Extension of the Mathematical Bridge of the Functional Operator Set: Closure Rank Function and Operative Stability Criterion

Key Points

• This research aims to extend previous work by introducing a closure rank function and deriving a stability criterion for functional operator sets.
• Developed the closure rank function for closed functional operator sets.
• Derived a locally verifiable stability criterion.
• Bridged theoretical concepts with V-layer implementation using a four-element certificate.
• Established an auditable stability criterion for functional operator sets.
• Demonstrated equivalence among algebraic, graph-theoretic, and category-theoretic stability conditions.
• Proposed a four-element certificate comprising D, B, delta, and epsilon_core for practical implementation.