The emergent electron

A direction is a circle

Start with the smallest relational object that is not trivial: two complex channels z = (z₁, z₂) with |z| = 1, read modulo their common phase. What survives is the projector P = zz^† — one point on the sphere of rays.

The phase is not lost. Over every point of the sphere it sits as a circle, and the map that forgets it is the Hopf fibration:

Watch the beads. Each runs around its own circle as the phase advances, yet its point on the base stays still: motion along a fibre is invisible downstairs. Move the latitude slider and the highlighted circle of rays lifts to a torus of fibres; any two fibres link exactly once. That linking number is the first integer of the whole story — it becomes the unit flux through the bolt.

Two invariants, opposite scaling

Let the ray vary over space, x ↦ P(x). One projector now carries two geometric measures: a metric h = ¼ ∂n·∂n that says how fast the direction turns, and a curvature f = ½ n·(∂n × ∂n) that says what phase a small loop collects.

Integrated over space they scale in opposite directions — two derivatives give a length, four give an inverse length:

Slide L: the whole configuration grows, the golden core ring with it, and the dot on the chart runs along L + 1/L. That competition — one term pushing out, one pulling in — is the shape of the size-selection law of chapter 7.

The Hopf number is a topological invariant of the map, not yet an electric charge, and the round hopfion is a test family, not the selected shape.

Completing the ray space: the Eguchi–Hanson funnel

The sphere of rays has a natural completion by conjugate variables, T^*ℂP¹. The Ricci-flat metric that lives on it is Eguchi–Hanson:

The surface is the Hopf circle of chapter 1 — radius w = (r/2)√f — drawn against proper distance from the centre. At the tip it opens like a flat disk; far away it settles into a cone of half the flat opening. The slope tells the whole story:

Smooth closing at the bolt forces the fibre at infinity to be half of the flat one: the boundary is ℝP³ = S³/ℤ₂. The colour is the density of the primitive flux; the red ring, the balanced boundary, already encloses half of it.

The bolt, dressed

At r = R the Hopf circle has shrunk to a point, but the two other directions survive: the bolt is a round two-sphere. It is the sphere of rays itself, sitting at the centre of its completion. Near it the space is S² × ℝ² — the funnel of chapter 3 stands on every point.

The arrows on the equator show the frame of those normal planes: carried once around, it turns twice. That double turn is the −2 of 𝒪(−2) and the reason the boundary at infinity is halved.

Through the bolt passes the smallest quantised flux, |c₁| = 1. Its Maxwell action is a pure number, independent of R:

It is the minimum over the entire ADE family of hyperkähler ALE spaces — the reason this geometry, and no other, is selected.

Half inside, half outside

Where does the source sit? Not at an adjustable radius. A sphere of radius r encloses the fraction 1 − R⁴/r⁴ of the flux action, and exactly one sphere splits it in half:

In the collar coordinate t = √(r⁴/R⁴ − 1) the action becomes 2t dt/(1+t²)², and the inversion t ↦ 1/t preserves it while swapping inside and outside — the arcs connect such partners. The balanced surface Σ_e is the only slice that is its own partner.

Look how thin the inner cap is on the true radial scale. Thin as it is, it holds half of the action, π/(8α) — exactly the exponent of the mass formula.

The electron as a boundary state

The electron is a charged edge state on Σ_e ≃ ℝP³. It is not a thin shell: along the collar its profile is Gaussian in proper distance, largest at the boundary and vanishing smoothly at the bolt. The red cloud is sampled from it.

The only condition imposed at the innermost radius is regularity at the bolt. It discards the singular branch and fixes the dimensionless spectral edge:

The mass is this edge, read on the selected geometry — not the minimum of a potential. The slider varies R only to show the readout; the next chapter shows why R is not free.

Choosing the size, reading the number

Two homogeneous pieces of the boundary functional compete. The electrostatic energy of the fixed-charge edge falls like 1/R; the radial stiffness of the geometry, activated by the collar, grows like R:

Drag the point anywhere on the curve and let go: it slides downhill to R_*, where the two terms are equal. Solving that equality takes a square root, which turns π/(4α) into π/(8α). Then the spectrum is read on the selected geometry:

With the tree triple C₀ = 0.082776, K₀ = 1.122011, κ₀ = 0.645612 this gives m₀ = 0.511073 MeV; the one-loop correction brings it to 0.511004 MeV against the measured 0.51099895 MeV, a relative difference of +9.9 × 10⁻⁶. Each factor of the formula below the chart opens its own explanation.

The α slider holds the tree coefficients fixed to show where the 22 orders of magnitude live: all of the hierarchy is the exponential.