Harmonic Operator Grammar for Biological-Information Syntax: Formal Operators, Admissibility Gates, and Syntax-to-Operator Transition

This paper formalizes Harmonic Operator Grammar as a typed admissibility framework for biological-information syntax. It defines the conditions under which a declared syntax-like structure may pass from grammatical admissibility to operator eligibility while preserving the distinctions among formal syntax, information processing, cybernetic control, biological comparison, empirical validation, and external realization. The central object is HOG_BI = (S_BI, Int, Op, Gate, Fail), where S_BI is the biological-information syntax base, Int the interval-assignment layer, Op the operator-family layer, Gate the admissibility-gate layer, and Fail the rejection-condition layer. The paper defines the syntax base S_BI = (Σ, Str_Σ, Rel_Σ, Lev, Bound), introduces the syntax-to-operator transition, and specifies the gate architecture required for operator assignment, route candidacy, interface candidacy, demonstration, and falsifiability. Harmonic Operator Grammar is positioned as a cross-domain formal interface among formal language theory, information theory, cybernetics, molecular biology, systems biology, network theory, and admissibility-based classification. Its purpose is not to reduce these domains to a common vocabulary, but to provide a controlled grammar through which biological-information syntax may be typed, constrained, upgraded, and rejected without collapsing syntax into mechanism, information into control, route into interface, or formal eligibility into empirical realization. The paper introduces a baseline set of operator families: interval, pulse, silence, contour, register, density, transition, closure, and transposition. Each family is treated as a formal operator class requiring declared jurisdiction, admissibility conditions, and failure modes. The framework remains prior to biological translation and does not claim empirical biological validation. It provides the grammar required for later operator-family refinement, coordinate derivation, controlled case studies, biological translation protocols, and comparative biological-information analysis.