Deterministic Descent via Witness Principles in Discrete Global–Local Systems

This paper develops an abstract, deterministic descent framework for discrete global–local systems based on a witness principle. The framework shows that whenever a global state exceeds a specified threshold, the existence of a local actuator is forced, producing an exact one-step descent. Iteration yields finite termination without probabilistic, averaging, or density-based arguments. The theory is formulated independently of arithmetic. Selmer structures for elliptic curves provide a substantive instantiation, demonstrating that the framework genuinely bites in deep arithmetic settings without reliance on Euler systems, auxiliary class constructions, or randomness. Additional non-arithmetic examples illustrate portability beyond number theory. A central conceptual contribution is a reassignment of the role of local sites: local components (such as primes in arithmetic instantiations) act as witnesses that enforce global descent rather than as passive sampling points. This reframes local data as an active forcing mechanism in deterministic descent and rigidity arguments.