Value--Anchored Natural--Law Fronts over Reversible Persistence--First Holographic Systems II

This preprint develops a value-anchored “theory of theories” for compute–performance laws and related natural-law models. Instead of proposing yet another specific scaling law, it builds a geometric framework on the space of theories themselves. Each theory is modeled as a law on a base space of compute budgets, environments, and outcomes, equipped with an entropy–transport–type metric. This induces a Wasserstein geometry first on laws, then on the theory space, and finally on probability measures over theories. On this four-layer structure, the paper introduces value densities on the base space and constructs value-anchored shortfall functionals on distributions over theories. Under explicit geodesic convexity assumptions, these functionals generate well-posed EVI_λ gradient flows on the Wasserstein space of theory distributions. The resulting dynamics are interpreted as “value-anchored natural-law fronts”: metric-gradient selection flows over competing natural-law or scaling-law hypotheses, driven by externally specified value criteria and constrained by compute and environment statistics. A key contribution is to show how several components of the author’s broader program embed into this value-anchored theory space. Fibered Bures–HK entropy–transport geometry (FBHK), image-EVI and interior Bures–HK control for compute-optimal design (DIR), holographic observation quotients (HOQ) and their infinite hierarchical extension, and (reversible) persistence-first holographic systems (PFHS, rPFHS) all appear as concrete families of theories or invariants inside the abstract framework. The paper also states a qualitative selection principle for reversible PFHS-generated theories, structurally reminiscent of Dobrushin/DLR-style arguments but formulated purely as a value-anchored selection statement inside a restricted family. Analytically, all curvature, PDE, and PFHS-level results are taken as explicit hypotheses and delegated to existing work on entropy–transport geometry and Wasserstein gradient flows. The contribution is structural and organizational: it provides a clean geometric language for comparing and selecting entire natural-law or scaling-law theories under compute and value constraints, with potential applications to scaling-law analysis, compute-optimal AI design, and alignment-relevant model selection.