Structural Rigidity in Multiplicative Systems: Extremality, Constraints, and Structural Obstructions in Divisor-Sum Ratios

This work presents a unified structural perspective on extremal behavior in multiplicative arithmetic systems through the divisor-sum ratio σ(n)/n. Instead of treating constrained multiplicative families independently, the article develops a common framework based on structural rigidity. Three main rigidity mechanisms are analyzed: 1. parity rigidity;2. truncation rigidity;3. rearrangement rigidity. The paper argues that multiplicative extremality depends simultaneously on:- access to small primes,- unrestricted exponent growth,- and optimal exponent allocation. A structural rearrangement principle is introduced together with explicit examples illustrating how arithmetic constraints reduce multiplicative efficiency. The work consolidates previous independent results into a coherent structural program centered on multiplicative rigidity and constrained multiplicative optimization.