A Unified Geometric Framework for BPS Flows: Split Attractor, Hessian, and Spectral Networks

We provide a systematic and rigorous geometric framework that relates three structures naturally associated to BPS central charges in $\mathcal{N}=2$ supersymmetric gauge theories: the split attractor flow (SAF) of $|Z|$, the Hessian flow (HF) of $\operatorname{Im}(e^{-i\vartheta}Z)$, and the spectral network (SN) on the base curve of the Hitchin fibration. Our main contributions are: (i) a concise proof of orthogonality between SAF and gradient Hessian flow using only the Kahler structure; (ii) a precise lift-projection duality showing that the spectral network projects to the *characteristic Hessian flow* (the Hamiltonian flow of $\operatorname{Im}(e^{-i\vartheta}Z)$) on the Hitchin base, clarifying a crucial distinction; (iii) a complete proof of the Kontsevich-Soibelman (KS) equivariance by induction on the SAF tree depth, with the geometric ordering provided by the characteristic Hessian flow. We illustrate the framework with detailed and nontrivial examples: $SU(2)$ pure and $N_f=4$ (including new BPS indices for higher flavour charges), $SU(3)$ pure (full BPS spectrum reconstruction), $SU(4)$, the Kronecker $3$-quiver, and we apply the induction to derive a closed-form BPS spectrum for the Argyres-Douglas $H_1$ theory, $\Omega(n\alpha_1+m\alpha_2)=\binom{n+m}{n}$, which is a new result. In the tropical limit we obtain an explicit generating function for disk counts in $SU(N)$ gauge theories, $Z_{\mathrm{disk}}^{SU(N)} = \prod_{\alpha\in\Phi_+}\prod_{k\ge1}(1-e^{-k\langle\alpha,y\rangle})^{-\binom{k+\mathrm{ht}(\alpha)-1}{\mathrm{ht}(\alpha)-1}}$, which follows directly from our recursion. These results demonstrate the computational power of the unified framework and provide new, verifiable predictions.