Section · Foundations
Foundations
The constitutional axioms of the A-GEOM / CPT framework. These are not assumptions that can be relaxed — they are the ontology freeze. Every other result builds on this ground.
The four constitutional axioms
S³ Universe
The universe is a 3-sphere. No external spatial infinity. No fundamental spacetime metric beyond S³.
Weber Force Only
The only fundamental interaction is Weber's electrodynamic force. Gravity, nuclear, and all other forces emerge from it.
Time = r/c
Time is not a fundamental dimension. It is defined by t = r/c where r is a characteristic radius and c is the speed of light.
Rule C
All energy gravitates — kinetic, potential, EM field energy. This follows from the EM–gravity double copy.
Pages in this section
The S³ Universe
The universe is a 3-sphere. This is the fundamental geometric substrate from which everything else derives. We establish the S³ embedding, its metric, and why it eliminates the need for an external spatial infinity.
Clifford Torus Particles
Particles are topological structures (torus knots) on the Clifford torus inside S³. The winding numbers (p, q) of the torus knot T(p,q) encode the particle's identity — its charge, spin, and interaction class.
Hopf Fibration
The Hopf fibration S³ → S² with fibre S¹ provides the geometric scaffolding. Each point on S² corresponds to a fibre S¹ — the seat of the particle. Exchange of two seats traces a path that picks up a topological phase.
Weber Force — Introduction
Weber's electrodynamic force law is the single interaction in this framework. It is velocity- and acceleration-dependent, unlike Coulomb's law. This page introduces the law and explains why it is the correct replacement for both Coulomb and Lorentz force.
Ontology Freeze
The constitutional constraints of the framework: Weber force only, no fundamental spacetime, time = r/c, Rule C (all energy gravitates), S³ universe, Clifford torus particles. These cannot be violated. Every result and community submission is checked against these.
Rule C — All Energy Gravitates
Rule C: every form of energy — kinetic, potential, EM field — contributes to the gravitational source term. This follows from the double-copy identification of gravity with the EM sector and has direct observational consequences.
8D / Octonion Structure
The full particle spectrum requires 8 real dimensions and G₂ ⊃ SU(3) flavour assignments from the octonion algebra. This page develops the 8D embedding and the G₂ → SU(3) decomposition that yields quark flavour.
Torus Knot Spectrum
The allowed torus knots T(p,q) with gcd(p,q)=1 form the particle spectrum. This page maps winding numbers to observed particles, explains why T(2,3) is the proton and T(3,2) is the neutron, and derives the spectrum skeleton.
Where foundations lead