Interactive 3D diagrams of the framework's core geometric structures
These diagrams are not illustrations. They are the geometry the three axioms produce — derived, not assumed. Each is the same structure viewed from a different angle. If correct, they are the first glimpse of the actual shape of the universe. Drag to rotate, scroll to zoom. Best viewed on desktop.
VII. General Relativity Forming
Gravity is not imposed. It accumulates. Each actualisation event at the Present curves spacetime slightly outward — the geometric egress of the syphon leaves a permanent imprint on the accumulated Actual. Over many crossings, \(G_{\mu\nu} = 8\pi G\,T_{\mu\nu}\) emerges. The spacetime grid below starts flat. As crossing events concentrate, the grid dimples and curves. The curvature is not a force acting on the grid from outside. It is the grid — the shape of accumulated becoming-time, pressed outward by the ongoing bloom.
Accumulating — each gold flash is an actualisation event. The spacetime grid deforms around concentrations of crossings. The curvature depth is proportional to the crossing density — the more frequently the syphon crosses at a location, the deeper the well. This is \(T_{\mu\nu}\) generating \(G_{\mu\nu}\). No force. No action at a distance. Just accumulated geometry.
Two bodies — two concentrations of crossing events. Each curves the grid around itself. Between them the curvatures overlap — the grid is pulled toward both. What looks like attraction is two wells of accumulated geometry sharing a slope.
Geodesic — a test particle following the straightest possible path through the curved grid. It does not feel a force. It follows the geometry — the path of least accumulated becoming-time through the deformed Actual. This is what falling means in the framework.
VIII. The Hopf Fibration
The three-sphere \(S^3\) — on which the fermion orbits — has a remarkable structure: it fibres as circles over the two-sphere \(S^2\). Every point on \(S^2\) corresponds to a circle in \(S^3\), and any two distinct circles are linked exactly once. This is the Hopf fibration. It is the topological basis of spin-\(1/2\) — the reason a fermion must rotate \(4\pi\) to return to its original state. Each fibre circle is one orbit. The linking is the entanglement. The base sphere \(S^2\) is the space of directions. Rotate the base and the fibres rotate with it — but with a half-angle, because of the Möbius identification.
Fibres — each coloured circle is one fibre of the Hopf fibration: a closed orbit of the fermion on \(S^3\), projected into three dimensions via stereographic projection. The fibres near the north pole of \(S^2\) project to large circles; those near the south pole to small circles; the equatorial fibres form tori. No two fibres intersect — but any two are linked exactly once.
Linking — select any two fibres and they thread through each other. This is not coincidence. It is forced by the topology of \(S^3\). In the framework, this linking is the topological basis of quantum entanglement: two fermions sharing a common bilateral mesh are topologically linked, and no local operation can unlink them without tearing the manifold.
IX. CPT Symmetry
CPT symmetry — the combined operation of charge conjugation C, parity inversion P, and time reversal T — is the deepest symmetry in physics, confirmed to one part in ten billion. In the framework it is not postulated. It is the functional equation of the zeta function, which is the symmetry of the torus under the identification \(s \leftrightarrow 1-s\). C swaps matter and antimatter strands. P reflects the spatial geometry. T reverses the becoming-time direction. Their composition is the identity — the universe is invariant under the full CPT rotation. The Riemann Hypothesis is the statement that this symmetry is exact.
C swaps matter (blue) and antimatter (red) — the two strands of the bilateral mesh exchange. The crossing remains. P reflects the spatial geometry — left becomes right, the Möbius twist reverses handedness. T reverses becoming-time — Future and Past swap, ingress becomes egress. CPT combined is the identity: the universe after all three operations is indistinguishable from before. This is not a coincidence. It is forced by the functional equation \(\zeta(s) = \chi(s)\zeta(1-s)\), which is the torus symmetry \(s \leftrightarrow 1-s\). The Riemann Hypothesis — all zeros on \(\mathrm{Re}(s)=1/2\) — is precisely the statement that this symmetry is exact.