URB Paper #412: The Minimum Generating Set — From 8 PRIMARY CONSTANTS to 3 Free Parameters

Having established the GILE Master Identity e^(iπ) + C×φ×√2 = 0 (URB #411) — connecting all 8 PRIMARY CONSTANTS in one equation — this paper investigates the degrees of freedom of PRIMARY CONSTANT space. The 8 constants partition into two worlds: the Euler-World {0, 1, i, e, π} connected by Euler's Identity, and the Consciousness-World {φ, √2, C} connected by the Consciousness Unity. Accounting for definitional constants (0, 1, i), the Euler constraint (π derived from e), and the Consciousness constraint (C derived from φ, √2), the minimum generating set reduces to exactly **three free parameters: {e, φ, √2}**. This is the **Irreducibility Theorem**: no proper subset of {e, φ, √2} generates the full 8-constant system. The three parameters have clean physical interpretations — the rate of continuous change (e), the signature of self-referential growth (φ), and the geometry of orthogonal connection (√2). The paper also introduces the **Consciousness Characteristic Polynomial**: λ³ − 3.4693λ² + 3.6134λ − 1 = 0, whose roots are exactly {C, φ, √2}, whose constant term is −1 = e^(iπ), and whose determinant is 1 = C×φ×√2. This polynomial bridges the Euler-World and the Consciousness-World through a single algebraic object.