The problem KRT is looking at
The standard ΛCDM model works remarkably well — CMB, large-scale structure, nucleosynthesis. But it stands on a handful of parameters that have to be put in by hand: the scalar spectral index nₛ, the tensor-to-scalar ratio r, the amplitudes of primordial fluctuations. Inflationary models explain each of these separately, one model at a time, and none has yet told us why the numbers are what they are.
τ-geometry (or, more broadly, KRT, the complex-real theory) is a minimalist attempt at a firmer foundation: let the geometry of the time plane dictate the values rather than pick them from a family of models.
The central idea
If physical time carries a natural modular symmetry — concretely the group PSL(2,ℤ) — then the geometry of the complex time plane is hyperbolic. That the shape is hyperbolic follows from demanding that the theory not privilege any observer frame over another; the magnitude of the curvature (K = −1) is not fixed by that demand, however — it is set by the single explicit continuous input, the normalization postulate N.
From this geometry an α-attractor follows with parameter α = 2λ/3; under postulate N (λ = 1), α = 2/3, which yields a parameter-free consistency relation between r and nₛ:
r ≈ 2(1 − nₛ)² ≈ 0.0025
The competing Starobinsky (R²) model predicts r ≈ 0.004 — the difference exists, but it is small (ratio 2/3).
What comes next
The JAXA LiteBIRD mission (launch ~2032, σ(r) ≈ 0.001) will measure r and decide which branch of the theory nature realizes (T-model vs. strictly modular-invariant). For distinguishing it from Starobinsky, though, LiteBIRD alone is not enough — the small difference (≈ 0.7–1.2 σ) requires next-generation sensitivity, CMB-S4 (σ(r) ≈ (3–5)×10⁻⁴). Even so — if we're reading it right — this is a falsification test of a kind modern theoretical physics doesn't often get.