Charge

You know what charge is now. It is the coupling of the break to the connection. But there is something remarkable about its structure — and it falls out of a theorem about circles.

6.1 — What charge is

Electric charge is the coupling of ε to the connection. ε is the break — the one element that has no σ-image. It breaks the 1:1. The connection Aμ expresses the pre-state’s global phase on the manifold. Charge is how strongly the break disturbs the phase reading.

The electron carries charge −e. This is ε itself. The positron carries charge +e. This is ε read from the other sector — σ(ε) does not exist in the algebra, but the antiparticle is the conjugate state: ψ* where ψ describes ε. The conjugation is σ (Axiom S). Light and Dark. The charge flips because the sector flips.

Charge conservation is a consequence of U(1) gauge invariance (Noether’s theorem). The global phase symmetry gives a conserved current. That current is the electric current. Charge conservation is the Born rule’s symmetry (ψψ* is invariant under phase rotation) read as a conservation law on the manifold.

6.2 — Why charge is quantised

Charge comes in discrete units: ±e, ±⅓e, ±⅔e (quarks). Never ½√e or 0.7e. Always rational multiples of e.

In standard U(1) gauge theory, charge quantisation is not automatic. The Lie algebra u(1) ≅ ℝ admits continuous representations — any real number is a valid charge. To force discrete charges, standard physics requires extra machinery: Dirac’s magnetic monopole argument, or embedding U(1) inside a compact non-abelian group (grand unification). The axioms require neither.

Theorem 5 (Charge Quantisation). All U(1) charges in the architecture are integer multiples of a minimum unit. The minimum unit is the charge of ε.

Proof. (i) The phase group is U(1) = S¹, the circle (section 1, KS-28 CLOSED). The Born rule gives |e^{iθ}ψ|² = |ψ|². The phase θ is an angle: θ and θ + 2π are the same phase. The symmetry group is compact.

(ii) The irreducible unitary representations of a compact group are discrete (Peter–Weyl theorem; standard mathematics). For U(1) specifically: the irreducible representations are labelled by integers n ∈ ℤ, each sending e^{iθ} ↦ e^{inθ}. There is no representation labelled by √2 or π or 0.7. Only integers. This is not a physical postulate. It is a mathematical fact about circles.

(iii) The bundle structure group is U(1), not its universal cover ℝ (section 3: the fiber at each point is S¹, the phase circle inherited from ℋ). Matter fields coupled to this bundle must transform in genuine representations of U(1), not representations of ℝ. The charge of any field is therefore an integer: q ∈ ℤ. This is where the architecture’s construction differs from the standard Lie-algebraic approach: section 3 constructs the bundle from the group (the circle), not from the algebra (the real line). The group is compact. Discrete representations. Integer charges.

(iv) ε carries the minimum non-zero charge. Axiom B: ε has valuation ν(ε) = 1. It is the minimum element. The minimum non-zero integer is 1. Therefore ε carries charge |q| = 1 in units of the fundamental charge.

(v) All other charged elements in the record algebra are compositions of ε-elements. Their charges are sums of ±1 contributions. Sums of integers are integers. Therefore all charges are integer multiples of ε’s charge. ∎

[Epistemic status: DERIVED USING STANDARD MATHEMATICS. Step (i) is derived (section 1). Step (ii) is the Peter–Weyl theorem (standard mathematics). Step (iii) follows from the bundle construction in section 3. Step (iv) uses Axiom B. Step (v) uses the compositional structure of the record algebra. No Dirac monopoles, no grand unification, no external mechanism is imported. The quantisation follows from: phase is a circle (derived) + circles have integer winding numbers (standard mathematics) + ε is one brick (Axiom B). Gap E: CLOSED.]

[Scope: Theorem 5 proves that all observable U(1)_EM charges are integer multiples of e. Quarks carry fractional charges (±⅓e, ±⅔e), but quarks are confined — they never appear as free charges, and all hadrons carry integer charge. The fractional charges arise from the SU(3) colour structure, which is beyond AP15’s scope. AP15 derives U(1)_EM. Within U(1)_EM, observable charges are integers. The quark substructure belongs to AP19.]