The Analogy Between Symbiosis Order and Euler's Formula: Theoretical Evolution and Self-Revision from Version 1.0/1.2 to 2.4

The previous article Capital Can Circulate but Not Accumulate — Geometric Proof of Euler's Formula and Symbiosis Order analogized Euler's formula eiπ+1= to Version 1.0 of Symbiosis Order, which contains a fundamental flaw of inconsistent geometric imagery. This paper conducts a self-clarification: the core mechanism of Version 1.0/1.2 lies in the zero-sum neutralization of currency and credits ($$M+P=$$), whose geometric essence is a 45° linear constraint incompatible with the rotational imagery of Euler's formula. The subsequently designed Version 2.4 abandons this linear constraint and adopts a well-defined orthogonal complex number model (currency = real axis/expansion-contraction, credits = imaginary axis/rotation), upgrading Euler's formula from an "inappropriate metaphor" to a self-consistent theoretical framework. Version 1.0/1.2 and Version 2.4 are not substitutive but function as an emergency system and a long-term mechanism respectively. The former can be rapidly implemented on the existing economic structure within several weeks, while the latter requires long-term network and institutional construction. Together, they form a complete institutional spectrum of Symbiosis Order spanning from short-term emergency response to long-term equilibrium.