Operator Families in Harmonic Operator Grammar for Biological-Information Syntax: Jurisdiction, Interaction, and Failure Conditions

This theoretical preprint refines the operator-family layer of Harmonic Operator Grammar for biological-information syntax. Building on the preceding formalization of HOG_BI = (S_BI, Int, Op, Gate, Fail), the paper defines the conditions under which an operator-eligible syntactic object may receive family-specific status as interval, pulse, silence, contour, register, density, transition, closure, or transposition. The central formal object of the paper is Fam_BI = (OpFam_BI, Prof, Jur, Gate_F, Fail_F, Interact). This object specifies how operator families are profiled, how their jurisdictions are bounded, how family-specific admissibility gates are declared, how family interactions are controlled, and how failed assignments may be weakened, suspended, retyped, reclassified, or rejected. The paper treats operator-family assignment as a formal classification layer rather than as biological mechanism, cybernetic control, information transmission, biological translation, implementation procedure, or external realization. It preserves the distinction between operator eligibility and family-specific status, and it defines failure conditions that prevent family vocabulary from collapsing into downstream claims. The framework is positioned between formal language theory, information theory, cybernetics, physical information theory, molecular biology, systems biology, network theory, falsifiability methodology, and admissibility-based classification. These domains function as boundary disciplines rather than proof domains. The paper therefore contributes an upstream formal architecture for later coordinate derivation, controlled case studies, biological translation protocols, and comparative biological-information analysis.