The Transport Principle: Total Division in Clifford Algebras via Minimal Completion

In a Clifford algebra of mixed signature, null vectors are zero divisors: division by them is impossible. This paper states and proves a transport principle for such algebras, namely that division by zero promotes the space rather than the element, in three parts. First, a no‑go theorem: the natural conjecture — that some enlargement of the algebra assigns each null vector a canonically chosen and symmetry‑respecting pseudo‑inverse — is false in every enlargement however large, whenever the base space has dimension at least three, because the parabolic null rotations stabilising a null direction shear every candidate partner. The obstruction is genuinely a mixed‑signature phenomenon, and the dimension bound is sharp: the two‑dimensional hyperbolic plane, which is self‑complete, is the lone exception. Second, a resolution at the level of points: the geometric reciprocal x → x/Q(x) admits a total, continuous extension when points are represented on the null cone of the completed algebra Cl(p+1, q+1), with the formerly impossible values landing on adjoined ideal points at null infinity. Third, a minimality theorem: this completion is unique and minimal among null‑lift completions; the signature increment (+1, +1) is forced, never (+2, 0) or (0, +2). The representational machinery of the second part is not new: it is the conformal model of geometric algebra, due to Hestenes and coworkers and developed extensively since Hestenes–Li–Rockwood (2001), Dorst–Fontijne–Mann (2007), Doran–Lasenby (2003), and Angles (2008). The contribution of this paper is the no‑go theorem, the minimality theorem, and the change of reading they jointly mandate: the conformal construction is not an available convenience but the unique minimal completion demanded by the failure of division itself, with the element‑level alternative provably closed. A coordinate normal form (“swap and rescale”) identifies the mechanism with the classical linearise‑then‑normalise pattern of homogeneous coordinates and Möbius matrices, now encountered as forced rather than chosen.