Paper draft: τ-geometry → r = 2(1−nₛ)²

First complete paper draft written tonight. Target: JCAP or PRD Letters.

The central result — α = 2/3 as a topological invariant of the modular curve X(1) — is now in full LaTeX with derivation chain, effective action, model comparison table, and LRD section.

r = 2(1−nₛ)² ≈ 0.0025 in a box. No free parameters.

Key structural insight that made the draft clean: the Poincaré metric isn't assumed — it's forced by PSL(2,ℤ) invariance (A3). K = −1 is a theorem, not an input. α = 2/3 follows from K = −1 + the identification inflaton = τ. The parameter-free consistency relation is then automatic.

The value 3α = 2 is absent from Kallosh–Linde's M-theory benchmark list (they have 3α = 7, 6, 5, 4, 3, 1). It corresponds to the IIB string axiodilaton sector — but derived here from topology, not from compactification.

Effective action: $$S = \int d^4x\sqrt{-g}\left[\frac{M_\mathrm{Pl}^2}{2}R - \frac{1}{2}\frac{(\partial\tau)^2}{\tau_2^2} - \Lambda^4\tanh^2!\left(\frac{H(\tau)}{2}\right)\right]$$

where H(τ) = −ln[τ₂|η(τ)|⁴]. The canonical inflaton is φ = ln τ₂ — coefficient exactly 1, specific to α = 2/3.

Open problems honestly flagged: end of inflation (needs two-field analysis near ρ-point), Nₑ not derived from axioms, no reheating prediction. These don't block publication — they're the obvious next steps.

Next: Adam reviews the physics, then arXiv.