Theoretical Cartography 4: Remote Isomorphism Generator—Signature Search, Adjoint Existence, and Archive Transport

The first three installments of theoretical cartography respectively provide a weighted hypergraph formalization of theoretical space, homological detection of missing bridges, and athree-tier criterion for rejecting illegal bridges, but none of them answers a generative question:where do candidate bridges come from? This paper establishes a remote isomorphism generator:a mechanical device that proposes candidate isomorphisms between remote subregions and performs archive transport along qualified bridges. The generator consists of four functional layers.The coarse-screening layer matches item by item by role signature over remote candidate domains and proposes proposals with high recall; the refinement layer refines signatures layer bylayer according to the declared comparison carrier and eliminates false positives that are locallyisomorphic but globally non-isomorphic; the convergence layer is an invariant relative to thatcarrier, not an absolute invariant of the graph. The existence layer handles the right-adjoint existence of a given candidate archive functor: in the finite lattice preorder case, a monotone maphas a right adjoint if and only if it preserves all joins, including the empty join, and the solutionset condition is automatically satisfied by finiteness; this layer is an offline theorem layer anddoes not occupy a pairwise filtering station on the preorder archive layer. The transport layercomputes left and right Kan extensions along the functor in the declared direction; the substanceof transport verification is fidelity replay—the transport back-end ledger must agree item by itemwith the ledger independently computed on the target side, rather than checking whether theKan extension exists; on the Pos-valued archive layer, left and right Kan extensions along anyfunctor exist unconditionally, and a fidelity failure does not negate existence. In controlled toycalibration, the end-to-end pipeline gives a ledger of 144 → 30 → 24 → 20 → 12; each of thefour operative stations performs substantive elimination, and the twelve pairs finally registeredagree item by item with the planted ground truth. The calibration ledger certifies recovery;the leakage-separation drill certifies, under pre-registered masks, information-flow isolation, andrecovery of all masks, that explicit ground-truth querying is excluded; neither certifies the capacity to discover unplanted structures. Validation of generative capacity belongs to leave-one-outvalidation at deployment scale and cannot be replaced by calibration-layer readings.