The starting tension
Relativity insists that no observer frame is privileged. Quantum mechanics, meanwhile, treats time as a parameter — not an operator — and thereby silently selects preferred frames when it comes to collapse, measurement, energy. That's not just a philosophical stumble; it's the point where the two formalisms cannot be physically merged without loss of information.
τ-geometry takes this tension seriously. Instead of a compromise it proposes a shift: time is complex. The real part is what clocks measure. The imaginary part is where quantum dynamics lives. Together the two components form a time plane that is itself hyperbolic — curved, not flat.
Why PSL(2,ℤ)
Identify a photon topologically with an elliptic curve (S¹ × S¹ with complex structure τ). The moduli space of elliptic curves over ℂ is then exactly ℍ²/PSL(2,ℤ) — a standard theorem of modular forms. M0 (observables factor through isomorphism classes) enforces this factorisation. Modular invariance is not an axiom; it is a theorem (Theorem 1).
The fundamental domain has three fixed points (i, ρ, i∞) and a global orientation of Im(τ). The classification of Fuchsian groups (Katok 1992) says that no other finite stabiliser orders in PSL(2,ℤ) are possible — for M ∈ SL(2,ℤ) with |tr(M)| ∈ {0, 1} only orders 2 or 3 arise in PSL(2,ℤ). Three fixed points + orientation = exactly four modes (proved 2026-04-18).
These four topological data are candidates for the four fundamental interactions: gravity ↔ cusp (parabolic, long-range), EM ↔ i (Z₂, spin ½), strong ↔ ρ (Z₃, three colours), weak ↔ orientation (chiral, flavour-changing). The attribution is physically natural; formal derivation is pending.
Testable prediction
Theorem 1 fixes the shape of the metric (Poincaré metric on ℍ²); its scale (K = −1, i.e. λ = 1 in Planck units) is not fixed by invariance — that is set by the normalization postulate N. Given N, in the α-attractor class (Kallosh-Linde 2013) α = 2λ/3 = 2/3 (= 2·Area(𝒻)/π). Consistency relation (the only continuous input is N):
r = 2(1 − nₛ)² ≈ 0.0025 for nₛ = 0.9649.
Numerics for T-model V = Λ⁴ tanh²(φ/2) confirm: r = 0.00239 at N = 57. For the canonical modular-invariant potential V_A = Λ⁴·|j(τ)|, Theorem 4 (closed 2026-04-21) additionally forces Nₑ = [PSL(2,ℤ):Γ(5)] = |A₅| = 60 topologically — the maximum-modulus principle fixes Configuration A with no Hessian, and A₅ combinatorics (20 order-3 elements → 10 axes → ×6 triangles) gives 60 with no open computation. The result is r = 2/900 ≈ 0.00222, fully parameter-free, with no slow-roll fit. LiteBIRD ~2032 (σ(r) ≈ 0.001) decides between branch A (T-model) and branch B (strictly modular-invariant, r ~ 10⁻⁵). Starobinsky (r ≈ 0.0037; ratio r_A/r_Star = 2/3, a difference of ≈ 0.7–1.2 σ) is distinguishable from branch A only with CMB-S4-class sensitivity (σ(r) ≈ (3–5)×10⁻⁴).
Side results
- Λ_obs ≈ 10⁻¹²²: emerges from Theorem 3 (S³ physical space) as a topological order-of-magnitude, not fine-tuning.
- QM limit P3: Schrödinger, Born, Heisenberg derived from τ-geometry + decoherence.
- Reflection operator: 4 reflections = four forces, residual (i^i)⁸ ≈ r.
What to read next
- Theory — formal path from axioms to prediction
- Synopsis for external review — 10-page filtered document
- Paper draft v1.1.7 — full write-up
- Research — essays including P8 bridge, causal centre, B3/B4
- Or switch to the formalista angle for a structured list of results.