xUniformStrain5_nonzero_orthogonal_component
plain-language theorem explainer
On the N=5 periodic Freudenthal 3-torus, the uniform x-directed edge strain is not purely conformal: it has a nonzero inner product against an explicit rectangle face shear that annihilates the whole conformal log-strain subspace. Gravity and discrete-geometry workers cite this to separate pure shear from vertex-potential modes. The proof is a short term construction: take the localized rectangle shear, reuse its conformal-orthogonality lemma, and evaluate the pairing by rewriting plus norm_num.
Claim. There exists a periodic edge perturbation $t$ on the $N=5$ Freudenthal 3-torus such that $t$ is orthogonal (under the periodic edge inner product) to every conformal log-strain, yet $\langle t,\, s_x\rangle \neq 0$ where $s_x$ is the uniform strain along the $x$-edges.
background
Lane 3 of the Seven-Gaps gravity development studies the edge (tensor) sector beyond the vertex-conformal ansatz. That ansatz assigns one real scalar per vertex and induces the symmetric log-strain $(\xi_u+\xi_v)/2$ on each edge ${u,v}$. The resulting image is a linear subspace of the full edge-perturbation space.
On the concrete $5\times 5\times 5$ periodic Freudenthal triangulation one has $n_V=125$ vertices and $n_E=875$ edges, so the conformal image has rank at most 125 and is a proper subspace of the 875-dimensional edge space. Two explicit witnesses live outside that slice: the localized rectangle face shear (strain $+1$ on the two $x$-edges and $-1$ on the two $y$-edges of the unit coordinate square at the origin) and the uniform $x$-strain.
The periodic edge inner product pairs two edge fields by summing products of their edge values. Orthogonality to the whole conformal subspace means the field has a pure shear component with no vertex-potential realization.
proof idea
Term-mode existence proof. The witness is the already-constructed rectangle face shear on the unit coordinate square. The first conjunct is the prior lemma that this shear is orthogonal to every conformal log-strain. The second conjunct rewrites the inner product of the shear against the uniform $x$-strain via the dedicated evaluation lemma, then closes by norm_num (the resulting rational is nonzero).
why it matters
This lemma sharpens the Lane 3 dimension-gap story: not only is the conformal slice proper inside the edge space (rank $\le 125 < 875$), but a globally uniform strain mode itself carries a nonzero pure-shear projection. The orthogonal witness is the same localized face shear used to realize the gap concretely in the module capstone (no vertex-conformal realization).
In the broader Recognition gravity program this separates tensor shear degrees of freedom from the scalar conformal ansatz on the eight-tick / $D=3$ discrete geometry (T7–T8 landmarks). Downstream the file records the capstone that the conformal slice is strictly smaller and that the gap is witnessed by an explicit non-conformal shear; the present pairing shows the uniform $x$-strain is not hidden inside the conformal range either. No further used-by edges are recorded yet; the result stands as a finished local obstruction inside the edge-tensor sector.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.