The Born rule — the postulate that measurement probabilities in quantum mechanics are given by ||² — is typically introduced as an axiom. We show that the functional form of the Born rule is already present in the semiconjugacy between the doubling map on the circle and the logistic map at r = 4: the semiconjugacy h () = ² (/ 2) is identically the Born probability for a real qubit parameter. The exponent 2 in ||² is forced by the degree of the doubling map, which is the minimal non-trivial degree for a self-map of S¹. The semiconjugacy is the unique continuous function satisfying three conditions: evenness (invariance under the Z/2 quotient that destroys phase), the intertwining relation with the logistic map, and homeomorphism on a fundamental domain. We then show that the full complex Born rule arises from the Hopf fibration S³ CP¹, which is the natural complexification of the real quotient, and that the complete U (1) -invariant content of the quotient is the density matrix = ||. The invariant measure transitions from the arcsine distribution (Beta\! (12, 12) ) in the real case to the uniform distribution (Beta (1, 1) ) in the complex case, with a geometric explanation in terms of the transition from fold to fibration. These results establish that the Born rule's functional form is a consequence of the degree-2 quotient structure rather than an independent postulate.
theurgi (Thu,) studied this question.