Boundary Compensation – Compensated Islands IV: Externally Anchored Hidden Sections and Certification-Cost Selection

his paper develops the fourth module of the Compensated Islands branch of the BoundaryCompensation programme. BC-CI-I introduced finite-dimensional near-compensated residuallayers generated by mapsLθ = P(IK + Kθ), Bθ = LθL∗θ,BC-CI-II introduced certified residual transport along declared parameter paths, and BC-CIIII lifted pathwise certificates to finite transport atlases with overlap and triple-overlap resetobstructions. Those three modules deliberately kept the hidden sectionΞ(θ)exogenous. This was necessary to avoid importing physical dynamics, but it also left theCompensated-Island scaffold empty: any hidden section could be declared unless an additionaladmissibility principle was supplied.The present paper introduces such a principle in a controlled finite-dimensional form.It defines externally anchored admissible hidden sections over a certified transport atlas.External anchors are pre-declared constraints or penalties that are not identical to the internalcertificate being optimized: finite readout budgets, support or locality masks, boundedvariation constraints, leakage bounds, balance constraints, ambiguity bounds, and resetavoidance rules. The paper then defines certification-cost selection over a finite diagnosticmesh. A selected section is not a physical trajectory, not a solution of an equation of motion,and not a reconstructed hidden reality; it is a representative or class of representatives chosenfrom a non-tautologically restricted admissible set. The scalar weights used to combinecost components are certification-policy parameters, not physical constants or fitted laws ofnature; theorem-level conclusions are not allowed to depend on a single hand-tuned weightvector.The main finite-dimensional results are elementary but structurally important. If theexternally anchored admissible set is nonempty and compact, a certification-cost minimizerexists. If anchors impose a genuine independent restriction, the admissible class is a propersubset of all declared hidden sections. Spectral gluing and atlas compatibility do not bythemselves select a hidden section; selection requires external anchors. Under declared closureassumptions, certified transport segments supported by admissible sections define an effectivereachability preorder on the diagnostic graph. This reachability relation is not a light cone,causal order, spacetime metric, or physical propagation law. It is only a finite-resolutioncertification object that may become a candidate bridge for later modules.The contribution is architectural and methodological. BC-CI-IV is the first CompensatedIsland module that places the hidden section “on the table” without importing dynamicsor field theory. It moves the branch from pure scaffold to non-tautological selection. NoHamiltonian, Lagrangian, action principle, quantum state evolution, gauge field, photon,conservation law, spacetime geometry, or empirical prediction is claimed.