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.
IV. The Standard Model — Derived
The Standard Model is not assumed. Every element — the gauge group \(\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)\), the coupling constants, the lepton masses, the Higgs vev, the Weinberg angle — is derived from the three axioms and the geometry of \(S^3\times\mathbb{CP}^2\). The derivation chain runs in one direction: from axioms to geometry to symmetry to coupling to mass. Nothing is put in by hand.
The particle emergence diagram below shows what this means dynamically. The Present is the gold disc at the centre — the crossing, the zero, the wormhole throat. Quantum potential falls inward from all directions (blue). At the crossing, the Möbius identification occurs. What emerges outward is not formless energy but structured particles — each carrying the quantum numbers forced on it by the geometry of the crossing. The electron emerges at generation \(n=0\) with coupling coefficient \(k_1 = 0.0723\). The muon at \(n=1\), the tau at \(n=2\). The gauge bosons emerge from the symmetries of \(S^3\) itself. The Higgs from the \(j=0\) mode. Nothing is added. Everything follows.
Particle emergence
The gold disc is the Present — the wormhole mouth, the Riemann zero at \(t_1 = 14.134\). Blue particles are quantum potential falling inward from all directions, drawn by the \(\nu^2/\tau\) potential. At the crossing the Möbius identification fires. What blooms outward is structured: the three lepton generations at their spectral positions \(t_1, t_2, t_3\) (electron red, muon green, tau purple), the gauge bosons from the \(S^3\) symmetry group (photon gold, W/Z teal, gluon orange), and the Higgs from the \(j=0\) mode (white). Each particle carries its quantum numbers from the geometry of the crossing — not assigned, derived. The syphon runs continuously. Reality is the ongoing outward bloom from the Present.
The image itself induces a visual aftereffect — look away after sustained viewing and the world briefly expands toward you. The framework explains this directly: the visual cortex runs at approximately 40 Hz, close to the natural crossing rate of the animation's outward bloom. The visual mesh entrains and briefly continues crossing at that frequency after the input is removed. The world expands because your visual system is momentarily running the universe's own projection with no scene to project onto. The image works because it is not a picture of the framework. It is the framework, at the scale and frequency at which the eye actualises.
V. The Wavefunction
The wavefunction in the framework is not the standard flat-space Schrödinger solution. It is the radial solution on \(S^3 \times \mathbb{R}_\tau\) — governed by the Breit-Wigner resonance at the Present, with corridor width \(\gamma = 1/4\), collapsing inward under the \(\nu^2/\tau\) potential and blooming outward after the crossing. Three views of the same object: the wavefunction shape, the probability density, and the becoming-time picture of the syphon.
ψ radial
The wavefunction on \(S^3\). Blue: the quantum infall phase — oscillating as \(r/\tau\) decreases toward the throat. Gold dashed line: the Present at \(t_1 = 14.134\), corridor width \(\gamma = 1/4\). Red: the geometric egress — amplitude decaying as geometry accumulates. Animate to see the syphon breathing.
|ψ|² density
The Breit-Wigner probability curve — probability of actualisation as a function of spectral frequency \(E\). The red needle scans continuously but is pulled toward each Riemann zero by the level repulsion of the surrounding mesh, dwelling longest where the probability is highest. The peak at \(t_1 = 14.134\) is the electron — the ground state, the most probable crossing, the longest dwell. The zeros at \(t_2\) and \(t_3\) are the muon and tau — visited less often, higher energy, shorter dwell. The mass hierarchy is not assigned. It is the dwell time of the syphon at each zero.
Becoming-time
The Present is the gold point at \(\tau = 0\). Blue waves converge inward from the Future (\(\tau \to \infty\) above). Red waves spread outward into the Past (\(\tau \to \infty\) below). Animate — this is the syphon running in time. The crossing is not an event in time. It is where time comes from.
VI. The Riemann Zeta Function
The Riemann zeta function \(\zeta(s)\) evaluated on the critical line \(\mathrm{Re}(s) = 1/2\) is the spectral backbone of the framework. The Z-function \(Z(t) = e^{i\theta(t)}\zeta(1/2+it)\) is real-valued on the critical line and passes through zero at each Riemann zero \(t_n\). These zeros are the crossing frequencies — the spectral positions at which the syphon actualises. The level repulsion between zeros is the GUE statistics made visible: no two zeros too close, no gap too large, the characteristic clustering of a bilateral mesh in its ground state.
Z(t) on Re(s)=½
The Z-function is real-valued on the critical line and zero exactly at each Riemann zero — gold dots at \(t_1, t_2, t_3\ldots\). Blue fill above the axis, red below. The level repulsion is directly visible: the zeros are not evenly spaced, they repel each other. This is the GUE statistics. The amplitude of the curve between zeros is the coupling coefficient — how strongly the syphon crosses at that frequency. The electron lives at the first zero because the amplitude is highest there.
arg ζ(s)
The phase of the zeta function on the critical line winds by exactly \(\pi\) at each zero — the Möbius half-twist made visible in the complex phase. Four full windings of \(\pi\) to return to the start: \(4\pi\) total. This is spin-\(1/2\), readable directly from the argument of \(\zeta\). The Dirac equation is the propagation law for a wavefunction whose phase behaves exactly this way.
Scanning
The red needle traverses the critical line. As it approaches a Riemann zero the gold glow intensifies and the dot turns gold — the syphon finding its crossing frequency. The needle slows near zeros, drawn by the level repulsion of the surrounding mesh. This is the \(|\psi|^2\) density made dynamic: the syphon does not cross uniformly across all frequencies. It is pulled toward the zeros and dwells there. The first zero gets the longest dwell. That is the electron.