[{"type":"thinking","thinking":"Now I need the actual target: the assembly bound |Sf n| ≤ Bn FAlt n. I need to understand what Bn FAlt n is in Lean (LiSkeleton), what the contour identity looks like (D-036/D-037's assembly), and what exactly the sides are. Let me look at:\n\n1. LiSkeleton: Bn, FAlt, Sf, SfBoundProp definitions.\n2. D-037's assembly: |Sf n| ≤ |λ_n(T)| + |sides| + |triv| = B(n) — from the session-014 discovery: \"Assembly TAIL-FREE: |Sf| ≤ |λ_n(T)| + |sides| + |triv| = B(n)\".\n3. D-036's structure: the contour estimate — H(T) = 3·Ĉ₀² + A·log + B₀; kernel bounds; I_I/II/III pieces.\n\nKey existing Lean pieces for ii′-6:\n- `rectInt'_Jlag` (LiSides, ii′-1b): RectangleIntegral' (Jlag n) z w = ResW1 n + Σ_{ρ∈Sz} (zeroMult ρ)·(1−(ρ/(ρ−1))ⁿ) on rectangles with 1 interior, ζ ≠ 0 on border.\n- `exists_zero_avoiding_T` (LiContour, ii′-4): clearance 1/(3·C0hat(a+1)) in [a, a+1].\n- L1 kernel bounds (LiContour): kFun_norm_le_two, kFun_shift_norm_le_exp_four_add_one, kFun_shift_norm_le_exp_sqrt_half_add_one, kFun_neg_two.\n- `logDeriv_riemannZeta_sub_nearZeros_le_tight (hZC) (hZC2) (hNLZ)` (LiTightR, 5f): the tight R-bound ≤ 10.5·log(t+2) + 48.\n- `sf_eq_resW1 (hZC)` (LiCrux, ii′-3c): (ResW1 n).re = Sf n.\n- `SfBoundProp` def (LiSkeleton:2438): ∀ n, 10⁵ ≤ n → (n:ℝ) ≤ (3×10¹²−1)² → |Sf n| ≤ Bn FAlt n.\n- Bn FAlt — need the definition.\n- RHUpTo (perimeter), LiVerifiedUpTo (perimeter).\n\nWhat remains for ii′-6 (per D-062 L0):\n|Sf n| = |ResW1 n| (via the crux: Sf n = (ResW1 n).re so |Sf n| ≤ ‖ResW1 n‖).\nAnd from rectInt'_Jlag on the zero-avoiding rectangle: ResW1 n = RectangleIntegral' (Jlag n) z w − Σ_{ρ∈Sz} mult·(1−(ρ/(ρ−1))ⁿ).\n\nSo ‖ResW1 n‖ ≤ ‖RectangleIntegral'‖ + ‖Σ zero terms‖ = |sides| + |λ_n(T)|.\n- |λ_n(T)| ≤ 2·FAlt(T) via hRH (zeros on the line to height T₀: |1−(ρ/(ρ−1))ⁿ| ≤ 2 since |ρ/(ρ−1)| = 1 on the line... wait: ρ = 1/2+iγ ⟹ |ρ/(ρ−1)| = |ρ|/|1−ρ| = 1 (conj symmetry) ⟹ |1−(ρ/(ρ−1))ⁿ| ≤ 2 ✓).\n- |sides|: the four rectangle sides bounded via L1 kernel bounds + tight R-bound + trivial residue. This is D-036's assembly: H(T) for horizontals + V for verticals.\n\nThe rectangle: Lagarias's contour (D-036): rectangle from σ₀ = ... Let me recall D-036's setup: the integrand Jlag n = k_n(w)·(−ζ′/ζ(w))? where k_n(w) = [wⁿ − (w−1)ⁿ]/(w−1)? or kFun. The rectangle: [−a, 2]×[−T, T]? D-037: rectangle with left side at re = −√n (or similar), right side at re = 2 (or 3), T zero-avoiding in [√n, √n+1].\n\nThis is a BIG sub-stage (1–2 sessions per D-064). Let me start by reading D-036 to get the exact assembly. D-036 is the contour-constants derivation. Let me check its structure.\n\nActually — before diving deep, this is a major design task. Per the sub-stage rule, I should break ii′-6 into sub-stages: \n- ii′-6-0: design doc (D-071) — the L0 assembly plan: exact statements, the rectangle, the four side bounds, the bridge, the final fold.\n- ii′-6-i: the sides identity (rectangle on zero-avoiding T → ResW1 = sides − zero-sum).\n- ii′-6-ii: the bridge |λ_n(T)| ≤ 2·FAlt(T) (hRH).\n- ii′-6-iii: the horizontal bounds (tight R-bound + Finding-2 repair) + vertical bounds (Dirichlet + FE).\n- ii′-6-iv: the fold |Sf n| ≤ Bn FAlt n on [10⁵, N₁].\n\nLet me read D-036 first — it's the analytic source of the assembly. Check its size/structure.","signature":""},{"type":"text","text":"Now let me study D-036 (the contour assembly source) and the Bn/FAlt definitions in LiSkeleton."},{"type":"tool_use","id":"Bash_163","name":"Bash","input":{"command":"cd D:/Python/Quantum/research/derivations && wc -l D-036*.md D-037*.md D-056*.md && grep -n \"^## \\|^# \" D-036*.md | head -25","description":"Survey D-036 structure"},"extra_content":null},{"type":"tool_use","id":"Grep_164","name":"Grep","input":{"pattern":"def Bn|def FAlt|noncomputable def Bn|noncomputable def FAlt|def C0hat|def Halt|def IIMaj","path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","output_mode":"content","-n":true,"-A":8},"extra_content":null}]