Foundations IV is a catalog of layer-agnostic structural laws built on the Foundations I/II/III apparatus. Its organizing question is what must be true, before choosing a substrate—whether analysis, arithmetic, physics, biology, chemistry, economics and finance, linguistics, cognitive and social science, or computation—for a layer to have lawful access, stable records, transport, symmetry, closure, and limits. The contribution is a uniform normal-form template applied across 52 laws; each row states its setup, theorem, proof spine, case structure, interaction status, and nonclaims, so recurring obstructions such as failed descent, hidden predictive structure, local-global failure, and record nonclosure can be compared in one register. The formalization status is explicit: the catalog contains 46 direct Lean theorems, 2 substrate-delegated anchors (No-Needles Stability and Closure–Reality Boundary), 2 partial wrappers (Hiddenness and Duality Fixity), and 2 obligation wrappers (CSL Formed-Closure and Spontaneous Symmetry Breaking); the four trust-base axioms used by the wrappers are disclosed through an opacity-guard pattern, each guarded by an opaque source predicate that prevents the axiom from being instantiated without the corresponding substrate or future-source certificate. Foundations IV is not a foundational reset of Foundations I/II/III, not a unification claim about the eight axes, and not an exhaustive catalog.
Ioannis Tsiokos (Wed,) studied this question.