Symplectic Geometry of the \(T_3\) Deformation of \(φ^4\): Leaf Structure, Single‑Leaf Kinks, and the Fate of the Shape Mode.

We extend the leafwise symplectic invariance programme to the Chebyshev \(T_3\) deformation \(f(ψ)=4ψ^3-3ψ\) of the \(φ^4\) scalar field theory. The deformed potential has four non‑degenerate vacua at \(ψ=-1,-12,12,1\). The critical points \(ψ=±12\) split the field space into three open leaves, on each of which the deformation is a local diffeomorphism. We prove rigorously that no finite‑energy BPS kink crosses a critical point; instead three elementary single‑leaf kinks exist, \(K_L: -1→ -12,\; K_C: -12→ +12,\; K_R: +12→ +1\). The correct inverse branches on the three leaves are identified, yielding the exact kink profiles. The leafwise symplectomorphism theorem does not apply to any of these kinks because each touches at least one leaf boundary. Nevertheless, the universal translational theorem holds: the reduced centre‑of‑mass symplectic form is \(dP da\) with \(P = Mγ v\). Exact masses are \(M_L = M_R = 11/180\), \(M_C = 19/90\). We analyse the linearised fluctuation operators. The naive pullback of the \(φ^4\) shape mode is non‑normalisable for all three kinks. For the central kink, \(K_C\), the effective potential is a symmetric single well with \(U(0)=-10/9\) and a depth of \(19/9\) below the continuum threshold \(ω^2=1\). A numerical finite‑element analysis shows that this well does not support a bound state; consequently, none of the three elementary kinks possesses a discrete shape mode. The paper thus provides a complete, rigorous example of a non‑integrable deformation where the only surviving collective coordinate is the translational one.