Erdős–Moser baseline: odd exponents and congruence restrictions

The response uses modular pairing to eliminate other odd-exponent solutions. For any further solution, it derives even k ≥ 2, m ≡ 3 (mod 8), restrictions modulo prime squares, and k < m log 2. These are necessary conditions, not a proof of nonexistence. The remaining gap is to exclude every pair satisfying those conditions. The full argument is preserved in the attached transcript; no novelty claim is made.