The Final Bottleneck: Structural Obstructions to Odd Perfect Numbers

This paper completes a three-part structural investigation into the odd perfect number problem.

Building on two prior works that introduced valuation conservation and the σ-graph framework, this paper demonstrates that all known structural strategies for resolving the existence of odd perfect numbers encounter a single, irreducible obstruction.

We prove that valuation volume arguments are capped by a sharp ceiling at the maximal prime divisor, ruling out resolution by accumulation of exponent mass. We then show that congruence-based “σ-chain” and recycling arguments fail due to a fundamental multiplicity blindness: exact valuation conservation does not penalize repeated use of the same arithmetic constraints.

The problem is reformulated in terms of cyclotomic smoothness and S-unit equations, revealing that existing finiteness results are quantitatively mismatched to the internal demands imposed by a hypothetical odd perfect number. Prime-power exponent towers and cross-tower collisions are analyzed and shown to remain compatible with σ-closure under current methods.

The paper isolates a single remaining bottleneck: a quantitative scarcity principle for cyclotomic smoothness (or equivalently, an exponent prime-factor explosion). We prove that if this principle holds in particular under the abc conjecture, then odd perfect numbers do not exist.

This work does not claim an unconditional proof of nonexistence. Instead, it provides a complete structural resolution of the problem, rigorously identifying the exact point where all known techniques fail and reducing the question to a precise Diophantine obstruction.

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.)