SNS-THEORY-001 Volume III: Algebraic Certificate Features, Hopf Routing, and Quantum-Group Invariants

This volume introduces algebraic certificate features for code-theoretic and machine-learning objects over finite rings, and studies their invariance under Hopf-algebraic and quantum-group symmetries. It gives a functional certificate abstraction, examines braid-closure and quantum-shadow diagrams, and records the quaternion guardrail counterexamples used to bound the scope of finite-ring certificate methods. Certified results, candidates, and open problems are labeled with an explicit proof-posture taxonomy. Patent rights are reserved; see license terms.