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.

Odd Perfect Numbers — Structural Resolution Series

This paper is part of a three-paper structural investigation of the odd perfect number problem:

1. Structural Constraints, Valuation Conservation, and σ-Graph Obstructions in the Odd Perfect Number Problem
   https://doi.org/10.5281/zenodo.18446275
   (Introduces the valuation conservation framework and σ-graph formalism, and proves the maximal-prime multiplier obstruction.)

2. The Resolution Hunt: Cyclotomic Smoothness, σ-Closure, and Structural Obstructions to Odd Perfect Numbers
   https://doi.org/10.5281/zenodo.18451695
   (Exhausts all valuation, congruence, and recycling strategies, reformulating the problem in terms of cyclotomic smoothness and S-unit constraints.)

3. The Final Bottleneck: Structural Obstructions to Odd Perfect Numbers
   https://doi.org/10.5281/zenodo.18453978
   (Completes the structural analysis, proving that all known approaches reduce to a single quantitative Diophantine bottleneck and establishing conditional nonexistence results.)