rectangleShearFace5_ne_zero
plain-language theorem explainer
The explicit unit-square shear on the 5×5×5 periodic torus is not the zero edge perturbation. Anyone citing the concrete nonzero vector in the conformal orthogonal complement needs this fact. The proof evaluates the shear on the bottom +x edge and reduces to 1 ≠ 0.
Claim. The rectangle shear edge field on the $N=5$ periodic Freudenthal 3-torus (strain $+1$ on the two $x$-edges and $-1$ on the two $y$-edges of the unit face with corners $(0,0,0)$, $(1,0,0)$, $(1,1,0)$, $(0,1,0)$) is not identically zero as a function on edges.
background
Lane 3 of the Seven-Gaps gravity work studies the edge (tensor) sector beyond the vertex-conformal ansatz. That ansatz assigns one scalar per vertex and induces the log-strain $(\xi_u + \xi_v)/2$ on each edge ${u,v}$. On the concrete $5\times 5\times 5$ periodic Freudenthal 3-torus one has $n_V=125$ vertices and $n_E=875$ edges, so the conformal image is a proper subspace of the full edge-perturbation space.
The explicit witness rectangleShearFace5 localizes a pure shear on the unit coordinate square in the $xy$-plane: $+1$ on the two $x$-directed edges and $-1$ on the two $y$-directed edges. The bottom $+x$ edge of that face is the named edge with base vertex $A=(0,0,0)$ and displacement class $0$. A companion evaluation lemma records that the shear equals $1$ on that edge.
The ambient setting is fully proved (no sorry, no silent hypotheses): rank bounds, dimension gap $125 < 875$, and non-conformality of this shear are already established elsewhere in the module.
proof idea
Assume for contradiction that the shear equals the zero function on all edges. Functional congruence at the bottom $+x$ edge yields that the shear's value there is $0$. Rewrite that value via the evaluation lemma for the shear on that edge, obtaining $1=0$. Discharge with the elementary fact $1\neq 0$. Four short tactic steps; no induction or linear-algebra machinery.
why it matters
This is the nonzero half of Deliverable 5 (orthogonal-split witness form). The parent theorem packages it with the already-proved inner-product vanishing against every conformal edge field, concluding that the orthogonal complement of the conformal slice inside the $875$-dimensional edge space contains a concrete nonzero vector.
In the Recognition gravity program this seals that the edge/tensor sector is strictly larger than the vertex-conformal ansatz on a real triangulation: shear modes are not gauge artifacts of the conformal log-strain. It sits downstream of the dimension-gap facts ($\mathrm{rank}\le 125<875$) and upstream of any claim that the seven-gaps edge sector carries independent dynamical content beyond conformal rescaling. No forcing-chain landmark (T5–T8) is invoked; the result is pure discrete geometry on the periodic torus.
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