The Arena
The framework begins with a claim about shape: that settled fact — actuality — occupies a specific geometric space. This page introduces that space, and the larger space of possibility it sits inside, then shows the two ways the same actuality space can be built.
Start with possibility. Before a quantum system is measured, its state is a blend of the outcomes it could still resolve into — the framework represents that space of possibility as CP³. A measurement settles the blend into one outcome, and settled outcomes live in a smaller space nested inside possibility: RP³, the actuality space, the part that has become real.
This page is about RP³. It can be built two ways from possibility, and both give the same space: keep only what a measurement leaves unchanged, or take a three-sphere and fold every point onto the point opposite it. The two panels below show each route.
Two constructions, one space
Each panel builds RP³ by a different route. The first is the antipodal quotient: on the three-sphere S³, glue every point to the one diametrically opposite it. The panel shows this exactly. The circles are true Hopf fibers of S³, drawn by stereographic projection, and −p really is a half-turn along the fiber through p — so the identification p ∼ −p acts along each circle. A pair of antipodal points slides along its own fiber, fuses, and becomes a single amber point: one point of RP³, one settled fact. The Hopf base never moves — the projection works purely along fibers — and each cycle deposits its facts somewhere new, visiting the manifold before the pattern repeats.
The second is complex conjugation, z ↦ z̄: apply it across CP³ and one set of points stays unmoved — those equal to their own conjugate. That set is RP³. This map is the framework's operator τ, and a measurement is a single firing of it, a point of complex possibility projected onto the real locus τ fixes. CP³ has six real dimensions, so this panel is schematic where it must be; its exact core is the inset, where conjugation acts on CP¹ and lands on RP¹.
The two routes land on the same space and give it its double character — the real locus fixed by the measurement operator, and a closed three-manifold in its own right. The quotient panel is geometrically exact throughout; the conjugation panel is exact in its one-dimensional inset and a labeled stand-in elsewhere. What is drawn exactly is drawn exactly, and what is a stand-in says so.
The same shape, reached independently
The same three-dimensional projective structure was arrived at independently in perceptual phenomenology — a model of the geometry of conscious experience built from depth, perspective, and the relation between viewpoint and viewed, and validated against human perceptual data (Rudrauf et al. 2017). On the framework's side, RP³ is the set left fixed by τ; on the perceptual side, the same space came out of the geometry of experience. The framework treats that convergence as a starting point worth pursuing, with the case still to be made.
The Explainer is the plain-language introduction; the vocabulary is in the math primer; the same geometry set in motion across the energy cascade is the Model; and the full development is in the technical reference.
You've now seen the actuality space built two ways. There's also an experiment — The Road Behind — that lets you move around inside it, walking its shape until you reach the point where it loops back on itself.
Step inside the space →