Weight-Basis Embeddings of the (2,6) Amplituhedron as a Logarithmic Pullback of the D₆ Weyl Arrangement

This paper constructs an explicit weight-basis embedding of the planar (k=2, n=6) amplituhedron into a D₆ ⊂ E₈ root system and demonstrates that its canonical differential form is a logarithmic pullback of the D₆ Weyl hyperplane arrangement. Using a fixed weight-basis projection and a D₆ root chain, we show that: • Every physical cyclic boundary corresponds to a simple root (‖α‖² = 2).• Non-cyclic minors correspond to composite roots (‖α‖² = 4) and yield zero residue.• The singular divisor of the canonical form equals the union of D₆ Weyl mirrors.• Residues at physical poles are identically unity. We further examine planar n = 8 MHV configurations and show that the integer Dₙ lattice remains sufficient in the planar case. The necessity of the E₈ spinor sector for NMHV (k = 3, n = 7) configurations is analyzed and formally stated as an open problem. This work establishes a concrete algebraic bridge between amplituhedron geometry and Weyl hyperplane arrangements, providing a structural interpretation of scattering singularities as root-system mirror crossings. 4️⃣ Keywords #Amplituhedron #Positive Grassmannian #Weyl_Group #D6_Root_System #E8_Lie_Algebra #Scattering_Amplitudes #Canonical_Differential_Forms #Plücker_Coordinates #Logarithmic_Singularities #Coxeter_Groups #Root_Systems #Planar_MHV_Amplitudes #Algebraic_Geometry_in_Physics #Representation_Theory #Geometric_Scattering_Theory