⧗ Geometric Singularity
hyperbola · projective closure · loop
The singularity of 1/x is not a hole — it's a point at infinity on the
projective closure of xy = 1. The loop around it encodes the residue.
xy = 1
asymptotes
evaluation point
contour loop
point at ∞
🧬 geometric nRAN state
relation : x · y = 1
finite value : 1/x
singular state :
● pole order = 1
residue = 1
asymptote = y-axis
pole
loop: 0
↻ involution loop
0.50
⇄
2.00
x ↔ 1/x
winding:
∞ projective closure
Homogeneous: XY = Z²
Points at infinity (Z=0):
(1 : 0 : 0)
(0 : 1 : 0)