PTRH Paper 26. We complete the resolution of Paper 22 Conjecture 1 in the PTRH framework. Paper 25 established that the nine-point configuration zₙ = zeta₉ⁿ is an exact saddle of the CHY scattering equations for all Z₉-symmetric massless kinematics and computed the Parke-Taylor factor and circulant-matrix structure of the CHY measure. The present paper supplies the amplitude half. A lemma classifies the 720 CHY saddles of M₀, ₉ by Z₉ orbit structure, identifying a unique Z₉-invariant fixed point. The PTRH encoding isometry (Paper 24) identifies sector-state external conditions with the Z₉-symmetric moduli locus, pinning each leg to zᵢ = zeta₉ⁱ; the CHY integral therefore reduces to the residue at the Z₉-invariant saddle. The explicit reduced bi-adjoint scalar amplitude is derived: Aₙ䃙 = -1/ (Deltaₘin¹8 * det'Phiₜilde), where Deltaₘin = 2 sin (pi/9) is the PTRH spectral gap and det'Phiₜilde is the reduced determinant of the Z₉-circulant CHY matrix of Paper 25. The structural form of Aₙ䃙 is shown to be analogous to the N=9 instance of the Guevara-Hu celestial qudit-chain construction. A corollary establishes BCFW bootstrap exactness: the full nine-particle amplitude is uniquely determined by recursion from the three-particle seed, and Paper 23 non-renormalization implies the tree-level sector-resolved structure is exact. Conjecture 1 is thereby resolved under the PTRH sector-state prescription.
George H. Bressler (Sat,) studied this question.