The Resolution Hunt: Cyclotomic Smoothness, σ-Closure, and Structural Obstructions to Odd Perfect Numbers

This paper continues a structural investigation into the odd perfect number problem, building on a valuation-conservation and σ-graph framework developed previously. We show that all known approaches based on valuation volume, congruence constraints, and local order arguments are subject to a fundamental limitation: the system can evade contradiction by concentrating exponent mass into smooth, low-complexity exponents, creating an “inbreeding” or recycling loophole. We prove that valuation volume at the maximal prime is strictly capped, ruling out brute-force accumulation arguments. We then formalize the σ-closure condition as a system of S-unit constraints on cyclotomic values and show that congruence-debt and lcm-based arguments are insensitive to multiplicity, explaining why recycling remains possible. The core contribution is a complete structural map of the obstruction landscape. We demonstrate that any proof of nonexistence must rely on quantitative smoothness scarcity for cyclotomic values, and we isolate a precise missing lemma, Exponent Prime-Factor Explosion (EPF), which would force the appearance of a large prime divisor in an exponent. Assuming the abc conjecture, EPF yields an immediate contradiction via known lower bounds for prime factors of exponential expressions. This work reframes the odd perfect number problem as a Diophantine scarcity problem rather than a valuation or combinatorial one, reducing nonexistence to a sharply defined question about smoothness of cyclotomic values in finite prime sets. Related Work This paper builds on the structural framework developed in the companion preprint: Structural Constraints, Valuation Conservation, and σ-Graph Obstructions in the Odd Perfect Number ProblemJ. Rodgers (2026)Zenodo DOI: https://doi.org/10.5281/zenodo.18446275 Readers encountering this work first are encouraged to consult the above paper, which establishes the valuation-conservation framework, σ-graph formalism, and maximal-prime multiplier obstruction used throughout the present analysis.