Zero Divisors as Signposts: A Transport Principle for Clifford Algebras, from Gravitational Singularities to a Candidate Theory of Everything

Abstract

Division by zero is mathematics' most famous forbidden move. This paper takes the opposite view: in the algebras that most naturally describe spacetime - Clifford algebras - the failure of division is not a defect but a signpost: dividing by zero does not break the mathematics, it transports you to a larger, higher-dimensional algebra where the impossible operation finally makes sense. We prove three things about this ascent. It can never be achieved by repairing the offending element - no enlargement can single out a repair that is not an arbitrary choice: a no-go theorem. It can be achieved by enlarging the space of points instead, exactly as a geometer's blow-up repairs a singular point. And there is only one minimal way to do it: the enlargement is forced, not chosen. We then apply the idea to the singularity inside a black hole - physics' most famous division by zero. There the forced enlargement carries four-dimensional spacetime up to a six-dimensional geometry, landing, unprompted, on Penrose's twistor theory - and Einstein's equations for empty space were living there all along, as the statement that a single ideal point at infinity never changes. Along the way, the singularity acquires quantum ladder structure with no quantisation imposed; counting its modes reproduces the scaling of black-hole entropy; projecting back yields Hawking's radiation exactly - its temperature and both of its statistics - given a standard state; and the twistor match is an identity, not an analogy. The information paradox dissolves into a category error. Information is lost in four dimensions and preserved in six - the same physics read at two levels - the deficit a fixed algebraic projection whose shadow is the thermal Hawking flux; granted two structural axioms, the radiation's entropy follows the Page curve exactly. And gravity's own action emerges as a fifty-year-old construction completed - one that can exist only because the cosmological constant is not zero. The two pillars themselves then turn out to be one object: quantum mechanics and general relativity are the two canonical presentations of the same Clifford relation - the metric is the anticommutator - two shadows of a single algebra, with spacetime itself the manifold of its quantum vacua. The algebra's eight parallel towers yield dark matter as ordinary matter in worlds next door, sharing one gravity - a dark-to-visible ratio of five against the observed 5.4 - while the filled twistor stakes the neutrino mass sum at the oscillation floor with normal ordering. Pressed to its end, the framework yields the Standard Model's algebra, content and charges - a candidate theory of everything, its remaining freedom - a ledger's length, an arrow's strength and sign - resolved into the date of the universe's own division.