[{"type":"thinking","thinking":"Wait — I'm making a mess. The `fiber_mult_count` lemma has a botched finiteness proof (the `rw [abs_of_pos h3] at *` is wrong — there's nothing to rewrite) and a sorry. Let me stop and write this properly.\n\nThe finiteness of {ρ : Z | dyadClass X ρ = j ∧ X < |im|}: members have |im| < 2^{j+1}X (dyadClass_upper) and |re| ≤ 1 ⟹ ‖ρ‖ ≤ |re| + |im| ≤ 1 + 2^{j+1}X < 1 + 2^{j+1}X + 1 ⟹ ⊆ {‖ρ‖ < 1 + 2^{j+1}X + 1} finite via finite_subtype_norm_lt. The subset proof:\n```lean\n        (finite_subtype_norm_lt (1 + (2 : ℝ) ^ (j + 1) * X + 1)).subset (by\n          intro ρ hρ\n          -- hρ : ρ ∈ {ρ : Z | dyadClass X ρ = j ∧ X < |im ρ|} — defeq to the And\n          have hup := dyadClass_upper (by linarith : (0 : ℝ) < X) ρ\n          have hnorm : ‖(ρ : ℂ)‖ ≤ |(ρ : ℂ).re| + |(ρ : ℂ).im| :=\n            Complex.norm_le_abs_re_add_abs_im _\n          have hre : |(ρ : ℂ).re| ≤ 1 := by\n            rw [abs_le]\n            exact ⟨by linarith [ρ.2.2.1], by linarith [ρ.2.2.2]⟩\n          have him : |(ρ : ℂ).im| < (2 : ℝ) ^ (j + 1) * X := hup\n          -- goal: ‖↑ρ‖ < 1 + 2^(j+1)*X + 1\n          have h2pow : (0 : ℝ) ≤ (2 : ℝ) ^ (j + 1) * X := by positivity\n          -- goal form: ρ ∈ {ρ : Z | ‖↑ρ‖ < 1 + 2^(j+1)*X + 1} — defeq to the <\n          show ‖(ρ : ℂ)‖ < 1 + (2 : ℝ) ^ (j + 1) * X + 1\n          linarith [hnorm, hre, him, h2pow])\n```\nWait — Complex.norm_le_abs_re_add_abs_im — check name: `Complex.norm_le_abs_re_add_abs_im (z : ℂ) : ‖z‖ ≤ |z.re| + |z.im|`? I believe it's `Complex.norm_le_abs_re_add_abs_im` ✓ (used in some files). If not: `by rw [Complex.norm_eq_abs]; exact abs_...` — no: ‖z‖ ≤ |re| + |im| : `Complex.norm_le_abs_re_add_abs_im` — I'm fairly confident it exists.\n\nHmm wait — the finiteness subset target: finite_subtype_norm_lt R : {ρ : Z | ‖↑ρ‖ < R}.Finite — the subset arg : {fiber set} ⊆ {‖↑ρ‖ < R}: intro ρ hρ: hρ : ρ ∈ {fiber set} — defeq to the And ✓ (setOf membership defeq) — obtain the parts: hρ.1 : dyadClass = j, hρ.2 : X < |im|. And the goal after intro: ρ ∈ {ρ : Z | ‖↑ρ‖ < 1+...} — defeq to ‖↑ρ‖ < ...: `show ‖(ρ:ℂ)‖ < ...` ✓.\n\nThen the count: the fiber-finset maps via Subtype.val into the union of the two zerosIn toFinsets. Let me write the full count proof:\n\n```lean\n  -- the count: the image under Subtype.val ⊆ the two windows\n  set W1 : Set ℂ := Zeta23.zerosIn ((2 : ℝ) ^ j * X - 1) ((2 : ℝ) ^ (j + 1) * X) with hW1\n  set W2 : Set ℂ := Zeta23.zerosIn (-(2 : ℝ) ^ (j + 1) * X) (-((2 : ℝ) ^ j * X - 1)) with hW2\n  have hW1fin : W1.Finite := Zeta23.zerosIn_finite _ _\n  have hW2fin : W2.Finite := Zeta23.zerosIn_finite _ _\n  -- the image inclusion\n  have himg : Subtype.val '' {ρ : Z | dyadClass X ρ = j ∧ X < |(ρ:ℂ).im|}\n      ⊆ W1.toFinset... \n```\nHmm — this is getting complicated. Simplify: bound the fiber sum by going through the image-finset:\nΣ_{ρ ∈ fiberFinset} (zeroMult ↑ρ : ℝ) = Σ_{z ∈ fiberFinset.image Subtype.val} (zeroMult z : ℝ) (Finset.sum_image — the injOn: Subtype.val_injective.injOn ✓ banked) ≤ Σ_{z ∈ W1.toFinset ∪ W2.toFinset} zeroMult z (Finset.sum_le_sum_of_subset_of_nonneg: the image ⊆ the union: for z = ↑ρ with ρ in the fiber: nontrivial ✓; X < |im|; dyadClass = j ⟹ |im| < 2^{j+1}X (upper) — and |im| > X... the windows: W1 = zerosIn (2^j X − 1) (2^{j+1}X): members have 2^jX−1 < im ≤ 2^{j+1}X: my positive-side fiber zeros: im > X ≥ 2^j X... for j ≥ 1: im ≥ 2^j X (dyadClass_lower needs X ≤ |im| ✓ we have X < |im|) ⟹ im ≥ 2^jX > 2^jX − 1 ✓ and im < 2^{j+1}X ≤ ... im < 2^{j+1}X ⟹ im ≤ 2^{j+1}X ✓ — so im ∈ (2^jX−1, 2^{j+1}X] ✓ ∈ W1 (with nontrivial ✓). For j = 0: dyadClass_lower gives 2^0·X = X ≤ |im| — but the fiber-0 members with nonzero tailTerm have |im| > X: im ∈ (X, 2X) ⊆ (X−1, 2X] ✓ = W1 with j=0 ✓. So positive members ⊆ W1 ✓; negative: im ≤ −X: −2^{j+1}X < im... im > −2^{j+1}X (from |im| < 2^{j+1}X) and im ≤ −X ≤ −2^jX (for j ≥ 1: im ≤ −2^jX ⟺ |im| ≥ 2^jX ✓; j=0: im < −X ⟹ im ≤ −X = −2^0X ✓): so im ∈ (−2^{j+1}X, −2^jX] = W2 ✓ (my W2 := zerosIn (−2^{j+1}X) (−(2^jX−1))?? — I wrote −(2^jX − 1) as the upper: (T₁, T₂] = (−2^{j+1}X, −(2^jX−1)]: im ≤ −2^jX vs −(2^jX−1) = −2^jX+1: im ≤ −2^jX ≤ −2^jX+1 ✓ ⊆ ✓ — but cleaner: use upper = −2^jX directly: W2 := zerosIn (−2^{j+1}X) (−2^jX): im ∈ (−2^{j+1}X, −2^jX] ✓ matches exactly. Let me use that.)\n\nThen Σ_image ≤ Σ_{W1.toFinset ∪ W2.toFinset} zeroMult = (by Finset.sum_union with disjoint) = Ncount... — the toFinset sums = the finsums = Ncount values ✓ via finsum_mem_eq_finite_toFinset_sum backwards:\nΣ_{z ∈ W1.toFinset} zeroMult z = ∑ᶠ z ∈ W1, zeroMult z = Ncount(2^jX−1, 2^{j+1}X) ✓.\n\nThen the two Ncount bounds via Ncount_le_of_interval_gen:\nNcount(2^jX−1, 2^{j+1}X] ≤ A₀·((2^{j+1}X − (2^jX−1)) + 1)·log(3·max(|2^jX−1|, |2^{j+1}X|)+5): b−a+1 = 2^{j+1}X − 2^jX + 2 = 2^jX + 2: max: |2^{j+1}X| = 2^{j+1}X; |2^jX−1| = 2^jX−1 (positive since ≥ 199): ≤ 2^{j+1}X ✓: ≤ A₀(2^jX+2)·log(3·2^{j+1}X+5).\nNcount(−2^{j+1}X, −2^jX]: b−a+1 = 2^jX+1: max = 2^{j+1}X: ≤ A₀(2^jX+1)·log(3·2^{j+1}X+5).\nSum ≤ A₀(2^{j+1}X + 3)·log(3·2^{j+1}X+5) ✓ matches the lemma statement.\n\nThe disjointness of W1, W2: im > 2^jX−1 ≥ 199 > 0 vs im ≤ −2^jX ≤ −200 < 0 ✓.\n\nOK — the fiber_mult_count is well-defined. Then the main lemma:\n\n```lean\ntheorem zero_tail_inv_sq_le {X : ℝ} (hX : 200 ≤ X) :\n    ∑' ρ : {ρ : ℂ // IsNontrivialZero ρ}, tailTerm X ρ\n      ≤ 16 * A0count * Real.log X / X := by\n  have hX0 : (0 : ℝ) < X := by linarith\n  classical\n  -- to ℝ≥0∞ and fiberwise\n  rw [← ENNReal.ofReal_le_ofReal_iff (by positivity?...)]\n```\nHmm — the RHS positivity: 16·A₀·log X/X ≥ 0: A₀ ≥ 1, log X ≥ log 200 > 0, X > 0: positivity can't (A0count, log). Explicit: `have hC0 : (0:ℝ) ≤ 16 * A0count * Real.log X / X := by ...` via div_nonneg (mul_nonneg ... (Real.log_nonneg (by linarith))) hX0.le — log X ≥ 0 ⟸ X ≥ 1: Real.log_nonneg (by linarith : 1 ≤ X) ✓.\n\nThen:\n```lean\n  rw [← ENNReal.ofReal_le_ofReal_iff hC0]\n  rw [ENNReal.ofReal_tsum_of_nonneg (fun ρ => tailTerm_nonneg ρ) (tailTerm_summable X)]\n  -- goal: ∑' ρ, ofReal (tailTerm X ρ) ≤ ofReal (16A₀ logX/X)\n  -- fiberwise over dyadClass X:\n  rw [← ENNReal.tsum_fiberwise (fun ρ => ENNReal.ofReal (tailTerm X ρ)) (dyadClass X)]\n  -- goal: ∑' j, ∑'_{fiber j} ofReal(tailTerm) ≤ ofReal(C)\n  calc ∑' j : ℕ, ∑' b : (dyadClass X) ⁻¹' {j}, ENNReal.ofReal (tailTerm X b)\n      ≤ ∑' j : ℕ, ENNReal.ofReal (F j) := by\n        apply ENNReal.tsum_le_tsum  -- check name: `ENNReal.tsum_le_tsum (h : ∀ i, f i ≤ g i)` ✓\n        intro j\n        -- the fiber bound\n        ...\n    _ = ENNReal.ofReal (∑' j : ℕ, F j) := by\n        rw [ENNReal.ofReal_tsum_of_nonneg (fun j => F-nonneg) (F-summable)]\n    _ ≤ ENNReal.ofReal (16 * A0count * Real.log X / X) := by\n        apply ENNReal.ofReal_le_ofReal  -- hmm: ofReal_le_ofReal : ofReal a ≤ ofReal b ↔ a ≤ b needs 0 ≤ b: `ENNReal.ofReal_le_ofReal_iff hC0`.mpr — the direct: `ENNReal.ofReal_le_ofReal (h : a ≤ b) : ofReal a ≤ ofReal b` ✓ exists (monotone)\n        exact the real bound\n```\n\nThe fiber bound: ∑'_{b : fiber j} ofReal(tailTerm X b) ≤ ofReal(F_j):\n- the fiber subtype {b : Z // dyadClass X b = j} — finite: the fiber set ⊆ {‖ρ‖ < 1 + 2^{j+1}X + 1} finite ✓ — Fintype via Set.Finite.fintype — then tsum_fintype: ∑' = ∑ over Finset.univ of the subtype — and the subtype's univ sum = the sum over the fiber-finset (Set.Finite.toFinset) — `Set.Finite.toFinset` + `Finset.sum_...`: the subtype-univ sum vs the toFinset sum: `Set.Finite.sum_toFinset`? hmm: for a finite SET s : Set Z: ∑ x : s, f x = ∑ x ∈ s.toFinset, f x — hmm: `Set.Finite.toFinset` : the subtype s's univ-finset is s.toFinset: `Finset.univ` of ↥s = s.toFinset: `Set.Finite.toFinset`... the sum over the subtype: `Finset.sum_univ...`? There might be friction. Alternative: avoid tsum_fintype: bound ∑'_{fiber} ofReal(tailTerm) ≤ ofReal(F_j) directly via: the fiber tsum = the if-form on Z... ugh.\n\nCleaner: ∑'_{b : fiber-subtype} ofReal(tailTerm X b) = ∑'_{ρ : Z} ofReal (if dyadClass X ρ = j then tailTerm X ρ else 0)?? — the fiber-subtype tsum vs the if-form on Z: `tsum_subtype`? hmm — is there `tsum_subtype (f) (s) : ∑' x : s, f x = ∑' x, s.indicator f x`?? — `tsum_subtype (s : Set α) (f : α → ℝ) : ∑' x : s, f x = ∑' x, s.indicator f x` — I recall `tsum_subtype` exists ✓ (maybe `tsum_subtype s f`). The fiber as a SET of Z: {ρ : Z | dyadClass X ρ = j}: the subtype (dyadClass X)⁻¹'{j} is defeq to ↥{ρ | dyadClass X ρ = j} ✓. So:\n∑'_{b : fiber} ofReal(tailTerm b) = ∑'_{ρ : Z} ({ρ | dyadClass = j}.indicator (ofReal ∘ tailTerm X)) ρ\n≤ ∑'_{ρ : Z} (ofReal(F_j · m-normalized...))... hmm — indicator bound: the indicator's tsum over Z: bound by ofReal of the real indicator tsum... this is circular-ish. Let me just do the finite-subtype route carefully:\n\nFor the finite fiber (Fintype): ∑'_{b : fiber} ofReal(tailTerm X b) = ∑_{b : fiber} ofReal(tailTerm X b) — the tsum over a finite type = the univ finset sum: `tsum_fintype (f) : ∑' b, f b = ∑ b, f b` ✓ (from the probe: tsum_fintype needs [Fintype β] ✓). Then ∑_{b : fiber} ofReal(tailTerm X b) = ofReal(∑_{b : fiber} tailTerm X b) (ENNReal.ofReal_sum_of_nonneg ✓). Then bound the real sum: ∑_{b : fiber} tailTerm X b ≤ (1/(4^jX²))·∑_{b : fiber} (zeroMult b : ℝ) [each term: tailTerm ≤ (1/(4^jX²))·m: for X < |im|: tailTerm = m/(1+γ²) ≤ m/(4^jX²) (the 1/(1+γ²) ≤ 1/(4^jX²) from |im| ≥ 2^jX via dyadClass_lower ✓ (needs X ≤ |im| — we have X < |im| ✓)); else tailTerm = 0 ≤ · ✓] ≤ (1/(4^jX²))·(the count bound) = F_j-ish.\n\nHmm wait — but the fiber here is the FULL dyadClass-fiber (includes |γ| ≤ X members whose tailTerm = 0) — the count bound I built (fiber_mult_count) is stated for {dyadClass = j ∧ X < |im|} — the sum ∑_{b : full-fiber} m vs ∑_{nonzero-fiber} m: I bound ∑_{b : full fiber} tailTerm: terms with |γ| ≤ X vanish: ∑_{full} tailTerm = ∑_{|γ|>X part} tailTerm — as a finset identity: filter! The full-fiber-finset sum = sum over the filtered (X < |im|) part: `Finset.sum_filter`: ∑_{full} tailTerm = ∑_{full.filter (X < |im|)} tailTerm + ∑_{not} tailTerm(=0) = the filtered sum ✓. And the filtered finset ⊆ the fiber_mult_count's set {dyadClass = j ∧ X < |im|} — equal actually ✓ (defeq sets). So ∑_{full fiber} tailTerm ≤ (1/(4^jX²))·Σ_{filtered} m ≤ (1/(4^jX²))·(fiber_mult_count bound) ✓.\n\nHmm — the \"full fiber\" as a finset: the Fintype's univ... Let me define the fiber finset explicitly as a SET-based toFinset: fiberFin X j := (show {ρ : Z | dyadClass X ρ = j}.Finite from ...).toFinset and relate its subtype-univ sum... The tsum_fintype route gives ∑_{b : ↥{ρ | dyadClass = j}} — and ↥{...} vs the toFinset: `Set.Finite.sum_toFinset (hs : s.Finite) : ∑ x ∈ hs.toFinset, f x = ∑ x : ↥s, f x`? — there might be `Set.Finite.sum_toFinset`?? hmm — actually for the UPPER BOUND I don't need the exact equality: ∑'_{b : fiber} ofReal(tailTerm) ≤ ofReal(F_j) — go through: ∑'_{fiber} ofReal(tailTerm) = ∑'_{ρ : Z} (fiberSet.indicator (fun ρ => ofReal (tailTerm X ρ))) [tsum_subtype ✓ if it exists] ≤ ∑'_{ρ : Z} ofReal(F-term_j ρ) where F-term_j ρ := if dyadClass X ρ = j then (1/(4^jX²))·m_ρ else 0 [indicator ≤ indicator pointwise ✓ ofReal monotone ✓] = ofReal(∑'_{Z} (if dyadClass = j then (1/(4^jX²))·m else 0)) [ofReal_tsum_of_nonneg: summable (of_norm_bounded vs (1/(4^jX²))·zero_sum_inv_sq-majorant... m ≤ ... hmm — m·(1/(4^jX²)) ≤ m·(1/(4^jX²))·... any bound: (1/(4^jX²))·m vs the dominator m/(1+‖γ‖²): m ≤ m·(1+‖γ‖²)... no: m/(1+‖γ‖²) ≤ m — so (1/(4^jX²))·m is NOT ≤ m/(1+‖γ‖²) (the former bigger when 4^jX² < 1+‖γ‖²... j=0, X = 200: 4^jX² = 40000; zeros with 1+‖γ‖² < 40000 — small zeros: m·(1/40000) vs m/(1+γ²) — for γ tiny: m/1 vs m/40000 — the dominator is BIGGER — so of_norm_bounded with the dominator FAILS for small γ. But the if-form (dyadClass = j) restricts to |γ| < 2^{j+1}X — the support is FINITE ⟹ summable via summable_of_hasFiniteSupport ✓!! (support ⊆ the finite fiber set ✓). Then the indicator tsum = the filtered finset sum ✓ via tsum_eq_sum (tsum_split_finset's pieces) or tsum_eq_sum_of... the if-form's tsum over Z = the finset sum over the fiber-finset via `tsum_eq_sum (s := fiberFin) (fun b hb => if_neg hb)` ✓ then Finset.sum_congr (if_pos) ✓.\n\nOK the plumbing is all standard now. And tsum_subtype — check exists: let me just avoid it: I need ∑'_{b : fiber-subtype} ofReal(tailTerm) — instead of tsum_subtype, use the FINITEness directly: tsum_fintype (Fintype of the fiber — haveI from Set.Finite.fintype). Then ∑_{b : fiber} = univ sum — and I need ∑_{b : ↥fiberSet} ofReal(tailTerm b) ≤ ofReal(F_j): via ofReal_sum_of_nonneg backwards: = ofReal(∑_{b : ↥fiberSet} tailTerm b) — and the subtype sum ∑_{b : ↥fiberSet} tailTerm b = ∑_{ρ ∈ fiberFin} tailTerm ρ (the univ-of-subtype vs toFinset: `Set.Finite.toFinset` sum bridge: `Finset.sum_subtype`? hmm — there IS: `Finset.sum_subtype (s : Finset α) {p} (h : ∀ x, x ∈ s ↔ p x) (f) : ∑ x : {x // p x}, f x = ∑ x ∈ s, f x`?? the form: `Finset.sum_subtype (s) (h : ∀ x, x ∈ s ↔ p x) f : ∑ x ∈ s, f x = ∑ x : subtype p, f x`? Let me just probe these small bridges... OR simpler: define the fiber finset AS the Fintype-univ: work with `Finset.univ` of the subtype throughout — the count bound (fiber_mult_count) is stated as a sum over the toFinset of a set — let me RESTATE fiber_mult_count as a sum over the SUBTYPE (Fintype.univ):\n\n`fiber_mult_count : ∑ ρ : {ρ : Z // dyadClass X ρ = j ∧ X < |im|}, (zeroMult ρ.1 : ℝ) ≤ ...` — hmm the subtype-of-subtype sum. The bridge from the subtype sum to the window counts: the image under (·.1.1) : Z → ℂ: Subtype.val... the sum over the subtype = sum over the image (Finset.sum_image with injOn — Subtype.val injective on the doubly-subtype ✓). \n\nOK — I keep going back and forth; the patterns are all banked. Let me just write it and fight the elaborator. Also: check `Complex.norm_le_abs_re_add_abs_im` and `ENNReal.tsum_le_tsum` and `tsum_subtype` existence quickly, then write §4-main in one go.","signature":""},{"type":"text","text":"Let me probe the remaining bridge names before writing the assembly."},{"type":"tool_use","id":"Bash_377","name":"Bash","input":{"command":"cd D:/Python/Quantum/research/lean/NguyenRH1 && cat > Probe.lean << 'EOF'\nimport Mathlib\nopen Complex Filter Topology Finset Real\n#check @Complex.norm_le_abs_re_add_abs_im\n#check @ENNReal.tsum_le_tsum\n#check @tsum_subtype\n#check @Finset.sum_subtype\n#check @Set.Finite.sum_toFinset\n#check @ENNReal.ofReal_le_ofReal\n#check @Set.Finite.fintype\nexample : ∀ (a b : ℝ≥0∞), a ≤ b → a ≤ b := fun a b h => h\nEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean Probe.lean 2>&1 | head -20","description":"Probe the bridge lemma names","timeout":300000},"extra_content":null}]