Outside the Closure: Reverse Polarisation, Internal Charges and the Many-Body Boundary of the Weyl Orbit

Volumes I and II fixed the identities of Patrick B. Reynaud's letter on Souriau's moment map with a dilatation, and closed the ten-dimensional Weyl orbit: a Darboux chart with mass plane, a stratification by causal type and spin, reduction to Souriau's space of motions, geometric quantisation that integrates the spin sphere and refuses an integer on the open plane, and a Liouville derivation of the Gibbs product measure. Volume III is the book of what lies outside that closure. The reverse map η, exchanging P with K and sending D to −D, is an outer automorphism of so(2,4) and a conformal inversion on the moment map; it preserves orbit dimension but does not preserve a Hamiltonian in the momentum polarisation. A compactification of the mass plane that would quantise mass replaces the Weyl orbit. The count 11 + 2 + 6 = 19 adds generic coadjoint orbits of su(2) and su(3) that the Standard-Model multiplets do not occupy. The tetraquaternion, axial torsion torque and tree-level g-factor are records or nouns, not ranks. A gas of Weyl bodies is either the free-mass ensemble already derived or a current-algebra orbit not yet built. The Bondi–Metzner–Sachs group shares the massless absence of a scale and is not a corollary. The script remains a debt. KSCCN review article (Volume III), 5 October 2026. Continues https://doi.org/10.5281/zenodo.23164930 and https://doi.org/10.5281/zenodo.23165689. Dr. Syed Muntasir Mamun, ORCID 0000-0001-6845-2853. Views are the author's.