Addendum VII to the Berg Theory

Addendum VII develops the cosmological module of the Berg Theory by treating cosmic evolution as one admitted closed-FLRW branch of a connected energy-web field Ψ. The branch is controlled by the series-fixed Jordan-frame prefactor F(φ) = κ⁻¹ − 2ξφ², the admissibility condition F(φ) > 0, and explicit drift, residual, and regularity monitors. Turning points are defined by H = 0 and classified by the sign of Ḣ, now derived directly from the action-reduced acceleration equation rather than from an ambiguous effective-fluid shorthand. This gives a clean operational distinction between turnaround and bounce while preserving the same sign conventions used throughout the Berg series. Version 6 substantially strengthens the module. The work is now organized as a main document with three companion appendices and a supplementary figures-and-sources package. The main text carries the physical line: one connected web, one admitted closed branch, cyclic phase order, Berg-state closure, neighbour-module interfaces, public-data discipline, and falsification-facing status language. Appendix A carries the action-to-background derivations, turning-point checks, regularity conditions, and equation-side bridges. Appendix B records the claim ladder, downgrade rules, same-branch ledgers, prediction locks, and no-private-rescue discipline. Appendix C provides the reduced numerical prototype, monitor contract, event tables, convergence ladder, and generated-output structure. The technical core fixes an action-checked closed-background reduction, a corrected turning-point classifier, a reduced numerical cycle run with repeated turnaround/bounce alternation on the same admitted branch, and a cycle-level entropy/reset ledger. The release also strengthens the interfaces to the surrounding Berg programme: Addendum V supplies the gravity and distance-readout bridge, Addendum VI supplies the retained-core matter-production window, Addendum VIII supplies phase and freeze-out discipline, and Addendum IX supplies the stronger cross-probe obligations. In this form, Addendum VII no longer treats cyclicity as narrative closure alone; it makes cyclic continuation conditional on admissibility, monitor control, same-identity preservation, reset bookkeeping, and explicit failure conditions. The status discipline remains deliberately strict. The reduced homogeneous branch and its numerical witnesses provide structured support for a cyclic same-branch cosmology, but they do not by themselves complete the full observational programme. Growth, lensing, slip, standard sirens, and local normalization remain separated as IX-facing five-lock obligations. Public-data routes, source-rights boundaries, prediction locks, and future likelihood-level tests are therefore documented as part of the release architecture without being promoted into a completed observational proof. This makes the module falsification-facing: a failed late-time packet, a broken same-identity condition, an inadmissible prefactor, an unstable turning-point classifier, or an unresolved five-lock conflict cannot be repaired by private cycle language. Together with the core article and the neighbouring addenda, this release advances the Berg programme from conceptual cosmological regime language toward a quantitative, auditable branch framework. It supplies the equations, monitors, numerical witnesses, status ledgers, and failure rules needed to confront the proposed closed energy-web cosmology with late-time distance data, local-gravity and drift bounds, growth/lensing consistency, standard-siren propagation, matter-production timing, and cycle-closure requirements. Public cosmological data products and comparison datasets mentioned in this release are cited through their original publications, repositories, or public release records. Raw third-party observational data are not redistributed as part of this package unless explicitly permitted by the respective source. The supplementary material contains derived outputs, source notes, templates, and reproducibility receipts for the Addendum VII workflow.