On the Reduction of Erdős Problem #271 for S(0,4) to a Discrete Dynamical Invariant

We study exact structural reductions for the Stanley sequence S(0,4), motivated by the growth questions in Erdős Problem 271. The specific target is the doubling inequality a₂ⱼ ≥ 2a_j for j ≥ 8. We give a conditional second-moment criterion and explain, using a regular sequence of Rolnick and elementary energy lower bounds, why the generic properties of greedy progression-free sequences do not establish this criterion. We formulate the greedy process as an exact discrete dynamical system on backward distances and forward forbidden distances. A seed-specific invariant records precisely three uncovered exceptions and characterizes the reachable finite prefixes. We prove a two-step recurrence for a signed median margin and reduce a first violation of doubling to a critical interval-covering condition. Finally, we encode critical violations by finite systems of Boolean polynomial equations; the doubling conjecture is equivalent to their universal unsatisfiability, and hence to the existence of associated Nullstellensatz certificates. Explicit counterexamples refute pointwise fresh-coverage and defect-count criteria. The critical covering condition remains an open problem. The reductions concern doubling and do not establish the asymptotic growth of S(0,4) or resolve Erdős Problem 271.