RightsSeed: Adjudication as Small-Step Semantics — Liability Algebra, Proof-Carrying Dispute Resolution, and Refinement-Preserving Justice for WorldSeed Worlds

RightsSeed is a mechanically checkable adjudication layer for WorldSeed-compliant programmable worlds and world markets. It elevates rights, obligations, harms, liability, and dispute resolution into first-class semantic objects, enabling justice to be enforced as an executable runtime invariant rather than a post hoc social process. RightsSeed defines a deterministic adjudication semantics: disputes are admissible only when all standing, jurisdiction, evidence, observer semantics, and action semantics are explicitly declared in the underlying WorldSeed world. Verdicts are produced by inference rules and compiled into proof-carrying verdict artifacts, preventing unverifiable judgments, jurisdiction laundering, and narrative-based enforcement. The framework introduces a liability algebra supporting compositional accumulation, bounded culpability, reparative restoration, and enforced compensation. Enforcement is specified operationally via small-step semantics with runtime rejection of enforcement bypass attempts. RightsSeed further defines refinement-preserving justice: if a world refines another world via a declared refinement mapping, admissibility, liability judgments, and enforcement obligations are preserved under refinement, preventing semantic drift of justice across world forks and modular compositions. RightsSeed is defined strictly as an extension layer of WorldSeed and assumes WorldSeed compliance as a prerequisite. All claims, artifacts, and certification predicates are world-indexed and mechanically auditable. This paper provides the formal axioms, admissibility rules, inference rules, small-step operational semantics, progress/preservation theorems, and refinement preservation theorems necessary to treat justice as a compositional, proof-carrying subsystem in open world markets. Reference: Zhang, X. WorldSeed: Axiomatic Specification and Infrastructure for Physically Learnable Worlds. Zenodo, 2025. https://doi.org/10.5281/zenodo.18102539