The reciprocal line was one real axis. Run it through the μ-hinge and it gains three things at once — a second axis (i), a point at infinity (the pole), and curvature. That is one emergence, D1 → D2. It does not reach D3; a round surface is still two-dimensional.
0 = south pole · ∞ = north pole (the point the μ-limit manufactures — it closes the open plane into a sphere).
The unit circle |z| = 1 is the equator — and that is where i lives. Our canon names it directly: i is the equator. It is also where B = 1, the balance maximum — the present, the rest frame, the only circle inversion holds fixed.
The real axis is one meridian (through 0, 1, ∞); the imaginary axis is the meridian at a right angle to it. Two axes now, not one: the complex plane, gained.
Stereographic projection preserves angles but not flatness. A straight-sided triangle on the plane sums to exactly 180° — the signature of zero curvature. Lift it to the sphere and its sides bow outward; the angles sum to more than 180°, by exactly the area it encloses (Girard's theorem, [A]). Shrink it toward a point and the excess vanishes — small patches of the sphere look flat, which is why the plane is the sphere's local portrait. Grow it to an octant and the three angles are all right angles: 270°. Watch the readout climb.
All inherited. The Riemann sphere, stereographic projection, and Girard's spherical excess are standard 19th-century geometry — used, not discovered [A]. The reading of this as the D1→D2 emergence of number (real → complex, +i, +∞, +curvature) is ours [S/I]; it invents no geometry.
The sphere is D2, not D3. A curved surface is still two real dimensions — curvature is a change of metric, not of degree of freedom. Adding i is the one real dimension gained over the line. (D3 is a different object entirely — the coupling law, Liebig's barrel.)
Kept in its own frame. This is the geometry sphere — the projection, the curvature, the axes. The operator sphere uses the same surface for the game; the two are the same S² read for different work, never fused.
The ladder of instruments: the floor → the line → the hinge → this sphere → the operators. And the numbers that live on it: the egg. Where a strong version of any of this overreached, its death is dated on the record. · If you can see directly, put this down →