Why Use Euler's Formula to Test the Symbiotic Point System Design ——Not Labeling, but Verifying the Feasibility of "Decentralization with Fluidity"

Abstract: Why Use Euler's Formula to Examine This Institutional Design—— Not for Labeling, but for Testing the Possibility of "Non-Concentration with Fluidity"Human society has long faced the fundamental paradox between "vitality and concentration": a pure market economy preserves the fluidity of capital but inevitably leads to irreversible concentration; traditional regulation or redistribution can prevent concentration but at the cost of suppressing vitality. The root of this dilemma lies in the geometric structure of a one-dimensional line — on a single axis, "fluidity" and "non-concentration" are mutually exclusive; you cannot have both. Euler's formula e^i=+i provides a minimalist testing criterion. It depicts circular motion on a two-dimensional complex plane, characterized by three key features: two orthogonal and independently operating dimensions (real axis and imaginary axis), a constant modulus that does not converge to a极点, and endogenous fluidity arising from continuous angular change. These three features together constitute the geometric prototype of "non-concentration with fluidity. "Using this framework to examine the three versions of the Symbiosis Points System yields clear results: · Pure market economy: Only a real axis (capital), no independent imaginary axis → one-dimensional line; vitality and concentration are two sides of the same process, inevitably leading to monopoly. · Versions 1. 0/1. 2 (zero-sum neutralization): Currency and points are linearly hedged against each other, still a zero-sum oscillation on a single dimension → reverse coercion on a one-dimensional line; concentration can be prevented, but fluidity is suppressed. · Version 2. 4 (dual-track currency + points): Currency (real axis) handles transactions and expansion; points (imaginary axis) handle behavioral guidance and credit entitlement; the two dimensions are orthogonal and independent → capital can grow freely (vitality), but accumulation does not automatically translate into privilege (non-concentration) ; fluidity arises from individuals' self-interested choices navigating between the two tracks. Boundaries must be clarified: the examination using Euler's formula is a geometric structure test, not a proof of engineering feasibility. It answers the question "Does the underlying logic of this institution allow 'non-concentration with fluidity' to coexist mathematically? " rather than "Can it resist capture and prevent vulnerabilities in reality? " Passing this test is a necessary but not sufficient condition — it filters out designs doomed to fail on a one-dimensional line, leaving those that must still undergo further engineering scrutiny. Therefore, using Euler's formula to examine this institutional design is not about attaching an elegant mathematical label, but about answering a more fundamental question: Has this institution truly broken free from the cage of one-dimensional thinking? Only by passing this geometric structure test does a design qualify to enter the next stage of discussion: "how to implement it. "