Imagine a ballet dancer in a hexagonal hall of mirrors. She spins once — a full 360 degrees — and sees six reflections. Each one is her. Each one is real. Each shows a different angle.
Now imagine the dancer is a photon. And one of those six reflections is our entire universe.
Time is complex. Literally.
What if time isn't just a line from past to future, but has a perpendicular, "invisible" dimension? A real part (we measure it with clocks) and an imaginary part (it shows up in quantum physics)?
Hawking worked with "imaginary time" — but as a mathematical trick. What if it's a fundamental fact?
The symmetry that changes everything
Complex numbers carry a natural symmetry: the modular group PSL(2,ℤ). Two basic operations: shift (T: add 1) and inversion (S: flip and mirror). If physics must be unchanged under this symmetry, the shape of the time-plane geometry is forced to be hyperbolic — curved like a saddle. That is a consequence of the symmetry. The magnitude of the curvature (K = −1 in Planck units) is not fixed by the symmetry, however; it is set by the theory's single continuous input — the normalization postulate N.
From curvature to gravitational waves
In cosmology, α-attractors describe inflation — the rapid expansion of the early universe — as a field moving on a curved surface. The parameter α sets the curvature and predicts the strength of primordial gravitational waves (r).
The shape of the fundamental domain (the smallest non-redundant piece of the complex time plane, with area π/3) gives α = 2λ/3. It's not a free fit — the shape is forced by the symmetry — but it isn't a pure topological invariant either: the value of the scale λ = 1 (and hence α = 2/3) is set by the normalization postulate N. Given N, α = 2 × (π/3) / π = 2/3.
From that follows: r = 2(1 − nₛ)²
For nₛ = 0.965 (Planck 2018): r ≈ 0.0025.
A prediction we can test
The Japanese space mission LiteBIRD (~2032) will measure the polarization of the cosmic microwave background with sensitivity σ(r) ~ 0.001. Our prediction r ≈ 0.0025 falls squarely in its range.
The famous Starobinsky model (R², 1980) predicts r ≈ 0.004.
But mind the resolving power: LiteBIRD will reliably decide between our two branches (T-model vs. strictly modular-invariant potential, where r ~ 10⁻⁵). For distinguishing it from Starobinsky (ratio 2/3, a difference of only ~0.7–1.2 σ), though, LiteBIRD is not enough — that requires next-generation sensitivity, CMB-S4 (σ(r) ≈ (3–5)×10⁻⁴).
If r comes out somewhere else entirely (r > 0.01 or r < 0.0005) — α = 2/3 falls.
That's how science works. Make a specific prediction, wait for the experiment.
Six reflections
Fundamental domain area: π/3. One photon rotation: 2π. Ratio: 6.
Six = 2 × 3: two time dimensions (Re/Im) × three spatial. That's why we live in a 3+1D universe and not a 2D or 7D one.
Cosmological evolution — the whole story from Big Bang to heat death — covers exactly one of those six domains. One sixth of a photon. Not metaphorically. Arithmetically.
This research is a working hypothesis at pre-print level, without peer review. The prediction for r is hard and falsifiable. Everything else is open for debate.