Cartan-Fejer Gram Tomography and Flop Covariance for BPS Resummed Gromov-Witten Potentials

We introduce a matrix-valued finite difference formalism for the genus zero BPS resummed local Gromov-Witten potential of a threefold flop. Factoring the central difference as $${\mathrm{\Delta }}_{\eta }={\mathrm{\nabla }}_{\eta }^2,{\mathrm{\nabla }}_{\eta }=T_{h\eta \mathrm{/}2}-T_{-h\eta \mathrm{/}2},$$ we prove that the mixed differences $\mathbf{H}=({\mathrm{\nabla }}_{{\eta }_i}{\mathrm{\nabla }}_{{\eta }_j}F{)}_{i,j}$ admit an exact rank one signed $q$-Gram decomposition. Primitive BPS classes are therefore detected by rank one coefficient matrices, and a matrix-valued Möbius inversion reconstructs all multiplicities on a primitive ray. Under a simple threefold flop the Gram forcing is covariant away from the flopped ray and acquires the universal logarithmic anomaly $${\kappa }_C(C^{+}\otimes C^{+})\log r,{\kappa }_C=\sum _{d\ge 1}{d}^3{n}_{dC}\mathrm{.}$$ After a natural renormalization it becomes flop covariant, while one classical logarithmic derivative recovers the cubic correction in the crepant transformation formula. We interpret ${\kappa }_C$ as the third moment of the GV width distribution: Toda’s noncommutative width is the second moment, and contraction algebra BPS invariants give a finite cohomological width series whose numerical realization contains all these moments. For root supported BPS theories, simple root Gram coefficients reconstruct the signed Cartan matrix. Applied to the standard ADE foldings, the theory distinguishes $B_n$ from $C_n$, detects the $F_4$ double bond and $D_4$ triality, and yields explicit global $q$-Gram discriminants.