Conjugation Equivariance and Transport Parity in the Newton Dynamics of C_α(z)

Scoped supersession notice (22 September 2026). This record is part of the source research consolidated in Paper A12, Dynamics of the Lambert W Newton Iteration: Basin Geometry, Routing Transitions, and Finite-Rank Period-Three Resonance (DOI: 10.5281/zenodo.21845665). For the conjugation-parity material from Paper 33 stated in that article, it supersedes the corresponding earlier programme presentation and is the current consolidated reference. Paper 33 also serves other programme work. Supersession is limited to its A12 contribution; other results, programme interpretations, and open problems in this record are not thereby replaced or resolved. In particular, the finite-matrix results do not assert a continuum resonance. The original description below and deposited files are retained for provenance. Paper 33 in the "Geometry of the Critical Line" programme. The Newton transport arc (Papers 20–32) established the exact empirical equality P_{C,-1} = P_{C,0} — equal bounce traffic through conjugate corridors — at every tested α, to machine precision. This paper derives the transport parity as a theorem. For real α, the Cover Function C_α(z) = z − exp(−α/z) is equivariant under complex conjugation: C_α(z̄) = C̄_α(z). The Newton map inherits this equivariance, which exchanges the basins of z_k and z_{-(k+1)}. Any conjugation-symmetric initial measure therefore produces equal bounce probabilities through conjugate corridor pairs. A precise structural correspondence with the SCT geometric framework (Papers 0–10) is identified: the functional equation s → 1−s is a point reflection that reduces to pure complex conjugation if and only if σ = 1/2 (Proposition 5.1). The critical line is the unique locus where the SCT symmetry reduces to pure complex conjugation, matching the involution that drives the Newton transport parity. The Newton-side theorem is fully proved; the formal coordinate map connecting the s-plane to the z-plane remains future work. Part of the Geometry of the Critical Line programme.