The Fallacy of Dispersity Metrics in Symmetric Macromolecular Networks: Reclaiming Number-Average Mass (Mn) as the Sole Determinant of Mechanical and Rheological Equilibrium

<title>Abstract</title> <p> For nearly a century, polymer science has relied on weight-average molecular weight (Mw) and the polydispersity index (PI = Mw/Mn) as foundational metrics to characterize macromolecular chain distributions. This paper exposes a fundamental mathematical and epistemological error embedded within these standard conventions. We prove that Mw, Mrms (Root-Mean-Square molecular weight), and standard variance (σ) parameters are artificial, human-fabricated impositions that fail to reflect the physical reality of a polymer system. By modeling a strictly symmetric Gaussian number distribution of chain lengths,at macroscopically relevant high-molecular-weight regimes (Mn = 10 ⁵ and 10 ¹⁰ g/mol), we demonstrate that as the system breadth broadens to extreme limits, the actual structural,ratio between the longest and shortest chains explodes 19-fold, while Mw registers an artificial, minor shift of less than 10%. Because the distribution remains perfectly symmetric, the longer chains on the right tail are naturally and thermodynamically balanced by the shorter chains on the left tail—a manifestation of a closed mass-energy budget reminiscent of spacetime topology and cosmological equilibrium. We establish that a polymer synthesis reaction conducted in an open system can never yield a dissymmetric energy distribution; such profiles are forbidden by non-equilibrium thermodynamics. Crucially, we present unassailable chromatographic proof demonstrating that ideal Gel Permeation Chromatography (GPC) detector peaks are inherently symmetric Gaussian curves, meaning that the perceived asymmetry of polymer distributions is entirely an artifact of a non-linear mathematical imposition inside computer software. Consequently, we show that tracking statistical deviations ( σ) is an exercise in error-analysis misapplied to a living, self-limiting thermodynamic system. For the practical application of achieving optimal material behavior, the target of the polymer scientist must shift exclusively to the Number-Average Molecular Weight (Mn). We provide a unifying mechanical-rheological framework showing that maximizing Mn linearly drives mechanical toughness, while minimizing Mn governs pure melt flow properties. The mid-value Mn represents a natural, analog balance of processability and toughness, eliminating the need for Mw entirely. </p>