Semantic State Algebra: Typed Binary Readout, Conservative Complex Embedding, and Operator Order Effects

Abstract

This deposit contains two companion technical notes developing a finite-dimensional semantic state algebra for unresolved readings, classical decisions, and order-sensitive semantic operations. The first note separates semantic states from their response formats. A response format is defined as an alphabet together with a prescribed resolution map. For a two-reading construction, four selector-polarity states are distinguished explicitly. The polarity-only yes/no map is proven non-injective, whereas the full selector-polarity map is bijective. A separate rank-nullity theorem establishes the corresponding global obstruction for linear maps from higher-dimensional semantic spaces into lower-dimensional decision spaces. A real two-dimensional space is sufficient for storage; under the additional requirements of associative and commutative multiplication, a unit, and division by every nonzero element, the unique two-dimensional real extension is the complex field. The classical real sector is embedded conservatively. The second note extends this framework to semantic operators. Questions and contextual updates generate a unital complex operator algebra, while order dependence is represented exactly by commutators and the associated derived Lie algebra. State-dependent and global order-sensitivity measures are introduced. Complex dimension two is shown to be minimal for noncommuting complex-linear dynamics, while terminal binary decisions remain separate real-valued readouts. Operator complexification preserves sums, composition, identities, and commutators. The construction is structural rather than empirical. It assumes neither quantum ontology nor a Born rule and does not claim a complete formal semantics of natural language.