The Arena
The anchoring argument lands on a specific space: RP³, real projective 3-space. This page asks what that space sits inside, and shows the two ways it can be built.
RP³ is three-dimensional, and the complex projective space that has it as its real locus is fixed by that — CP³, and no other. The map that picks RP³ out of CP³ is complex conjugation, which the framework calls τ: apply it across CP³ and the points that stay put are exactly RP³. Among complex projective spaces exactly one has RP³ as its real locus, so the anchor names the space and the space names what contains it.
CP³ is also where quantum mechanics already works: the space of possible states for a quantum system is a complex projective space, and CP³ is the one belonging to a four-state system. Possibility is CP³; actuality is RP³; measurement is the process that settles the first onto the locus τ fixes.
The two panels below build the same actuality space by different routes: keep only what conjugation leaves unchanged, or take a three-sphere and fold every point onto the point opposite it.
τ in the one dimension where it can be drawn exactly. Conjugation acts on CP¹, the Riemann sphere, and the points it leaves fixed are the real circle RP¹ — the amber equator. The same map on CP³ fixes RP³, by the same argument in three dimensions instead of one. CP³ has six real dimensions and no honest picture, so this is the whole of it that can be shown rather than asserted.
Two routes, 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 it draws are Hopf fibers — S³ can be filled completely by circles, no two of which meet, and each of those circles is one fiber. Every point sits on exactly one. They are drawn here 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 of RP³. The identification is timeless; the cadence, and the spread of it across the manifold, are there to make it legible. The Hopf base never moves — the identification 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. It is an involution, not a projection: it does not carry possibility anywhere, it fixes where actuality lies. The process does the carrying; τ says where it lands. RP³ sits inside CP³ as a totally geodesic Lagrangian submanifold, which is what the fixed locus of an antiholomorphic isometric involution always is.
What that process is — the dynamics carrying possibility onto the locus τ fixes — is the open construction, and it is where the framework's current work sits. Naming the operator and naming the target is not the same as having the map. Everything quantitative waits on it, which is why there are no numbers on this site.
CP³ has six real dimensions and cannot be drawn, and a picture that pretends otherwise teaches the picture rather than the map. So the second panel drops to CP¹, where conjugation is drawable exactly and the fixed set is a circle you can see. Nothing about the argument changes with dimension: the fixed points of z ↦ z̄ are the points with real coordinates, in one complex dimension or in three.
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. Both panels are geometrically exact. Where something cannot be drawn exactly, this page does not draw it.
Where this structure came from
The projective structure entered this framework from perceptual phenomenology — Rudrauf and colleagues' model of the field of consciousness (Rudrauf et al. 2017) is where it came from, and the debt runs that way round rather than being a second line of evidence. The anchor page gives that its proper accounting, along with what the identification does and does not establish.
The Explainer is the plain-language introduction; the argument that leads here is on the conditions; the same geometry set in motion is the Model; and the papers are on the papers page.
You've now seen the actuality space built two ways. The Road Behind lets you move around inside it, walking its shape until you reach the point where it loops back on itself — the two-valuedness the argument leans on, made walkable.
Step inside the space →