concreteWeakFieldReggeData
plain-language theorem explainer
Packages a concrete local Regge star into the eight-vertex weak-field linearization data the gravity bridge consumes. Area-response coefficients are normalized to 1; deficit-response coefficients are the graph Laplacian of the regular hinge-area weights. Downstream matrix identities, the component-comparison object, and the second-variation/Dirichlet equality all instantiate this packaging. One-line application of the Laplacian Regge-data constructor.
Claim. Given a concrete local Regge star $S$ (positive background edge length and nonnegative hinge area), form the weak-field Regge linearization data on eight vertices whose area-response matrix is identically $1$ and whose deficit-response matrix is the graph Laplacian of the concrete area weights $A_{ij}$ ($A_{ii}=0$, $A_{ij}$ equal to the background hinge area for $i\neq j$).
background
The module builds a fully concrete finite flat-sector / Freudenthal-local model that the weak-field bridge can consume without new geometric axioms. A concrete Regge star records only background scales: a positive edge length and a nonnegative hinge area. The eight-vertex index set matches the local star bookkeeping used throughout the comparison.
Concrete area weights put zero on the diagonal and the background hinge area off-diagonal; symmetry is immediate. Upstream, laplacianReggeData turns any symmetric weight matrix $A$ into weak-field data by setting area response to $1$ and deficit response to the graph-Laplacian coefficient of $A$. Only the product of the two responses enters the second variation, so the unit normalization is harmless.
The ambient weak-field structure is the first-order linearization of a flat-background Regge configuration under conformal vertex perturbations: paired symmetric matrices for hinge-area and deficit responses.
proof idea
One-line definitional wrapper. Feed the concrete area-weight matrix and its symmetry proof into the upstream Laplacian Regge-data constructor. No further algebra: the resulting record has unit area response and Laplacian deficit response on eight vertices.
why it matters
This is the single packaging step that turns geometric hinge areas into the weak-field interface the bridge already understands. Every concrete matrix identity in the module unfolds through it: the second-variation coefficient matrix, the off-diagonal equality $M_{ij}=-A_{ij}$, exact zero row sums, and the component-comparison object. From there the second-order Regge action collapses to one-half the geometric Dirichlet form on the concrete weights, which is the module's stated closure goal.
The certificate structure records the same package alongside the regular-triangle area derivative and the vanishing uniform-scale dihedral derivative. In the broader Recognition geometry stack this is the first fully concrete finite model the weak-field conformal Regge bridge can cite; a future full Cayley-Menger / arbitrary-dihedral derivative computation is expected to target the same interface. The fixed eight-vertex arity aligns with the eight-tick local octave used elsewhere in the forcing chain.
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