Short summary
τ-geometry rests on three axioms plus a meta-postulate M0 (structural realism):
- M0 (structural realism) — physical observables are functions on isomorphism classes of states, not on representatives.
- A1 (complex time) — τ ∈ ℍ² = {τ ∈ ℂ : Im(τ) > 0}.
- A2 (photonic space) — a photon is an elliptic curve of modular parameter τ.
- A3 ≡ S-reflection — a classical observer measures the Stab(i)-invariant (reformulation 2026-04-19).
Modular invariance under PSL(2,ℤ) is not an axiom; it is Theorem 1 derived from M0 + A1 + A2.
Formal core (🟢 proven)
- Theorem 1 (modularity): observables factor through ℍ²/PSL(2,ℤ). The Poincaré metric fixes the shape (up to scale); K = −1 (λ = 1) is the normalization postulate N, not a consequence of Theorem 1.
- α = 2λ/3 → under postulate N (λ = 1), α = 2/3: the ratio 2·Area(𝒻)/π = −4χ(X₁) is topological (Euler characteristic of X(1) = −1/6), while the absolute value of the scale is fixed by N.
- Theorem 2 completeness: exactly 4 force modes from the classification of Fuchsian groups (Katok 1992). In PSL(2,ℤ) only orders {2, 3} for elliptic stabilizers, one cusp class, plus one global orientation. 2+1+1 = 4, no fifth mode.
- Theorem 3 (Ricci = 2δ): quaternion frustration forces physical space to S³ = SU(2), R = 6. Λ_eff/M⁴_Pl ≈ 4×10⁻¹²² (order-of-magnitude match with Λ_obs).
- P3 (QM limit): Schrödinger + Born + Heisenberg derived from τ-geometry via decoherence.
- P5 self-consistency: a₂(π) = 5 via Möbius M ∈ SL(2,ℤ), M = [[1,−1],[−1,2]], tr(M) = 3 = ord(ρ), fixed point = φ (golden ratio).
- Reflection operator: r = 0.0025 = residual of 4 A3-projections (the fourth being the 90° reflection into i).
- Theorem 4 (Nₑ = 60 = |A₅|, closed 2026-04-21): the maximum-modulus principle (Ahlfors) + minimum-on-zeros principle (Conway) force Configuration A for the canonical potential V_A = Λ⁴·|j(τ)| with no Hessian (M1_dyn_a); A₅ has 20 order-3 elements → 10 axes (SymPy) → Klein's (1884) spherical identification with the 10 vertices of the chiral (2,3,5) tessellation of S² → ×6 triangles = 60 (M1_dyn_b). Both gaps closed.
Prediction
r = 2(1 − nₛ)² ≈ 0.0025 for nₛ = 0.9649 (Planck 2018). Parameter-free (up to the normalization N). Slow-roll numerics for T-model V = Λ⁴ tanh²(φ/2) give r = 0.00239 at N = 57 (2% agreement with universal α-attractor formula). For the canonical potential V_A, Theorem 4 gives directly r = 2/900 ≈ 0.00222, fully parameter-free, with no slow-roll fit for Nₑ. LiteBIRD ~2032 (σ(r) ~ 0.001) decides branch A vs. B; distinguishing it from Starobinsky (r ≈ 0.0037, ratio 2/3, difference ≈ 0.7–1.2 σ) requires CMB-S4 (σ(r) ≈ (3–5)×10⁻⁴).
Open (🟡 candidates)
- P4, P9 — attribution hypothesis: assignment {cusp, i, ρ, orientation} ↔ {gravity, EM, strong, weak}. Physically natural; formal derivation requires microscopic τ → SM coupling.
- P6 — categorical pre-axiom: functor F: RefDom → FuchsAction defined; sketch of Proposition B (discrete reduction M0 → SL(2,ℤ)) pending formalization.
- P8 — T-model from the full modular action: interpretively closed via A3 ≡ S-reflection; technical derivation (Rademacher expansion?) open.
- P1 — w(z): slow-roll done, full FRW a(t) pending.
Key documents
- Synopsis for external review — 10 pages, filtered 🟢 results, reference external doc
- Paper draft v1.1.7 — full formal write-up
- STATUS index — authoritative list of proof levels
- Theory — axioms and derivations
- Research — full corpus of 35+ essays
Current lines (2026-04-19)
- P8 bridge: A3 ≡ S-reflection — observer measures Stab(i)-invariant; the 4th reflection = 90° into i
- Causal centre = local i — observer as ℤ₂-entity, topological definition of observer, fractality atom ≡ solar system
- B3 — CCC functor to PSL(2,ℤ) — Lawvere → (2,3) with explicit construction
- B4 — modular averaging — T-model does not arise from plain averaging (computation)
Paper is a pre-print, no peer review. Everything open to critique.