[{"type":"thinking","thinking":"Kernel bounds and dirichletC2 confirmed. Note: `kFun_shift_norm_le_exp_four_add_one (hσ : s.re ≤ 3) (hT : n ≤ s.im²)` gives ‖kFun n (s+1)‖ ≤ e⁴+1 — the shift form: kFun n (s+1) with s.re ≤ 3 and n ≤ s.im². For the horizontal at im = T, s+1 = w has re ≤ 4, im = T... the kernel at w: kFun n w with w = s+1, s = w−1, re s ≤ 3 ⟺ re w ≤ 4, and n ≤ (im s)² = T² ✓. So on the mid-segment re w ∈ [−5/2, 3] ⊂ [−..., 4] the e⁴+1 charge applies ✓ (this is the (6.11) bound: needs re(w−1) ∈ [−3,3] ⟺ re w ∈ [−2,4] — our [−5/2,3] ⊂ [−2,4] ✓).\n\nWait — hmm, kFun n (s+1) with the (6.11)-style bound: |(s+1)/s|ⁿ at re s ∈ [−3,3]: the charge e⁴+1 needs the max of (σ+1)/(σ²+T²) over re s... earlier I worked this out (max at σ = 3: 4/(9+T²)) — the Lean lemma presumably proves e⁴ via the coarser 4/T² route... whatever, it's proved. The charge applies for re w ∈ [−2, 4] (s = w−1 ∈ [−3, 3]) — and im s = im w ≥ T with n ≤ T² ✓.\n\nFor the dog-leg:\n- V_left at w = 3+it: |kFun n (3+it)| = |(3+it)/(2+it)|ⁿ−1: |(3+it)/(2+it)| = ((9+t²)/(4+t²))^{1/2} = (1+5/(4+t²))^{1/2}: ≤ e^{5n/(2(4+t²))} ≤ e^{5/2} for t² ≥ n ✓. New lemma needed (DL-2) — or does an existing lemma cover it? kFun_norm_le_two needs re ≤ 1/2 (no); the shift lemmas are at s+1 form with e⁴+1/e^{√n/2}+1 (too coarse). DL-2 needs the exact e^{5/2}+1 computation — new lemma, but elementary (the |1+1/(2+it)|² identity).\n- H_high at w = σ+iU: kernel ratio per step — needs the (6.12) ratio lemma (exists? The old plan for ii′-6-c used kFun_shift_norm_le_exp_sqrt_half_add_one for the right tail — the D-073 Lemma 4.1 is the NEW thing to formalize in DL-3).\n\ndirichletC2 = ∑' ‖LSeries.term Λ 2 n‖ — the s=2 term-norm sum ✓ (= Σ Λ(m)/m² ✓).\n\nNow — the conjugation for the lower side. Is there a Jlag conj lemma? The grep found none with that name. Let me check what conj machinery exists (zeroMult_conj, analyticOrderAt_riemannZeta_conj, isNontrivialZero_conj were built). For the lower-side dog-leg transfer, options: (a) an analogous theorem proved directly for the lower side (mirror of the upper), or (b) a Jlag conjugation identity: Jlag n (conj s) = conj (Jlag n s) — true? kFun n (conj s) = (conj s/(conj s −1))ⁿ − 1 = conj((s/(s−1))ⁿ − 1) = conj(kFun n s) ✓; −ζ′/ζ(conj s) = conj(−ζ′/ζ(s)) (ζ conj-symmetric: riemannZeta_conj exists — used in session 036's zeroMult_conj) ✓. So Jlag n (conj s) = conj (Jlag n s) — a short new lemma, then the lower-side integrals are conjugates of the upper-side ones. The directive prefers this route \"provided the existing integrand conjugation theorem supports it\" — it doesn't exist yet; we build it (small).\n\nNow the last audit item: where \"3.3\" enters Bn — confirmed: Bn's horizontal share 2·(e⁴+1)·(7·Ĥ T + 3.3) with T = √n+1 (and marginTilde identically; the mK constant block −2(e⁴+1)(7(3·2.9²+48)+3.3)).\n\nAlso the bridge: zeroSum_norm_le_four_FAlt_add_one (LiAssembly:443) — takes hTle : T ≤ 3e12−1 as hypothesis ✓ parameterized.\n\nI have everything for the audit table. Also let me quickly confirm the wide pigeonhole's exact statement (LiContour:404): exists_zero_avoiding_T_wide (hZC) (ha : 3 ≤ a) : ∃ T ∈ [a, a+1], clearance 1/(9·C0hat(a+2)) against zeros with im ∈ (a−1, a+2) — need exact form. Let me read it.","signature":""},{"type":"text","text":"All key pieces confirmed. Checking the wide pigeonhole's exact form (the T_cut anchor) to complete the audit."},{"type":"tool_use","id":"Read_138","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiContour.lean","offset":395,"limit":30},"extra_content":null}]