Two-Projection Reconstruction of Pure Axial Torsion in a Torsionful Twistor Setting

We study the reconstruction problem for pure axial torsion in a torsionful twistor setting on a four-dimensional Lorentzian spin manifold with metric-compatible torsion. Building on the pure-axial reduction of the torsionful α-surface integrability condition, we introduce a fixed primed dyad (oA', ιA') and consider the two projected axial variables φ_A := oA' SAA', φ'A := ιA' S{AA'}. These determine the full axial one-form pointwise through SAA' = φ_A ιA' − φ'A o{A'}. The main question is whether both projections can be obtained from twistor residual transport equations. We show that this is possible in a dual aligned sector of the primed Weyl spinor, characterized by the simultaneous conditions Ψ̃₀ = 0 and Ψ̃₄ = 0, for which the two residual systems reduce to transport equations for φ_A and φ'_A, driven respectively by Ψ̃₁ and Ψ̃₃. In this way, the full axial torsion one-form becomes reconstructible from two observation directions rather than one transport-visible projection plus complementary boundary data. We formulate a minimal dual-source model on a plane-wave-type background, with aligned source profiles in the Ψ̃₁ and Ψ̃₃ channels, and derive explicit integral expressions for the two projected torsion variables. This yields an explicit reconstruction formula for the full axial torsion one-form within the reduced model. The independence of the present paper rests on the dual dynamical accessibility of both projected variables from residual transport equations; no coupling between the two channels is assumed here. Any genuine coupling should be regarded as a further strengthening and is left for future work. This provides a first step toward a dyad-resolved tomographic reconstruction in a torsionful twistor setting. This paper is a companion to: Hiroyuki Shioiri, "A Pure Axial Specialization of the Torsionful Twistor Correspondence," Zenodo (2026), doi:10.5281/zenodo.19017912.