Born's Rule Level 2: Derivation of U(1) Phase Freedom from the K₄ Pulsation Structure in the PDL Framework

This paper resolves Open Problem OP4 of the Projective Dynamic Logo (PDL) programme by deriving the U (1) phase freedom of quantum mechanics from the combinatorial structure of the complete signed graph K₄, without presupposing complex amplitudes, Hilbert spaces, or gauge symmetry. Document D34 of the PDL corpus established Born's rule at Level 1: the modulus identity P = |ψ|² follows from the Gleason uniqueness theorem applied to the coherence-violation cost χ (τ;σ) ∈ +1, −1 of the K₄ structure. Level 1 left the origin of complex phases as an open problem. The present document closes this gap. Three propositions are established, each with complete proof. Proposition 1 shows that the global sign equivalence s ∼ −s of K₄ coherent configurations, combined with the half-cycle identity T² = −I₂ proved in D33, generates a canonical U (1) action on the amplitude space Hcyc ≅ ℂ²; the orbits are the fibres of the Hopf fibration S¹ ↪ S³ → S². Proposition 2 proves, by explicit algebraic computation verified over all 8 coherent configurations of K₄, that the (A) ∧ (B) stability criterion is U (1) -invariant: the cross-products P₁, P₂ ∈ +1, −1 are independent of any complex phase factor applied to ψ. Proposition 3 proves that the unique U (1) -equivariant probability measure on S³ whose marginal under the Hopf projection satisfies the Level 1 Gleason axioms is the round measure on S³, yielding Born's rule P = |⟨θ|ψ⟩|² on the base ℙ¹ (ℂ). Together, these three propositions establish that the U (1) phase freedom of quantum mechanics is not a postulate but a structural consequence of the K₄ pulsation: a state ψ and a phase-rotated state e^iαψ are physically equivalent because they correspond to the same physical K₄ configuration related by the global sign flip s ↦ −s. A connection to the Ginzburg–Landau order parameter of superconductivity is identified: the phase φ in Ψ = |Ψ|e^iφ is the fibre coordinate of the Hopf bundle generated by the K₄ pulsation, providing a combinatorial foundation for long-range phase coherence. This result completes the quantum dynamics layer of the PDL programme. The Schrödinger equation (D32), the Dirac equation (D33), Born's rule including the origin of complex phases (D34 + D46), the spin-½ double cover (D33), and the U (1) gauge freedom (D46) all follow from four combinatorial axioms on finite signed graphs, without free parameters and without additional postulates. The paper is self-contained: all necessary results from D29, D32, D33, and D34 are reproduced in full in Section 2.