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 S³ universe entry places the spatial world as a round 3-sphere of cosmological radius R_U, the geometric home of the working one-force programme. Compactness supplies a finite volume and a finite total mass, so Mach closure and the Hopf seats share one well-posed stage for every later book.
Clifford Torus Particles
Clifford torus particles give the one-force framework its matter seats: each particle is a single torus knot T(p,q) on the flat Clifford torus inside S³. The coprime winding pair carries the charge class, the spin class, and the self-linking number that later spectrum and spin pages read as identity.
Hopf Fibration
The Hopf fibration entry supplies the fibre geometry every later particle seat uses. It writes S³ as a continuous family of linked circles over S², so an exchange of two seats has a definite phase, and the Clifford torus appears as the preimage of the equator.
Weber Force — Introduction
This introduction places Weber's 1846 force as the single direct interaction of the one-force framework. Charge pairs act through relative separation, radial velocity, and radial acceleration, and the same law is the electrodynamic root that the double-copy reading later squares into gravity.
Ontology Freeze
The ontology freeze is the constitution of the working one-force framework: six axioms, OA1 through OA6, that every later page and every community submission is asked to keep. Together they name the stage, the interaction, the particle seats, charge, time, and the emergence of gravity.
Rule C — All Energy Gravitates
Rule C states that every reservoir of energy gravitates with the same weight inside the one-force framework. Kinetic energy, potential energy, and electromagnetic field energy join the rest masses in one gravitational source, once gravity is read as the square of the electromagnetic interaction.
8D / Octonion Structure
The 8D structure entry lifts the observable 3-sphere into the octonionic setting the one-force framework uses for internal labels. Hurwitz's list ends at the octonions, and this page sets S³ as an equator of S⁷ in eight-dimensional space so colour and flavour have a geometric home beside the spatial seat.
Torus Knot Spectrum
The torus-knot spectrum is the working catalogue that turns coprime windings on the Clifford torus into the particle skeleton of the one-force framework. Low-lying knots are matched to the electron class, a pion candidate, and the nucleon pair, giving later particle pages a shared (p, q) language.
Where foundations lead