couplingWeightCross
plain-language theorem explainer
Polarized cross weight for one unit-cell edge-pair coupling in the closed 4D midpoint Bloch symbol: half the coupling strength times the sum of mixed edge-strain products of two metric matrices H and K. Anyone expanding the TT first variation of the SRS continuum face cites this term as the linear-in-t coefficient along H + t K. The body is a pure algebraic definition (polarization of a product of strains), not a proved identity.
Claim. For $4\times 4$ matrices $H,K$ and a unit-cell edge-pair coupling $c$ with strength coefficient $s$ and integer edge directions $D_e,D_{e'}$, the cross coupling weight is $\frac{s}{2}\bigl(\varepsilon(H,D_e)\varepsilon(K,D_{e'})+\varepsilon(K,D_e)\varepsilon(H,D_{e'})\bigr)$, where the edge strain is the quadratic form $\varepsilon(M,D)=\sum_{i,j}M_{ij}D_i D_j$.
background
The module derives the directional first variation of the closed 4D midpoint Bloch symbol in the Euclidean weak-field TT sector, then transports the torus-normalized continuum face via the banked SRS-to-EH convergence on $H+K$ and $H-K$ plus polarization. Honesty constraints: theorem status only in that sector; not a source equation, not Ricci/null focusing, and not GAP1 closure.
A Coupling packages one unit-cell edge pair: rational strength data and two integer direction vectors $D_e$, $D_{e'}$. Edge strain of a matrix $M$ along $D$ is $\sum_{i,j} M_{ij} D_i D_j$, matching the Regge flat Hessian Bloch strain. Mat4 is the continuum $4\times 4$ matrix type from the Regge preflight layer.
The ordinary coupling weight is quadratic in those strains. The cross weight is the associated bilinear polarization, so the weight along the ray $H+tK$ expands as $W(H)+t,W_\times(H,K)+t^2 W(K)$.
proof idea
Definitional body only. It multiplies half the coupling strength by the sum of the two mixed products of edge strains of $H$ and $K$ on the pair $(D_e,D_{e'})$. No lemmas fire; the expression is the standard polarization of a product of two forms linear in the metric.
why it matters
Used by the indexed wrapper over the coupling table and by the algebraic line identity that expands the coupling weight of $H+t\bullet K$ into homogeneous degree-0/1/2 pieces. That expansion is the local engine for the TT first variation of the midpoint Bloch continuum face in this module.
Framework role is narrow and honest: it supports the Euclidean weak-field TT cross-term analysis that feeds the banked continuum transport, not a sourced Einstein or null-focusing equation. The module flags the missing future object as a Recognition-derived Freudenthal exact-$J$ metric refinement identifying sourced response with this midpoint variation, followed by Lorentzian null-dyad Ricci/stress transport. Do not cite PixelAreaModel, LocalNullPatch, or the MODEL exact-$J$ mesh action as justification here.
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