conformal_ansatz_cannot_recover_gravitational_waves
plain-language theorem explainer
A nontrivial rectangle shear (horizontal strain unequal to vertical) cannot arise from any assignment of scalar potentials at the four vertices under the conformal edge ansatz. Anyone arguing that ledger-to-geometry passage must go beyond pure conformal edge lengths will cite this. The proof is a one-line wrapper of the rectangle/shear obstruction already proved in the tensor shear sector.
Claim. For real strains $h$ and $v$ with $h \neq v$, there do not exist vertex potentials $\xi_a,\xi_b,\xi_c,\xi_d \in \mathbb{R}$ 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
The module records the honest status of the bridge from the discrete recognition ledger to effective geometric (hinge) data. One of its two headline findings is that the conformal edge ansatz is insufficient for the transverse-traceless gravitational-wave sector.
Under that ansatz each vertex carries a scalar potential, and each edge length variation is the average of its two endpoint potentials. That map is exactly the vertex-conformal log-strain map. A pure rectangle shear mode is the strain pattern with equal horizontal edges at value $h$ and equal vertical edges at value $v$.
Upstream, TensorShearSector already proves that whenever $h \neq v$ no four-tuple of vertex potentials realizes that pattern. Transverse-traceless gravitational-wave modes are pure shear modes, so the same obstruction blocks any conformal recovery of the GW sector.
proof idea
One-line term wrapper: the claim is definitionally the same as nontrivial_rectangle_shear_not_vertexConformal from the tensor shear sector, applied to the given $h$, $v$, and the hypothesis $h \neq v$. No extra algebra is performed here.
why it matters
This declaration is the named bridge-level packaging of the shear obstruction: it is why the conformal edge ansatz cannot be the actual connection between the ledger substrate and effective geometry. The module doc states that finding explicitly, and tags the broader ledger-to-hinge map itself as an assumed structure rather than a derived theorem.
Downstream, the ledger energy bridge constructs an explicit rectangle shear potential and proves it realizes the pure-shear strain pattern (horizontal $+h$, vertical $-h$). That construction lives outside the conformal ansatz and is the positive counterpart to the negative result recorded here: shear degrees of freedom exist, but they require a non-conformal potential.
In the Recognition gravity stack this closes the conformal route for TT modes and forces any complete ledger-to-geometry bridge to carry genuine tensor/shear structure, not merely vertex scalars averaged onto edges.
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