From Finite-Node Conifold Geometry to BPS Structures III: Mediated Triangle Transport and Graded Interaction Data

Abstract

In previous work, we extracted from a finite-node conifold degeneration the state-data package $A_\Sigma=(V_\Sigma,E_\Sigma,c_\Sigma)$ and then constructed the support-level interaction package encoded by a binary incidence structure and finite quiver-theoretic skeleton \cite{RahmanQuiverDataI,RahmanQuiverDataII}. The present paper introduces the next layer: a graded pairwise interaction package refining binary support. Since the support matrix records where a mediated channel is present, but not its derived size, cohomological degree, or exact-triangle behavior, we introduce \emph{mediated triangle transport} (MTT). An MTT datum combines bulk-mediated schober transport, localized probes, corrected-extension shadow compatibility, and derived interaction profunctors. For each ordered pair $(i,j)$, it produces $\mathbb T_{ij}(X,Y):=\RHom_{\mathcal C_{p_j}}(\Psi_j\Phi_i(X),Y)$ and the probe interaction complex $\mathsf H_{ij}:=\mathbb T_{ij}(L_i,L_j)=\RHom_{\mathcal C_{p_j}}(\Psi_j\Phi_i(L_i),L_j)$. We prove exactness and long exact interaction sequences, isolate a triangle-visible nonvanishing criterion, and formulate a conditional bridge theorem showing that supported channels yield nontrivial pairwise interaction complexes under the stated probe, content, and detector hypotheses. Under a bounded Hom-finite convention, the cohomology of $\mathsf H_{ij}$ defines $P_{ij}(q)=\sum_m \dim H^m(\mathsf H_{ij})q^m$, and these polynomials assemble into $I_\Sigma^{\mathrm{gr}}$. Thus $(A_\Sigma,I_\Sigma^{(0/1)},I_\Sigma^{\mathrm{gr}})$ provides the first graded interaction input for later stability, BPS, and wall-crossing theory.