BC-CI-IV introduced externally anchored hidden sections and certification-cost selection. BC-CI-V then replaced single-weight selection by weight-robust and Pareto-stablecertification, thereby separating structural persistence from hyperparameter tuning. Thepresent paper develops the sixth module of the Compensated Islands branch. It uses therobust section classes of BC-CI-V to define a finite-dimensional reachability relation on adiagnostic graph or mesh.Let G = (V, E) be a finite directed diagnostic graph over a certified transport atlas. Adirected edge e = (p, q) is called robustly certified when it is supported by declared transportcertificates, readout margins, reset-free atlas compatibility, and at least one hidden sectionwhose selection status is weight-robust in the sense of BC-CI-V. The certified reachabilityrelation is then defined by existence of a finite path of robustly certified edges,p ⪯cert q ⇐⇒ ∃ p = v0 → v1 → · · · → vk = q with all edges robustly certified.The main theorem-level claim is deliberately modest: under the declared concatenationconvention, ⪯cert is a preorder. It becomes a partial order only after adding an explicitacyclicity certificate, for example a strictly monotone finite potential along certified edges, orafter passing to the quotient by mutual reachability. Cycles are not interpreted as time loops.They are audited as finite graph phenomena and assigned statuses such as CERTIFIED CYCLE,CYCLE FRAGILE, TUNED LOOP, RESET CYCLE, or QUOTIENT COMPONENT.The construction is finite-dimensional, operator-theoretic, and certification-based. Itdoes not define physical causality, a light cone, spacetime order, Hamiltonian dynamics, fieldpropagation, photon trajectories, an arrow of time, or empirical predictions. The phrase“pre-causal” denotes only a candidate finite-resolution order structure built from robustcertification data. Its role is to test whether a stable reachability skeleton can be extractedfrom Compensated-Island transport without importing physical dynamics or claiming aphysical precursor to causality.
A. A. Malachevsky (Sun,) studied this question.