[{"type":"thinking","thinking":"I have the full design context now. Key content for ii′-4:\n\n**L3: zero-avoiding T (pigeonhole on perimeter C₀/FAlt)**\n- Input: zero-count bound. From the perimeter: Trudgian N(T)/S(T) — need to check the exact form declared in LiSkeleton.lean. D-036 §2: zeros in [√n, √n+1]: ≤ C₀(√n+1) where C₀(t) is presumably a bound on zeros in [t−1, t+1]... wait actually \"Zeros in [√n, √n+1]: ≤ C₀(√n+1)\" — hmm, C₀(T) is a bound on the number of zeros with |γ − T| < 1, i.e. in a unit window. Wait, [√n, √n+1] is a unit interval, so zeros in it ≤ C₀(√n+1) where C₀(t) bounds zeros in [t−1, t+1]. Pigeonhole: split the unit interval into sub-intervals; some gap ≥ 1/(C₀+1)... the clearance: place T at gap center ⟹ d_min ≥ 1/(3C₀(√n+1))? Hmm wait, D-036 says \"Pigeonhole: some gap ≥ 1/(C₀+1); choose T at a gap center ⟹ distance to nearest zero d_min ≥ 1/(3C₀(√n+1))\". Hmm, that's slightly off arithmetically (1/(2(C₀+1)) would be the gap-center clearance) but D-036 uses 1/(3C₀(T)) as a safe bound. Whatever — the Lean statement needs:\n\nFor each n (in range), there exists T ∈ [√n, √n+1] such that every nontrivial zero ρ = β + iγ with γ ∈ [√n, √n+1] satisfies |γ − T| ≥ 1/(3·C₀(√n+1)) — actually more precisely: no zero ordinate within d_min of T.\n\nWhat's the perimeter form? Let me check LiSkeleton.lean for the Trudgian declaration (FAlt form, named hyp). Need to see what exact zero-count hypothesis is available: maybe \"N(T) ≤ FAlt(T)\" type bound, or a unit-window count bound C₀. Let me read LiSkeleton.lean.\n\n**L1: kernel bounds**\n- k_n(s) — what is the kernel exactly? From D-036: |1+1/s|² ... k_n seems related to (1+1/s)ⁿ − 1 or similar. The Lagarias kernel in Jlag (from ii′-1b LiSides.lean). Kernel bounds:\n  - On left vertical σ₀ = −5/2: |k_n| ≤ 2 (since |1+1/s| ≤ 1 when Re s ≤ −1/2).\n  - On horizontals: |k_n| ≤ e⁴ + 1.\n  - On right vertical σ₁ = 2√n: |k_n| ≤ e^{√n/2} + 1.\n- Trivial-zero residue at −2: value 1−(2/3)ⁿ... wait STATE.md says \"the trivial-zero residue value 1−(2/3)ⁿ at −2\" but D-036 §3 says \"|(1+1/(−2))ⁿ − 1| = 1 − 2^{−n} ≤ 1\". Hmm — (1 + 1/(−2)) = 1/2, so (1/2)ⁿ − 1 absolute = 1 − 2^{−n}. And in LiSides the zero term is (1−(ρ/(ρ−1))ⁿ): for ρ = −2: ρ/(ρ−1) = −2/−3 = 2/3, so 1−(2/3)ⁿ. Both forms appear; the Lean sum in rectInt'_Jlag uses (1−(ρ/(ρ−1))ⁿ). So the trivial-zero term is 1−(2/3)ⁿ with |1−(2/3)ⁿ| ≤ 1. And zeroMult(−2) = 1 needed so the sum over Sz splits correctly with multiplicity 1.\n\nzeroMult(−2) = 1 route: FE + Gammaℝ_eq_zero_iff + ζ(3) ≠ 0.\n- zeroMult ρ is presumably the order of the zero of ζ (or of xiL?) at ρ. Let me check LiSides.lean for zeroMult's definition.\n- Simplicity at −2: order of ζ at −2 = order of [ΓR·Λ...]. Via the FE: ζ(s) = ... completedRiemannZeta... Standard: Λ(s) = ΓR(s)ζ(s) is entire; at s = −2, ΓR has a simple pole (ΓR = π^{−s/2}Γ(s/2), Γ has simple pole at −1), and Λ(−2) = Λ(3) ≠ 0 by FE (Λ(s) = Λ(1−s): Λ(−2) = Λ(3) ≠ 0 since both ΓR(3) and ζ(3) ≠ 0). So ζ = Λ/ΓR has a simple zero at −2 (Λ nonzero finite, ΓR simple pole ⟹ ζ zero of order 1 = pole order of ΓR).\n\nIn Mathlib: Gammaℝ_eq_zero_iff — wait, ΓR (Gammaℝ) has no zeros; it has POLES at −2n. `Gammaℝ_eq_zero_iff`?? Hmm — D-066 says \"`Gammaℝ_eq_zero_iff` (zeros of ΓR = {−2n} — also the future trivial-zero simplicity input)\". Hmm, actually Mathlib's Gammaℝ is π^{−s/2}·Γ(s/2)... it has no zeros. Maybe Gammaℝ_eq_zero_iff says Gammaℝ s = 0 ↔ false? No wait. Hmm, maybe it's about 1/Gammaℝ? Or maybe Mathlib has a lemma that Gammaℝ s ≠ 0 always (Gammaℝ_ne_zero?). D-066 lists both `Gammaℝ_ne_zero_of_re_pos` and `Gammaℝ_eq_zero_iff`. Hmm, weird. Whatever — the route: order of pole of ΓR at −2 is 1 (simple poles of Γ), Λ(−2) ≠ 0.\n\nLet me now read the Lean files to see the definitions: LiSkeleton.lean (perimeter, FAlt, Sf, etc.) and LiSides.lean (Jlag, zeroMult, rectInt'_Jlag).\n\nThese are big files. Let me grep for the key definitions first.","signature":""},{"type":"text","text":"Design context absorbed. Now the Lean side — perimeter declarations and `zeroMult`/`Jlag` definitions."},{"type":"tool_use","id":"Grep_41","name":"Grep","input":{"-n":true,"output_mode":"content","path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","pattern":"^(def|theorem|lemma|abbrev|noncomputable|structure|inductive)|Trudgian|FAlt|Czero|C₀"},"extra_content":null}]