nontrivial_rectangle_shear_not_vertexConformal
plain-language theorem explainer
A pure rectangle shear with unequal horizontal and vertical first-order log strains cannot arise from any four vertex potentials under endpoint averaging. Gravity and weak-field workers cite it to separate the conformal scalar slice from genuine tensor/shear modes. The proof is a one-line contrappositive of the rectangle obstruction that forces the two strains equal.
Claim. For real strains $h$ and $v$ with $h \neq v$, there do not exist vertex potentials $\xi_a,\xi_b,\xi_c,\xi_d$ such that the four edge averages equal $(\xi_a+\xi_b)/2 = h$, $(\xi_c+\xi_d)/2 = h$, $(\xi_b+\xi_c)/2 = v$, and $(\xi_d+\xi_a)/2 = v$.
background
Track 1.D isolates the tensor/shear sector of the weak-field metric from the older Track 1.B conformal ansatz. That ansatz places one real scalar at each vertex and induces an edge log-strain by averaging the two endpoint values. The module doc states the limitation directly: the scalar slice cannot represent pure shear, so it cannot cover transverse-traceless gravitational-wave modes by itself.
The elementary witness is a four-cycle (rectangle) whose opposite edges carry constant horizontal strain $h$ and constant vertical strain $v$. Vertex-conformality means those four numbers are endpoint averages of four potentials. The upstream lemma records the algebraic obstruction: if such potentials exist, then necessarily $h = v$. A nontrivial shear is precisely the case $h \neq v$.
proof idea
Term-mode contrappositive. Assume four potentials realizing the two horizontal averages equal to $h$ and the two vertical averages equal to $v$. Feed those equalities into the upstream rectangle obstruction, which concludes $h = v$. That contradicts the hypothesis $h \neq v$. No further algebra is needed.
why it matters
This is the packaged non-existence form of the rectangle obstruction, the form downstream results actually quote. The ledger-to-geometry bridge theorem "conformal ansatz cannot recover gravitational waves" restates the same statement and cites the shear obstruction as the reason a pure conformal edge map misses TT modes. On the $N=5$ periodic Freudenthal torus, the typed non-conformality of uniform $x$-strain instantiates the four averages with $h=1$, $v=0$ and invokes this fact. The Track 1.D scaffold endpoint also lists the separation of independent edge perturbations from vertex-conformal ones among its discharged obligations. In the broader gravity track it marks why a scalar vertex potential is insufficient and why an independent edge/tensor sector must be carried.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.