A Descent-Based Structural Analysis of Goldbach Counterexamples

Assuming the existence of a smallest even integer E that fails to admit a Goldbach decom-position, this paper develops a structured descent framework based on decomposing the oddsummands of E into primes plus even remainders. Every nontrivial decomposition producesa strictly smaller even integer, which must satisfy Goldbach by minimality. This forces rigidstructural constraints on any hypothetical counterexample. We show that E must fall into one oftwo highly constrained configurations—the Isolated or Successive case—each of which imposesdeterministic composite patterns incompatible with the statistical distribution of primes. Thekey arithmetic lemmas underlying the descent mechanism have been formally verified in Lean 4.