The Parametric Family x² − x ± 1: Axioms, Algebraic Duality, and the Born Rule

We prove that the parametric family x² − x + c (c = ±1) is uniquely forced by three axioms: conservation, minimality, and binary coupling. The two polynomials generate the simplest real and imaginary quadratic number fields — ℚ(√5) and ℚ(√−3) — and classify trace-1 dynamics in GL(2,ℤ) into hyperbolic and elliptic regimes. The elliptic regime determines a unique positive-definite inner product via Schur’s lemma. Under continuous-time propagation, the hyperbolic regime forces complex-valued amplitudes. The Born rule p(z) = |z|² is derived as the unique measurement map respecting the dynamically determined inner product and tensor product compositionality. Ten independently verified physical domains — spanning condensed matter, quantum foundations, dynamical systems, and statistical mechanics — are cataloged as empirical instances of this algebraic structure at critical boundaries. No free parameters appear in the algebraic structure itself.