rectangleShearPotential
plain-language theorem explainer
Defines the four-cell rectangle pure-shear potential: vertices 0 and 2 sit at height 0, vertices 1 and 3 at height −h. Its coboundary yields horizontal strains +h and vertical strains −h, the traceless shear mode used as the canonical witness that shear is visible to both the J-ledger and the quadratic hinge energy. Anyone citing the corrected ledger-to-geometry bridge or the shear-visibility gate uses this field. The body is a one-line piecewise assignment on Fin 4.
Claim. For each real amplitude $h$, the rectangle shear potential is the map $f_h:\{0,1,2,3\}\to\mathbb{R}$ with $f_h(0)=f_h(2)=0$ and $f_h(1)=f_h(3)=-h$. Its coboundary strains are $f_h(i)-f_h(j)$, giving $+h$ on the two horizontal edges and $-h$ on the two vertical edges.
background
Lane 1b of the Seven Gaps program rebuilds the ledger-to-geometry bridge after the signed-deficit form was ruled out. Ledger costs are nonnegative and even in the deformation, while signed Regge deficits are odd; the honest geometric target is therefore the nonnegative quadratic hinge energy $\sum_h A_h\delta_h^2$.
The ledger side is scoped to coboundary strains $s_{ij}=f_i-f_j$ of a cell potential $f$. Only then do the ratios satisfy the cocycle identity and the Recognition Composition Law gate (via the d'Alembert form of $J$). General antisymmetric strains can violate that gate.
This potential is the pure-shear mode on a four-cell rectangle: horizontal strain $h$, vertical strain $-h$. As an edge pattern with $h\neq v$ it has no vertex-conformal (averaging) realization, yet as a difference field it is a coboundary, so the corrected bridge applies.
proof idea
Definitional one-liner: assign $-h$ on Fin-4 indices 1 and 3, and 0 elsewhere. No lemmas are invoked. Downstream strain identities are proved by unfolding this piecewise map and discharging the four Fin-4 literal disequalities with decide.
why it matters
This is the canonical shear witness for the corrected bridge. It feeds four local results: the strain-pattern theorem (horizontal $+h$, vertical $-h$), the shear-visibility gate (strictly positive ledger total cost for $h\neq 0$), the matching geometric positivity of quadratic hinge energy, and the packaged bridge instance under the small-strain bound $|\varepsilon h|\le 1$.
Framework role: it exhibits the transverse-traceless sector on which the conformal-average ansatz is blind, so shear is visible to the J-ledger after the signed-deficit no-go. It sits inside the coboundary-scoped matching of ledger total cost to quadratic curvature energy; the remaining open item is the Hessian-symbol comparison against independently derived Regge geometry.
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