periodicTTNormalEquationLoad5
plain-language theorem explainer
Defines the load (right-hand side) of the finite transverse-traceless normal equations on the N=5 periodic Freudenthal torus: each component is the edge-space inner product of an input edge perturbation against one fixed combined conformal-plus-longitudinal generator. Gravity and discrete-geometry workers cite it when assembling the Gram system that splits shear from gauge. The body is a one-line application of the N=5 edge inner product to the generator map.
Claim. For an edge perturbation $\varepsilon$ on the typed periodic Freudenthal edges and a combined normal-equation index $i$ (either a conformal vertex-delta slot or a longitudinal vertex-vector slot), the load component is $\langle \varepsilon,\, g_i\rangle_E$, where $g_i$ is the corresponding fixed generator and $\langle\cdot,\cdot\rangle_E$ is the finite $N=5$ edge inner product $\sum_e \varepsilon(e)\,\eta(e)$.
background
Track 1.D builds the tensor/shear sector missing from the Track 1.B conformal ansatz. Vertex scalars induce only averaged endpoint length changes and cannot carry pure shear, so they miss transverse-traceless weak-field modes. This module treats independent edge perturbations and proves elementary rectangle obstructions for the conformal slice.
An edge perturbation here is a real function on the typed periodic Freudenthal edges of the $N=5$ torus. The combined normal-equation index runs over conformal vertex-delta generators together with longitudinal vertex-vector (gauge) generators. The edge inner product is the plain finite sum $\sum_e \varepsilon(e)\eta(e)$ on that edge space; it is the inner product used for the tensor/shear decomposition.
The load is the right-hand side of the concrete TT normal equations: pair the input edge field with each generator in that fixed combined family, producing one real residual per index.
proof idea
One-line definitional wrapper. Evaluate the $N=5$ periodic edge inner product on the pair $(\varepsilon, g_{\mathrm{idx}})$, where $g_{\mathrm{idx}}$ is the image of the combined index under the fixed generator map (conformal vertex-delta or longitudinal gauge generator). No further algebra or tactics.
why it matters
This load is the geometric forcing term for the finite TT Gram system on the periodic torus. Downstream structures package it as image/range data (every load from an edge perturbation must lie in the image of the Gram operator), as load-solver data (solve the Gram system for every such load), and as explicit Gram-system solution data (a coefficient projector whose Gram apply recovers the load). Sibling lemmas use it to show the load annihilates the Gram kernel and that load-coefficient pairings equal generator-map inner products.
In the Recognition gravity track this is scaffolding for isolating pure shear (TT) modes from conformal and longitudinal gauge junk on a discrete Regge-like complex, complementary to the conformal ansatz that cannot represent gravitational-wave polarizations. It does not yet invoke the forcing chain (T5–T8) or the mass ladder; it is local linear algebra on the edge space of the $N=5$ periodic Freudenthal triangulation.
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