rectangleShearFace5_inner_conformal_eq_zero
plain-language theorem explainer
The face-localized rectangle shear on the 5×5×5 periodic torus is orthogonal to every conformal log-strain. Anyone citing the concrete shear complement of the vertex-conformal ansatz needs this pairing identity. The proof expands the four nonzero edge values as endpoint averages of a vertex potential and cancels by algebraic telescoping.
Claim. For every edge perturbation $c$ on the $5\times 5\times 5$ periodic torus that lies in the conformal log-strain subspace (induced by some vertex potential via endpoint averages), the discrete edge inner product of the unit-face rectangle shear $s_{\mathrm{rect}}$ with $c$ vanishes: $\langle s_{\mathrm{rect}}, c\rangle = 0$.
background
On a 3D Regge triangulation the vertex-conformal ansatz assigns one real scalar per vertex and induces the log-strain $(\xi_u+\xi_v)/2$ on each edge ${u,v}$. This module measures how small that conformal slice sits inside the full edge-perturbation space of the $5\times 5\times 5$ periodic Freudenthal 3-torus ($n_V=125$, $n_E=875$), and exhibits an explicit shear complement.
The rectangle shear places strain $+1$ on the two opposite $x$-edges and $-1$ on the two opposite $y$-edges of the unit coordinate square with corners $A=(0,0,0)$, $B=(1,0,0)$, $C=(1,1,0)$, $D=(0,1,0)$, and zero on the remaining edges. The four face-edge definitions and their endpoint lemmas identify those edges with the ordered pairs $(A,B)$, $(D,C)$, $(B,C)$, $(A,D)$.
Membership in the conformal subspace means there exists a vertex potential $\varphi$ such that every edge value of $c$ equals the average of its two endpoints.
proof idea
Unpack the conformal hypothesis to a vertex potential $\varphi$ via the endpoint-form characterization of the conformal subspace. Rewrite the four face-edge values of $c$ using the endpoint lemmas for the bottom $x$-edge, top $x$-edge, right $y$-edge, and left $y$-edge, obtaining $c=(\varphi_u+\varphi_v)/2$ on each. Substitute into the specialized left-hand formula for the inner product against the rectangle shear, which reduces to $c_{AB}+c_{DC}-c_{BC}-c_{AD}$. The resulting combination of endpoint averages telescopes to zero by ring.
why it matters
This is the orthogonality half of Deliverable 5: packaged with nonvanishing of the shear, it shows the orthogonal complement of the conformal slice inside the 875-dimensional edge space contains a concrete nonzero vector. Downstream, non-conformality of the shear itself follows by self-pairing (the self inner product equals 4, which cannot vanish under the orthogonality identity). The uniform $x$-strain is then shown to have a nonzero component orthogonal to the conformal subspace by pairing against this same shear witness.
In the Seven-Gaps gravity lane the result exhibits an explicit tensor/shear mode beyond the pure conformal (scalar-potential) ansatz on the discrete 3-torus, supporting the proved rank gap that the conformal image has finrank at most 125, strictly less than the 875-dimensional edge space.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.