T3/Z2 Cosmic Topology
The universe is a 20.6 Gpc orbifold with built-in chirality
eta(T3/Z2) = Z2 = 32pi/3 = 33.510
The eta invariant of the spatial manifold IS the master constant Z2
What is T3/Z2?
T3 is the 3-torus: space that wraps around in all three directions. Travel far enough in any direction and you return to your starting point.
Z2 is an involution (reflection symmetry) acting on the torus. Points related by the Z2 action are identified as the same point.
T3/Z2 is an orbifold: a quotient space with 8 special fixed points where the Z2 symmetry acts trivially.
T3 (3-Torus)
Periodic boundary conditions
b1 = 3 independent cycles
Z2 Action
Point reflection x to -x
Creates chirality
8 Fixed Points
Vertices of fundamental cube
= CUBE = 8
The Eta Invariant
The eta invariant measures the spectral asymmetry of a manifold - roughly, the difference between left-handed and right-handed modes of the Dirac operator.
Atiyah-Patodi-Singer (1975):
eta(M) = Sum sign(lambda_n) |lambda_n|^(-s) at s=0
Key Result:
For T3/Z2 with the standard flat metric:
eta(T3/Z2) = 8 x (4pi/3) = 32pi/3 = Z2
The 8 fixed points each contribute 4pi/3 to the eta invariant
The Fundamental Domain
The T3/Z2 orbifold has a characteristic scale - the size of the fundamental domain (the smallest region that tiles to fill all of space).
Fundamental Domain Scale
Lc = 20.6 Gpc
~67 billion light-years
1Light wraps around the universe
2High-z quasars have ghost images
3CMB has finite-size cutoff
4Parity-odd correlations are global
Global Chirality
The Z2 action defines a single chirality axis for the entire universe. This is the direction connecting opposite fixed points of the involution.
Predicted Chirality Direction:
(l, b) = (287 deg, 9 deg) +/- 5 deg
Galactic coordinates of the Z2 axis
T3/Z2 Prediction
Parity-odd 4PCF should show r ~ 1 correlation between any two sky regions (global chirality)
Local Physics Prediction
Parity-odd 4PCF should show r ~ 0 correlation (independent random processes)
DESI DR1 Result (May 2026):
r = 0.9986 +/- 0.0002
Decisive confirmation of T3/Z2 topology
Why Topology Matters
1
Z2 is not arbitrary
It is the eta invariant of the spatial manifold - a topological constant that cannot be changed without changing the topology itself.
2
Dark matter is topological
The T3 torus has b1 = 3 independent 1-cycles, each supporting 2 winding modes. These 6 topological modes ARE the dark matter - no particles required.
3
Observable predictions
Ghost quasars, CMB circles, 4PCF chirality axis - all testable consequences of finite topology that differ from infinite flat space.
4
Unifies physics
From Z2 = eta(T3/Z2): fine structure constant, Weinberg angle, cosmological densities, MOND scale, particle masses - all from topology.