[{"type":"thinking","thinking":"Continue to ii′-6-b: perimeter Props + extended pigeonhole + sides identity + bridge.\n\nPlan (D-071 §4):\n1. **Perimeter Props** (in LiContour.lean near ZeroCountProp2/NoLowZeros):\n   - `ZeroCountGlobalProp : Prop := ∀ T : ℝ, 2 ≤ T → ((mult-weighted one-sign count of nontrivial zeros with 0 < γ ≤ T : ℝ)) ≤ FAlt T`\n   - `MZeroProp : Prop := ∀ t : ℝ, |t| ≤ 10 → ‖logDeriv riemannZeta (-5/2 + t * I)‖ ≤ 1.3352`\n   \n   For ZeroCountGlobalProp I need the zero-counting set: nontrivial zeros with 0 < im ≤ T. There's zeroAbsWindow (|im − t| ≤ 1) and zeroWindow (unit interval [t, t+1]). For the global count I need a new window def: `zeroGlobalWindow (T : ℝ) : Set ℂ := {ρ | IsNontrivialZero ρ ∧ 0 < ρ.im ∧ ρ.im ≤ T}` with finiteness (via zerosIn_finite).\n\n2. **Extended pigeonhole** `exists_zero_avoiding_T_wide (hZC : ZeroCountProp) {a : ℝ} (ha : 3 ≤ a) : ∃ T, a ≤ T ∧ T ≤ a + 1 ∧ ∀ ρ, IsNontrivialZero ρ → a - 1 < ρ.im → ρ.im < a + 2 → 1 / (9 * C0hat (a + 2)) ≤ |ρ.im - T|`.\n   \n   Clone exists_zero_avoiding_T's proof with the 3-unit window. The count: zeros with im ∈ (a−1, a+2) = union of three unit windows [a−1, a], [a, a+1], [a+1, a+2] — the zeroWindow t def is {IsNontrivialZero ρ ∧ t ≤ ρ.im ∧ ρ.im ≤ t+1}? Let me check zeroWindow's def. Each unit window's mult-count ≤ C0hat(t+1) via hZC (t ≥ 2; need a−1 ≥ 2 i.e. a ≥ 3 ✓). Total ≤ C0hat(a) + C0hat(a+1) + C0hat(a+2) ≤ 3·C0hat(a+2) (C0hat monotone — C0hat_mono exists? used in LiGapCircles ✓).\n   \n   Then the midpoint pigeonhole `exists_avoiding_midpoint` (the ii′-4 machine): with k+1 midpoints in [a, a+1] and k = count... Let me look at exists_zero_avoiding_T's proof to clone it.\n\n3. **Sides identity** (in a new file LiAssembly.lean or in LiSides?): given T from the wide pigeonhole (a := √n), the rectangle z = −5/2 − iT, w = 2√n + iT:\n   - rectInt'_Jlag requires: 1 interior, ζ ≠ 0 on border.\n   - Produce: `sides_identity : ResW1 n = RectangleIntegral' (Jlag n) z w - ∑ ρ ∈ Sz, (zeroMult ρ) * (1 - (ρ/(ρ-1))^n)` — wait, rectInt'_Jlag's statement: RectangleIntegral' (Jlag n) z w = ResW1 n + Σ_{ρ∈Sz} mult·(1−(ρ/(ρ−1))ⁿ). So ResW1 = RectInt − Σ.\n   - Then |Sf n| = |(ResW1 n).re| ≤ ‖ResW1 n‖ ≤ ‖RectInt‖ + ‖Σ‖. And ‖Σ‖ ≤ |λ_n(T)| + |triv(−2)|.\n\n   The rectangle's Sz = zeros inside (the Finset from rectInt'_Jlag's statement — let me check what Sz is exactly in rectInt'_Jlag).\n\n4. **The bridge** `lambdaT_norm_le_four_FAlt (hRH : RHUpTo (3×10¹²)) (hZCG : ZeroCountGlobalProp)`: |λ_n(T)| ≤ 4·FAlt T.\n   - ‖Σ_{ρ ∈ Sz_nontriv} mult·(1−(ρ/(ρ−1))ⁿ)‖ ≤ Σ mult·2.\n   - On the line: ρ.re = 1/2 (hRH, |ρ.im| ≤ T ≤ 3×10¹²) ⟹ |ρ/(ρ−1)| = 1: |ρ|² = 1/4 + γ²; |1−ρ|² = 1/4 + γ² ✓ equal ⟹ |ρ/(ρ−1)| = 1 ⟹ |1−(ρ/(ρ−1))ⁿ| ≤ 2.\n   - Count: both-signs mult count = 2·(one-sign) — the map ρ ↦ conj ρ is a bijection between {0 < γ ≤ T} and {−T ≤ γ < 0} (isNontrivialZero_conj ✓, zeroMult_conj ✓ — zeroMult conj ρ = zeroMult ρ from LiTightR/LiMoments §4's zeroMult_conj).\n   - One-sign count ≤ FAlt T via ZeroCountGlobalProp.\n\nLet me start by reading rectInt'_Jlag's exact statement and the zeroWindow def + exists_zero_avoiding_T's proof (to clone for the wide version).","signature":""},{"type":"text","text":"Continuing to ii′-6-b. Reading the exact forms I need to clone: `rectInt'_Jlag`, `zeroWindow`, and `exists_zero_avoiding_T`'s pigeonhole."},{"type":"tool_use","id":"Read_239","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSides.lean","offset":420,"limit":105},"extra_content":null}]