Track1DTensorShearScaffoldEndpoint
plain-language theorem explainer
Packages the Session 215 Track 1.D tensor/shear scaffold as one proposition: a conformal map from vertex potentials to edge strains exists on every finite 3D triangulation, the five-edge encoded and periodic perturbation spaces are equivalent, and unequal rectangle side-strains cannot arise from four vertex averages. Gravity workers cite it as the handoff that pure shear lives on independent edge degrees of freedom, not the scalar conformal slice. The body is a pure conjunction of three already-established facts.
Claim. The Track 1.D scaffold endpoint is the conjunction of: (i) for every finite 3D Regge triangulation there is a map from vertex conformal potentials to edge-length perturbations; (ii) the encoded five-edge perturbation space is equivalent to the periodic five-edge perturbation space; (iii) if $h \neq v$ are real, no four reals $\xi_a,\xi_b,\xi_c,\xi_d$ satisfy $(\xi_a+\xi_b)/2 = h$, $(\xi_c+\xi_d)/2 = h$, $(\xi_b+\xi_c)/2 = v$, $(\xi_d+\xi_a)/2 = v$ (rectangle shear is not vertex-conformal).
background
Track 1.B assigns one scalar potential to each vertex of a finite 3D Regge triangulation and induces edge-length variations by averaging the two endpoint potentials. That conformal slice cannot represent pure shear, so it cannot by itself cover transverse-traceless weak-field modes. This module opens the tensor/shear track by treating independent edge perturbations as the natural carrier and by isolating the elementary rectangle obstruction for the conformal ansatz.
A triangulation records abstract incidence (vertices, edges, tetrahedra) with nondegenerate squared-edge data. Vertex potentials are real assignments on vertices; edge perturbations are real assignments on global edges. The conformal log-strain map sends a vertex potential to the averaged endpoint strains on each edge. The five-edge periodic torus supplies a concrete finite model on which encoded and periodic edge perturbations are compared.
Upstream geometry supplies the triangulation type, the vertex-potential abbreviation, and the edge-perturbation surface used for anisotropic shear.
proof idea
Definitional packaging only: the proposition is the conjunction of three clauses. The first is nonemptiness of the type of maps from vertex potentials to edge perturbations on an arbitrary triangulation (witnessed downstream by the conformal log-strain map). The second is nonemptiness of an equivalence between the encoded five-edge and periodic five-edge perturbation types. The third is the universal rectangle obstruction: unequal opposite-side averages cannot be realized by four vertex values. No tactic work lives here; discharge is deferred to the companion existence theorem.
why it matters
Closes the Session 215 Track 1.D scaffold and is re-exported verbatim as the master-theorem handoff integration endpoint, which Track 7 consumes. Downstream documentation states the intended reading: edge-level perturbations are the right carrier for tensor/shear work, and nontrivial rectangle shear is not vertex-conformal. That separation is the prerequisite for treating transverse-traceless gravitational-wave modes on the Regge surface rather than forcing them through the scalar conformal ansatz of Track 1.B. Within Recognition gravity, it marks the move from isotropic vertex potentials toward the full weak-field metric sector on the discrete geometry.
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