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偏元 e——方向偏好场中不可约的生长形态(Partial-Deviation e: The Irreducible Growth Form in a Direction-Biased Field)

Year: 2026
Type: preprint
Abstract: 本文尝试在偏元数学框架(0-∞.0)内重新阐释数学常数e。经典e定义为极限lim_{n→∞} (1+1/n)^n,代表在无方向、完美对称场中的生长。我们提出一个替代定义:e是一个方向偏好场中不可约的生长形态,在任何具有非零原点偏差ε的物理系统中,它都无法被简化为一个标量常数。在方向偏好场中,每一步复利都携带方向印记:标准复利表达式变为e_ε = lim_{n→∞} (ε + 1 + ε/n)^n,产生一个向量值常数而非标量。e_ε的模长近似等于经典e加一个ε的一阶修正,但其方向编码了系统的内在偏好。当ε → 0时,向量e_ε退化为标量e ≈ 2.71828…。本文提出,e不是一个普适常数,而是一个自组织极限——系统从具有偏好的原点出发,在维持结构稳定的前提下所能达到的唯一生长速率。我们提供一个可证伪条件:如果发现一个自组织系统的生长速率精确等于经典e,但其原点偏好被独立测量为严格为零,则本文对e的偏元数学解释被证伪。 This paper attempts to reinterpret the mathematical constant e within the framework of Partial-Deviation Mathematics (0-∞.0). Classical e is defined as the limit lim_{n→∞} (1+1/n)^n, representing growth in a directionless, perfectly symmetric field. We propose an alternative definition: e is a direction-biased growth form, irreducible to a scalar constant in any physical system with a non-zero origin bias ε. In a direction-biased field, each compounding step carries a directional preference: the standard compounding expression becomes e_ε = lim_{n→∞} (ε + 1 + ε/n)^n, producing a vector-valued constant rather than a scalar. The magnitude of e_ε approximates the classical e plus a first-order correction in ε, but its direction encodes the system's inherent bias. When ε → 0, the vector e_ε degenerates to the scalar e ≈ 2.71828… . This paper proposes that e is not a universal constant but a self-organization limit—the unique growth rate at which a system, starting from a biased origin, can grow while maintaining structural stability. We provide a falsification condition: if a self-organizing system is found whose growth rate exactly equals the classical e but whose origin bias is independently measured to be strictly zero, the Partial-Deviation interpretation of e is falsified.
DOI: 10.5281/zenodo.21074326
Author: Chen, Song
Language: English
Open Access status: green
OpenAlex ID: W7166662238
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