The Complete Circle: Extending the Semicircle Constraint to 2π with Conjectured CPT Correspondence

We extend the semicircle constraint (q - 1/2)² + C_qc² = 1/4 to the complete circle by including both roots C_qc^(±) = ±√(q(1-q)). We PROVE: (1) the semicircle arc length equals π via the Fisher information metric; (2) the complete circle has information-theoretic circumference 2π; (3) the angular parameterization q(φ), C_qc(φ) covers the full circle for φ ∈ [0, 2π). We CONJECTURE: (4) the negative root C_qc^(-) corresponds to the CPT-conjugate sector; (5) this 'missing π' represents antimatter and time-reversed processes; (6) the value π emerges fundamentally from quantum probability geometry and propagates to circular structures at all scales. The visibility formula V(q) = 2q(1-q) is derived as a testable prediction. This work establishes the mathematical foundation while proposing physical interpretations connecting quantum geometry to particle-antiparticle duality.