Two-Branch Transport — Phase Portrait in (ρ, Φ)

ρ = ½ ln(r₁/r₂)  ·  Φ = θ₁ + θ₂  ·  ρ̇ = −κusinh(2ρ)cosΦ  ·  Φ̇ = 2ω − 2κucosh(2ρ)sinΦ

1.00
0.60
0.25
Constructive (locked, persistent)
Frustrated (locked, decaying)
Inverted / anti-phase
Dephased (unlocked)
Persistence boundary
Locking boundary
Stable node
Saddle / unstable
Boundaries. The persistence boundary (dashed teal) is κucosh(2ρ)cosΦ = γ; inside the green band the transport amplitude grows (particle-like); outside it decays. The locking boundary (solid red) is |ω| = κucosh(2ρ); it is a vertical line at ρ = ±ρL = ±½arccosh(ω/κu), visible only when ω > κu. Inside the locking boundary the state is dephased (no phase-locked fixed point).   Try dragging ω above κu to see the dephased band open at small ρ.

Fixed points. When ω ≤ κu: two fixed points at ρ = 0, sinΦ = ω/κu. The stable node (filled, Φ ≈ arcsin(ω/κu)) is the Constructive attractor — both branches phase-lock and the amplitude persists; by the orientation axiom (κu > 0, Reu(AB)|* > 0) this is the direct observable readout state. The unstable node (open circle, Φ ≈ π − arcsin(ω/κu)) divides basins. When ω > κu: the two roots merge onto the locking boundary at Φ = π/2 (degenerate).

Classification. Constructive: κu > 0, locked, κucosh(2ρ)cosΦ > γ. Frustrated: locked, |κucosh(2ρ)cosΦ| < γ (amplitude decays slowly). Inverted: locked, κucosh(2ρ)cosΦ < −γ (anti-phase; decays rapidly). Dephased: |ω| > κucosh(2ρ) (no phase locking; incoherent transport).