[{"type":"thinking","thinking":"All verified:\n- `Finset.card_eq_sum_ones (s : Finset ι) : #s = ∑ _ ∈ s, 1` ✓ (in a Group namespace? It's in Algebra/BigOperators/Group/Finset/Basic — might be in a namespace. Line 958: `lemma card_eq_sum_ones (s : Finset ι) : #s = ∑ _ ∈ s, 1` — probably `Finset.card_eq_sum_ones`. The `#s` notation = s.card. It's in namespace Finset presumably ✓.\n- `AnalyticAt.analyticOrderAt_ne_zero` ✓.\n- `ENat.toNat_pos (hn0 : n ≠ 0) (hxt : n ≠ ⊤) : 0 < n.toNat` at Data/ENat/Basic.lean:192 ✓ (namespace ENat).\n\nNow make the edits.\n\nEdit 1: LiContour.lean — add one_le_zeroMult before ZeroCountProp and change the def. The def is at line 56-61 (from earlier read: lines 44-61 show zeroWindow def, zeroWindow_finite, ZeroCountProp def at 56-61 with docstring at 53-55... let me get exact text). From the earlier read:\n\n```\n/-- [PERIMETER — Trudgian 2014 N(T)/S(T), D-062 §4] The unit-window\nzero-count bound: at most Ĉ₀(t+1) nontrivial zeros have ordinates in\n[t, t+1] (t ≥ 2). Declared counting input for the L3 pigeonhole;\nĈ₀ = `C0hat` is D-056 §2's honest majorant (LiSkeleton). -/\ndef ZeroCountProp : Prop :=\n  ∀ t : ℝ, 2 ≤ t → (Nat.card (zeroWindow t) : ℝ) ≤ C0hat (t + 1)\n```\n\nReplace with one_le_zeroMult + new def.","signature":""},{"type":"text","text":"All verified. Making the three edits: `one_le_zeroMult` + strengthened `ZeroCountProp` in LiContour, then the two consumer adaptations."},{"type":"tool_use","id":"Edit_93","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiContour.lean","old_string":"/-- [PERIMETER — Trudgian 2014 N(T)/S(T), D-062 §4] The unit-window\nzero-count bound: at most Ĉ₀(t+1) nontrivial zeros have ordinates in\n[t, t+1] (t ≥ 2). Declared counting input for the L3 pigeonhole;\nĈ₀ = `C0hat` is D-056 §2's honest majorant (LiSkeleton). -/\ndef ZeroCountProp : Prop :=\n  ∀ t : ℝ, 2 ≤ t → (Nat.card (zeroWindow t) : ℝ) ≤ C0hat (t + 1)","new_string":"/-- A nontrivial zero has multiplicity ≥ 1 (zeroMult is the analytic\norder). Bridges the mult-weighted perimeter count to the plain window\ncardinality (card ≤ Σ mult). -/\ntheorem one_le_zeroMult {ρ : ℂ} (hρ : IsNontrivialZero ρ) : 1 ≤ zeroMult ρ := by\n  have hρ1 : ρ ≠ 1 := by\n    rintro rfl\n    have h := hρ.2.2\n    rw [Complex.one_re] at h\n    exact (lt_irrefl (1 : ℝ)) h\n  have htop := zeta_order_ne_top_of_ne hρ1\n  have hne : analyticOrderAt riemannZeta ρ ≠ 0 :=\n    (analyticAt_riemannZeta hρ1).analyticOrderAt_ne_zero.mpr hρ.1\n  exact ENat.toNat_pos hne htop\n\n/-- [PERIMETER — Trudgian 2014 N(T)/S(T), D-062 §4] The unit-window\nzero-count bound: the multiplicity-weighted count of nontrivial zeros\nwith ordinates in [t, t+1] is at most Ĉ₀(t+1) (t ≥ 2). Declared\ncounting input for the L3 pigeonhole and the tight R-bound; Ĉ₀ =\n`C0hat` is D-056 §2's honest majorant (LiSkeleton).\n[Erratum 2026-08-23 (session 035): strengthened from the unweighted\nset-card form `(Nat.card (zeroWindow t) : ℝ) ≤ Ĉ₀(t+1)` to the\nmult-weighted form — Trudgian's N(T) counts zeros with multiplicity,\nand the mult-weighted form is what the 5f/ii′-6 near sums\n(zeroMult-weighted, from `seriesFun`) consume; the set-card encoding\nunder-captured the declared source (a second ii′-1a-class encoding\ndefect, found at the point of consumption). The set-card bound follows\nfrom this form via `one_le_zeroMult` (card ≤ Σ mult); the two\nconsumers (exists_zero_avoiding_T, exists_gapCircle_radius) adapted.] -/\ndef ZeroCountProp : Prop :=\n  ∀ t : ℝ, 2 ≤ t →\n    (((zeroWindow_finite t).toFinset.sum fun ρ => zeroMult ρ) : ℝ) ≤ C0hat (t + 1)"},"extra_content":null}]