Zero-State Axioms: Minimal Boundary Structure, Structural Consequences, and Semantic Models

ZSA is a first-order bookkeeping framework for boundary-governed admissibility in systems where extension, coherence, valuation, and partial measurement interact.

We study semantic universes equipped with a configuration preorder, a coherence relation, a valuation into a partially ordered value domain, and a partial measurement operator. Under explicit closure and stability assumptions, we prove an Infinity–Measurement Boundary Theorem: if unbounded extensibility and nontrivial partial measurement, with propagation and eventual stability along coherent increasing chains, both hold, then there exists a \(\preceq\)-least boundary configuration \(Z\), unique up to \(\equiv_{\preceq}\), such that every measured configuration extends \(Z\), and measurement is stable along coherent increasing chains above \(Z\) whenever a limit exists.

Motivated by this, we introduce the Zero-State Axioms (ZSA) in a first-order language naming a distinguished constant \(Z\). We show that a boundary fragment of ZSA is forced in the induced expansion extracted from any universe satisfying the IMB hypotheses. We further show that one axiom is derivable from another over the fixed semantic background, and that the remaining core axioms are independent via explicit countermodels. We also supply reference-chain and product-style model constructions demonstrating consistency and flexibility.

Finally, under an invariance hypothesis for admissible reassignment on a fixed underlying frame, we derive a semantic pinning property: no strict predecessor of \(Z\) can serve as the boundary while preserving the relevant valuation and measurement profile off its downward cone.