Addendum II to the Berg Theory

Addendum II extends the Berg Theory core article (collection DOI: 10.5281/zenodo.17937466) into the dark sector by treating dark matter not as an additional particle species, but as a geometric and dynamical regime of the same underlying energy-web field Ψ / Psi used throughout the programme. The physically realized vacuum remains a nonzero, homogeneous background state of this field, while the formal zero-field limit is reserved for the non-physical absence of the web.

The central claim of the addendum is narrow and test-facing. A dark halo is defined as a structured, gravitating regime of the field that remains below the activation threshold for visible matter formation. In operational terms, the regime is characterized by coarse-grained zero-mean transport, low phase order, and a strict separation between web strength ϱ≡∣Ψ∣2\varrho\equiv |\Psi|^2ϱ≡∣Ψ∣2, which controls activation, and effective gravitating density, which enters dynamics and lensing bookkeeping. This yields a single-field embedding of vacuum background, dark halo structure, and matter activation without introducing a separate dark particle sector.

Two conservative realization routes are developed. Route A uses rapidly oscillating amplitude dynamics with pressureless averaging in the standard Turner sense; Route B uses a dephased outer-halo regime in which the amplitude stays near baseline while the stored energy is carried by disordered phase structure. The addendum does not treat these as arbitrary alternatives, but as two admissible dark-regime realizations subject to the same closure discipline and the same activation boundary.

The observational content is organized as a local cross-probe fingerprint. A declared halo realization must simultaneously account for rotation-curve structure, finite outer-halo handoff, lensing consistency, and merger collisionlessness, with no hidden repair moves introduced between channels. Any mismatch between dynamical and lensing potentials is not allowed to float freely across the halo: if present at all, it must remain localized to the transition shell where the field profile changes most rapidly. If a fit survives only by approaching or crossing the activation threshold, the regime is no longer counted as dark-halo closure and must be handed over to the matter-formation sector developed in Addendum VI.

Version 5 substantially strengthens the empirical side of the addendum. In addition to the analytical weak-field and Jordan-frame derivations, it now includes public benchmark passes against named systems. NGC 3198 is treated as a weighted spiral-benchmark success under the declared fixed-baseline route-A protocol; NGC 2403 is retained as a genuine stress benchmark rather than being overstated as an equal success; and the Bullet Cluster is used as a merger-and-lensing anchor, now sharpened with both classical and newer public benchmark material. This turns the addendum from a purely theoretical regime proposal into an empirically engaged, explicitly falsifiable module within the larger Berg programme.

This Version also tightens the methodological discipline. The regime boundary to visible matter is made more explicit, the one-halo one-reconstruction rule is enforced throughout, and the appendix is expanded into a practical audit toolkit with pass/fail gates, shell-consistency diagnostics, benchmark-target summaries, and prefactor-margin checks. The working potential is presented as a minimal effective form for the present dark-sector analysis rather than as a uniquely derived microscopic law, and the weak-field positivity corridor is reported explicitly so that halo fits cannot be rescued by hidden proximity to the prefactor boundary.

The result is a more mature dark-sector module: still ambitious in scope, but now much harder, clearer, and more accountable to data. Its role within the Berg Theory is not to claim final confirmation, but to show that the dark-halo sector of the one-field programme can be formulated as a concrete, benchmark-tested, and sharply falsifiable regime with a distinctive local closure fingerprint across dynamics, lensing, and mergers.

This version DOI: 

Cite-all-versions DOI: 10.5281/zenodo.17980636