-translated generator space inside the fixed conformal-plus-gauge image, both thread through this extraction.
In the broader Recognition gravity track this is bookkeeping on the discrete weak-field sector that must eventually host TT waves once the conformal obstruction is left behind. It does not itself force $D=3$ or the eight-tick structure; it only keeps gauge coefficients addressable inside the combined solve.
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