PV Chain Assembly

Assembles the residue side and modular side of the PV chain using Tendsto statements for the ε-truncated integrals.

Main Results

• cpv_residue_side_tendsto — the ε-truncated integral of f'/f around fdBoundaryH H tends to 2πi · Σ gWN · ord.
• cpv_modular_side_tendsto — the ε-truncated integral of f'/f around fdBoundaryH H tends to -(2πi · (k/12 - ord_∞(f))).

Modular side sub-lemmas

The modular side decomposes into:

1. Vertical seg1+seg4 integrands cancel pointwise for each ε (T-invariance)
2. Arc seg2+seg3 integral tends to -(2πi·k/12) (S-transformation)
3. Horizontal seg5 integral tends to 2πi·ord_∞ (q-expansion)
4. Combine via Tendsto.add

theorem cpv_modular_side_tendsto {k : ℤ} (f : ModularForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma 1)) k) (hf : f ≠ 0) (S : Finset UpperHalfPlane) (_hS : ∀ p ∈ S, p ∈ ModularGroup.fd) (hS_complete : ∀ p ∈ ModularGroup.fd, orderOfVanishingAt' (⇑f) p ≠ 0 → p ∈ S) : ∃ (H₀ : ℝ), √3 / 2 < H₀ ∧ ∀ {H : ℝ}, H₀ ≤ H → Filter.Tendsto (fun (ε : ℝ) => ∫ (t : ℝ) in 0..5, pvIntegrand f (fdBoundaryH H) (sArcOfS S ∪ sVertOfS S) ε t) (nhdsWithin 0 (Set.Ioi 0)) (nhds (-(2 * ↑Real.pi * Complex.I * (↑k / 12 - ↑(orderAtCusp' f)))))

Modular side: ε-truncated integral of f'/f tends to -(2πi·(k/12 - ord_∞)).