Internal Equilibria on Formal Concept Lattices in Finite No--Meta Law Spaces

This work develops a general framework for analysing internal equilibria in finite spaces of dynamical laws and observers, using tools from formal concept analysis and lattice-theoretic fixed point theory. Each law–observer pair is equipped with internal structural scalars, such as intrinsic (state) dimension, visible dimension, visibility gap, and computational cost. An internal visibility criterion induces a formal context and its associated concept lattice of maximally consistent observer–law pairs. On top of this structure, the paper defines law-side and observer-side valuation families that obey a strict “no-meta” discipline: all scores and decisions must be computable from internal scalars and visibility patterns only, with no external reward signals or designated target laws. Under an antitone design principle for best responses, the induced update operator becomes a monotone self-map on the concept lattice. By Tarski’s fixed point theorem, its fixed points form a nonempty complete sublattice, interpreted as the internal equilibrium set. Extremal equilibria can be computed by finite iteration from the bottom and top concepts. The framework also introduces profile equivalence and quotient contexts, showing how to compress structurally identical laws and observers while preserving visible layers and equilibria. Several worked examples illustrate visible layer decompositions, multiple equilibria, and a medium-sized context suitable for experimentation. The paper is intended as a transparent, auditable module for no-meta model selection and internal law management in finite systems.