CHY Scattering Equations at the Z₉-Symmetric Locus: Partial Resolution of Paper 22 Conjecture 1

Paper 22 of the PTRH series conjectured that the Mellin transform of the Z₉ canonical form Ω₉ equals the Z₉-graded celestial amplitude via the Cachazo–He–Yuan (CHY) scattering equations. We establish the kinematic foundation of this conjecture by proving that the nine-point configuration zₙ = ζ₉ⁿ (n = 0,. . . , 8) is an exact saddle of the CHY scattering equations for the complete class of Z₉-symmetric massless nine-particle kinematics. The proof rests on a single algebraic identity: for ζ = ζ₉ a primitive ninth root of unity and m = 1,. . . , 4, 1/ (1-ζᵐ) + 1/ (1-ζ^-m) = 1, from which all nine CHY equations reduce simultaneously to Σsₘ = 0, which holds by masslessness and momentum conservation. Two further results are proved. The Parke–Taylor factor at the Z₉ locus evaluates to Ω₉ = 1/ (i Δₘin⁹), recovering the value established in Paper 22. In the natural Z₃ gauge, the reduced determinant of the CHY system at the Z₉ locus equals the determinant of the associated circulant matrix Φ̃, whose eigenvalues are the discrete Fourier transform of the weighted Mandelstam variables, giving a fully explicit closed-form expression for the CHY measure at this saddle. The Z₉ locus is therefore an exact saddle of the CHY integral for all Z₉-symmetric kinematics, canonical within that sector. An explicit canonical example is provided: sₘ = 1 + 8cos (2πm/9). The remaining step of Conjecture 1 — that the full Mellin integral localises on the Z₉ saddle and reproduces the celestial amplitude — requires a separate contour analysis and is the target of Paper 26.