[{"type":"thinking","thinking":"§18 compiles clean. Now the final piece: §19 (the piece-3 assembly). Commit §18 first, then write the assembly:\n\n```\ntheorem far_series_le (hZC : ZeroCountProp) (hNLZ : NoLowZeros) {t : ℝ} (ht : 10 ≤ t)\n    {s : ℂ} (hσlo : -5/2 ≤ s.re) (hσhi : s.re ≤ 3) (him : s.im = t)\n    (hζ : riemannZeta s ≠ 0) :\n    ‖∑' ρ, if ρ ∈ nearZerosSub t then (0:ℂ) else seriesDiffTerm s (2+tI) ρ‖\n      ≤ 12.1284 * C0hat (t+1) + 7.2425\n```\n\nProof structure:\n1. norm ≤ ∑' farG (far_series_norm_le_farG_tsum).\n2. ∑' farG = ∑_{mainFinset} farG + ∑' (if ∉ mainFinset then farG else 0) (tsum_split_finset — the LiCircleBound ℝ version).\n3. ∑' (if ∉ mainFinset then farG else 0) = ∑' farGRight + ∑' farGLeft (pointwise split with hNLZ).\n4. main ≤ farG_main_le = 4.5·(2C₀+0.55)·1.3476; right ≤ farGRight_le; left ≤ farGLeft_le.\n5. Fold: 4.5·(2C₀+0.55)·1.3476 + Tail_R + Tail_L ≤ 12.1284·C₀ + 7.2425.\n\nFor step 3, the pointwise identity: for every ρ:\n(if ρ ∉ mainFinset then farG t ρ else 0) = farGRight t ρ + farGLeft t ρ\n- ρ ∈ mainFinset: LHS = 0. RHS: farGRight (x-cond fails since x < ⌊t⌋₊+1) + farGLeft (γ<0 impossible by hNLZ: γ < 0 ⟹ |γ| ≥ 14 ⟹ x = t+|γ| ≥ t+14 ⟹ ⌊x⌋₊ ≥ ⌊t⌋₊+14 > ⌊t⌋₊ ⟹ x ≥ ⌊t⌋₊+14 > ⌊t⌋₊+1 ⟹ ¬main — contradiction ⟹ γ ≥ 0 ⟹ farGLeft = 0). So RHS = 0 ✓.\n  Formalize farGLeft_mainFinset_eq_zero (hNLZ) (hmem : ρ ∈ mainFinset t) : farGLeft t ρ = 0.\n- ρ ∉ mainFinset:\n  - near (x < 1): LHS = farG = 0 (farG if_pos near? — near means ρ ∈ nearZerosSub ⟹ farG = 0 ✓); RHS: farGRight (x < 1 < ⌊t⌋₊+1 ⟹ x-cond fails ⟹ 0) + farGLeft (γ < 0 ⟹ farG = 0 ✓ near; γ ≥ 0 ⟹ 0) = 0 ✓.\n  - far, x ≥ ⌊t⌋₊+1: LHS = farG. RHS: γ ≥ t ⟹ farGRight = farG (x-cond ✓ t-cond ✓), farGLeft = 0 (γ ≥ t ≥ 10 > 0 ⟹ ¬γ<0 ⟹ 0); γ < t ⟹ x = t−γ > t ⟹ γ < 0 ⟹ farGRight = 0 (t ≤ γ fails), farGLeft = farG (γ < 0 ✓). ✓\n  - far, x < ⌊t⌋₊+1: ρ far with x < ⌊t⌋₊+1 ⟹ ρ ∈ mainFinset (contradiction with ∉). ✓ (mainFinset's filter is exactly 1 ≤ x ∧ x < ⌊t⌋₊+1 ⟹ far + this ⟹ mem ✓).\n\nSo the identity holds. Formalize:\n```\nlemma farG_tail_eq (hNLZ : NoLowZeros) {t : ℝ} (ht : 10 ≤ t) (ρ : {ρ : ℂ // IsNontrivialZero ρ}) :\n    (if ρ ∈ mainFinset t then (0 : ℝ) else farG t ρ) = farGRight t ρ + farGLeft t ρ := by\n  classical\n  by_cases hmem : ρ ∈ mainFinset t\n  · rw [if_pos hmem]\n    -- farGRight = 0 (x-cond fails), farGLeft = 0 (NoLowZeros)\n    have hright : farGRight t ρ = 0 := by\n      rw [farGRight, if_neg (fun h => by\n        have hx := (mem_mainFinset (by linarith : (0:ℝ) ≤ t)).mp hmem\n        exact absurd h.1 (by\n          have h1 : ((Nat.floor t : ℝ) + 1 ≤ |ρ.1.im - t|) := by exact_mod_cast h.1\n          linarith [hx.2, h1] : ¬ Nat.floor t + 1 ≤ |ρ.1.im - t|))]\n    have hleft : farGLeft t ρ = 0 := by\n      rw [farGLeft, if_neg (by\n        intro hγ\n        have h1 : 14 ≤ |ρ.1.im| := hNLZ ρ.1 ρ.2\n        have hx := (mem_mainFinset (by linarith : (0:ℝ) ≤ t)).mp hmem\n        -- x = t + |γ| ≥ t + 14 ⟹ x ≥ ⌊t⌋₊+14 > ⌊t⌋₊+1 contradiction with hx.2\n        have h2 : |ρ.1.im - t| = t + |ρ.1.im| := by\n          rw [abs_of_neg (by linarith [hγ, ht] : ρ.1.im - t < 0)]\n          have h3 : |ρ.1.im| = -ρ.1.im := abs_of_neg hγ\n          rw [h3]\n          ring\n        have h4 : t + 14 ≤ |ρ.1.im - t| := by rw [h2]; linarith [h1, ht]\n        -- ⌊x⌋₊ ≥ ⌊t+14⌋ ≥ ⌊t⌋₊+14: x ≥ t+14 ⟹ ⌊x⌋₊ ≥ ⌊t⌋₊+14 (floor_mono with the +14 shift)\n        have h5 : Nat.floor t + 14 ≤ Nat.floor |ρ.1.im - t| := by\n          have h6 : Nat.floor (t + 14) ≤ Nat.floor |ρ.1.im - t| := Nat.floor_mono h4\n          have h7 : Nat.floor (t + 14) = Nat.floor t + 14 := by\n            rw [show (14 : ℝ) = ((14 : ℕ) : ℝ) from by norm_num, Nat.floor_add_natCast]\n          rwa [h7] at h6\n        -- ⌊x⌋₊ ≥ ⌊t⌋₊+14 ⟹ x ≥ ⌊t⌋₊+14 > ⌊t⌋₊+1 ⟹ contradicts hx.2\n        have h8 : ((Nat.floor t + 14 : ℕ) : ℝ) ≤ |ρ.1.im - t| :=\n          (Nat.floor_le (abs_nonneg _)).trans (by exact_mod_cast h5)\n        have h9 : ((Nat.floor t + 14 : ℕ) : ℝ) = (Nat.floor t : ℝ) + 14 := by push_cast; ring\n        rw [h9] at h8\n        linarith [hx.2, h8] : ¬ ρ.1.im < 0)]\n    rw [hright, hleft, add_zero]\n  · rw [if_neg hmem]\n    -- case analysis on γ vs t and the near condition\n    by_cases hγ : t ≤ ρ.1.im\n    · -- γ ≥ t: x = γ − t\n      by_cases hx1 : ρ ∈ nearZerosSub t\n      · rw [farG, if_pos hx1]\n        rw [farGRight, if_neg (fun h => by\n          have h2 := mem_nearZerosSub.mp hx1\n          have h3 : |ρ.1.im - t| < 1 := h2\n          have h4 : (Nat.floor t : ℝ) + 1 ≤ |ρ.1.im - t| := by exact_mod_cast h.1\n          linarith [h3, h4] : ¬ (Nat.floor t + 1 ≤ |ρ.1.im - t| ∧ t ≤ ρ.1.im))]\n        rw [farGLeft, if_neg (by linarith [hγ, ht] : ¬ ρ.1.im < 0)]\n      · rw [farGRight, if_pos ⟨?_, hγ⟩]\n        · rw [farGLeft, if_neg (by linarith [hγ, ht] : ¬ ρ.1.im < 0)]\n        · -- x ≥ ⌊t⌋₊+1: ρ far (∉ nearZerosSub) and ∉ mainFinset\n          have hnotmain := hmem\n          by_contra hsmall\n          apply hnotmain\n          -- x < ⌊t⌋₊+1 and far ⟹ mem mainFinset\n          rw [mem_mainFinset (by linarith : (0:ℝ) ≤ t)]\n          refine ⟨?_, ?_⟩\n          · -- x ≥ 1 (far)\n            have h1 := hx1\n            rw [mem_nearZerosSub] at h1\n            push Not at h1\n            exact h1\n          · exact hsmall\n    · -- γ < t: x = t − γ\n      push Not at hγ\n      by_cases hx1 : ρ ∈ nearZerosSub t\n      · rw [farG, if_pos hx1]\n        rw [farGRight, if_neg (fun h => by\n          have h2 := mem_nearZerosSub.mp hx1\n          have h3 : |ρ.1.im - t| < 1 := h2\n          have h4 : (Nat.floor t : ℝ) + 1 ≤ |ρ.1.im - t| := by exact_mod_cast h.1\n          linarith [h3, h4] : ¬ (Nat.floor t + 1 ≤ |ρ.1.im - t| ∧ t ≤ ρ.1.im))]\n        rw [farGLeft, if_neg (by\n          -- γ < 0 possible: farGLeft = farG if γ < 0; near ⟹ farG = 0 anyway\n          by_cases hγ0 : ρ.1.im < 0\n          · rw [farG, if_pos hx1] at *\n            -- goal is to show ¬(γ < 0)?? no — farGLeft's if is (γ < 0): if TRUE, farGLeft = farG = 0 (near)\n            -- so I should NOT negate here — restructure below\n          ...)]\n      ...\n```\nHmm — the γ < t case with near: farG = 0 (near) so LHS = 0; farGLeft = farG if γ < 0 — farG = 0 (near ✓) so farGLeft = 0 either way ✓. Actually — for the near case BOTH sides are 0 trivially regardless of γ:\n- near: LHS = if ∉ main then farG = farG = 0 (near ✓). Wait — LHS is `(if ρ ∈ mainFinset then 0 else farG t ρ)` — for ρ ∉ mainFinset (we're in the ¬mem branch): LHS = farG t ρ = 0 (near ✓). RHS: farGRight + farGLeft: farGRight = if (x-cond ∧ t≤γ) then farG: x-cond fails (x < 1 < ⌊t⌋₊+1) ⟹ 0; farGLeft = if γ<0 then farG = 0 (near ✓) else 0 ✓. Sum = 0 ✓. So near case: all three are 0. ✓\n\nRestructure the ¬mem branch:\n```\n  · rw [if_neg hmem]\n    by_cases hnear : ρ ∈ nearZerosSub t\n    · have hz : farG t ρ = 0 := farG, if_pos hnear — via: rw [farG, if_pos hnear]\n      rw [hz]\n      rw [farGRight, if_neg (fun h => by\n        have h2 := mem_nearZerosSub.mp hnear\n        have h4 : (Nat.floor t : ℝ) + 1 ≤ |ρ.1.im - t| := by exact_mod_cast h.1\n        linarith [h2, h4] : ¬ (Nat.floor t + 1 ≤ |ρ.1.im - t| ∧ t ≤ ρ.1.im))]\n      rw [farGLeft]\n      by_cases hγ0 : ρ.1.im < 0\n      · rw [if_pos hγ0, hz]\n      · rw [if_neg hγ0]\n    · -- far (∉ nearZerosSub): x ≥ 1; ∉ main ⟹ x ≥ ⌊t⌋₊+1\n      have hbig : (Nat.floor t : ℝ) + 1 ≤ |ρ.1.im - t| := by\n        by_contra hsmall\n        apply hmem\n        rw [mem_mainFinset (by linarith : (0:ℝ) ≤ t)]\n        refine ⟨?_, ?_⟩\n        · have h1 := hnear\n          rw [mem_nearZerosSub] at h1\n          push Not at h1\n          exact h1\n        · -- x < ⌊t⌋₊+1 from hsmall: hsmall : ¬(⌊t⌋₊+1 ≤ x): x < ⌊t⌋₊+1\n          have h2 : |ρ.1.im - t| < (Nat.floor t : ℝ) + 1 := by\n            have h3 : ¬((Nat.floor t : ℝ) + 1 ≤ |ρ.1.im - t|) := by\n              intro h4\n              apply hsmall\n              have h5 : ((Nat.floor t + 1 : ℕ) : ℝ) ≤ |ρ.1.im - t| := by exact_mod_cast h4\n              have h6 : ((Nat.floor t + 1 : ℕ) : ℝ) = (Nat.floor t : ℝ) + 1 := by push_cast; ring\n              rwa [h6] at h5\n            exact (not_le).mp h3\n          exact h2\n      by_cases hγ : t ≤ ρ.1.im\n      · rw [farGRight, if_pos ⟨by exact_mod_cast (by\n          have h6 : ((Nat.floor t : ℕ) : ℝ) + 1 ≤ |ρ.1.im - t| := by\n            have h7 : ((Nat.floor t + 1 : ℕ) : ℝ) ≤ |ρ.1.im - t| := by exact_mod_cast hbig\n            ... — hbig is already (⌊t⌋₊:ℝ)+1 ≤ x ✓ just use it\n          exact hbig), hγ⟩]\n        rw [farGLeft, if_neg (by linarith [hγ, ht] : ¬ ρ.1.im < 0)]\n      · push Not at hγ\n        -- γ < t: x = t−γ > t ⟹ γ < 0 ⟹ farGLeft = farG; farGRight = 0\n        have hγ0 : ρ.1.im < 0 := by\n          have h2 : |ρ.1.im - t| = t - ρ.1.im := by\n            rw [abs_of_neg (by linarith [hγ] : ρ.1.im - t < 0)]\n            ring\n          have h3 : t < t - ρ.1.im := by\n            have h4 : t ≤ |ρ.1.im - t| := by linarith [hbig, ht]\n            rw [h2] at h4\n            linarith [h4, ht]\n          linarith [h3]\n        rw [farGRight, if_neg (fun h => hγ h.2), farGLeft, if_pos hγ0]\n```\nHmm wait — hbig : (⌊t⌋₊:ℝ)+1 ≤ x — the farGRight's if-condition is `Nat.floor t + 1 ≤ |ρ.1.im − t|` — as a ℕ-inequality? I defined farGRight's if as `(Nat.floor t + 1 ≤ |ρ.1.im - t| ∧ t ≤ ρ.1.im)` — hmm — `Nat.floor t + 1 ≤ |...|` — the LHS `Nat.floor t + 1` is ℕ + ℕ = ℕ, and ≤ |...| : ℝ — mixed ≤ with a cast: `↑(Nat.floor t + 1) ≤ |...|` — so the condition is `(↑(Nat.floor t + 1) : ℝ) ≤ |...|`. And hbig : (Nat.floor t : ℝ) + 1 ≤ |...| — the bridge: ↑(⌊t⌋₊+1) = (⌊t⌋₊:ℝ)+1 (push_cast). For the if_pos ⟨xcond, hγ⟩: xcond : ↑(⌊t⌋₊+1) ≤ |...| := by exact_mod_cast-or-bridge from hbig: have : ((Nat.floor t + 1 : ℕ) : ℝ) ≤ |...| := by rw [show ((Nat.floor t + 1 : ℕ) : ℝ) = (Nat.floor t : ℝ) + 1 from by push_cast; ring]; exact hbig.\n\nOK — this pointwise split is fiddly but mechanical. Also `Nat.floor_add_natCast` — check name: Nat.floor_add_natCast (n : ℕ) : ⌊a + n⌋₊ = ⌊a⌋₊ + n ✓ probably exists (used for the t+14 shift).\n\nNow the main assembly:\n```\ntheorem far_series_le (hZC : ZeroCountProp) (hNLZ : NoLowZeros) {t : ℝ} (ht : 10 ≤ t)\n    {s : ℂ} (hσlo : -5 / 2 ≤ s.re) (hσhi : s.re ≤ 3) (him : s.im = t)\n    (hζ : riemannZeta s ≠ 0) :\n    ‖∑' ρ : {ρ : ℂ // IsNontrivialZero ρ},\n      if ρ ∈ nearZerosSub t then (0 : ℂ) else seriesDiffTerm s (2 + t * Complex.I) ρ‖\n      ≤ 12.1284 * C0hat (t + 1) + 7.2425 := by\n  classical\n  have hnorm := far_series_norm_le_farG_tsum ht hσlo hσhi him hζ\n  have hsplit := tsum_split_finset (farG_summable (by linarith : (0 : ℝ) ≤ t)) (mainFinset t)\n  have htail : (∑' ρ : {ρ : ℂ // IsNontrivialZero ρ},\n      if ρ ∈ mainFinset t then (0 : ℝ) else farG t ρ)\n      = (∑' ρ : {ρ : ℂ // IsNontrivialZero ρ}, farGRight t ρ)\n        + ∑' ρ : {ρ : ℂ // IsNontrivialZero ρ}, farGLeft t ρ := by\n    rw [← Summable.tsum_add (farGRight_summable (by linarith : (0 : ℝ) ≤ t))\n      (farGLeft_summable (by linarith : (0 : ℝ) ≤ t))]\n    apply tsum_congr\n    intro ρ\n    rw [farG_tail_eq hNLZ ht ρ]\n  have hmain := farG_main_le hZC hNLZ ht\n  have hright := farGRight_le hZC ht\n  have hleft := farGLeft_le hZC hNLZ ht\n  calc ‖∑' ρ, if ρ ∈ nearZerosSub t then (0:ℂ) else seriesDiffTerm s (2+tI) ρ‖\n      ≤ ∑' ρ, farG t ρ := hnorm\n    _ = ∑ ρ ∈ mainFinset t, farG t ρ + ∑' ρ, (if ρ ∈ mainFinset t then (0:ℝ) else farG t ρ) := hsplit\n    _ = ∑ ρ ∈ mainFinset t, farG t ρ\n        + ((∑' ρ, farGRight t ρ) + ∑' ρ, farGLeft t ρ) := by rw [htail]\n    _ ≤ (4.5 * (2 * C0hat (t + 1) + 0.55) * 1.3476)\n        + (4.5 * (0.433 * (0.833 * 0.1 + 0.3303) + 0.886 * (0.3475 * 0.1 + 0.1216) + 2.9 * 0.1)\n          + 4.5 * (0.433 * (0.1942 * (1/23) + 0.1687) + 0.886 * (0.0736 * (1/23) + 0.0621) + 2.9 * (1/23))) := by\n          apply add_le_add hmain (add_le_add hright hleft)\n    _ ≤ 12.1284 * C0hat (t + 1) + 7.2425 := by\n      -- the numeric fold\n      have hC0nn : (0 : ℝ) ≤ C0hat (t + 1) := by\n        have h1 := two_le_C0hat (by linarith [ht] : (1 : ℝ) ≤ t + 1)\n        linarith\n      nlinarith [hC0nn]\n```\nThe final fold: 4.5·(2C₀+0.55)·1.3476 = 12.1284·C₀ + 3.33531; Tail_R = 4.5·(0.433·(0.0833+0.3303)+0.886·(0.03475+0.1216)+0.29) = 4.5·(0.433·0.4136+0.886·0.15635+0.29) = 4.5·(0.179088+0.138526+0.29) = 4.5·0.607614 = 2.734263; Tail_L = 4.5·(0.433·(0.0084435+0.1687)+0.886·(0.0032+0.0621)+0.126087) = 4.5·(0.433·0.1771435+0.886·0.0653+0.126087) = 4.5·(0.0767032+0.0578558+0.126087) = 4.5·0.260646 = 1.172907. Sum: 3.33531+2.734263+1.172907 = 7.24248 ≤ 7.2425 ✓ (margin 2e-5 — TIGHT!). Hmm — let me recompute precisely:\n- 4.5·0.55·1.3476 = 2.475·1.3476 = 3.33531.\n- Tail_R: 0.433·0.4136 = 0.1790888; 0.886·0.15635 = 0.13852610; +0.29 = 0.6076149; ×4.5 = 2.7342671.\n- Tail_L: 0.1942/23 = 0.0084434783; +0.1687 = 0.1771434783; ×0.433 = 0.0767031262; 0.0736/23 = 0.0032; +0.0621 = 0.0653; ×0.886 = 0.0578558; 2.9/23 = 0.1260869565; sum = 0.2606458827; ×4.5 = 1.1729064722.\n- Total: 3.33531 + 2.7342671 + 1.1729065 = 7.2424836 ≤ 7.2425 ✓ margin 1.6e-5.\n\nUgh — that's thin (1.6e-5). nlinarith should verify exactly (rational arithmetic ✓ no approximations). OK.\n\nLet me also double-check 4.5·2·1.3476 = 12.1284 ✓ (9·1.3476 = 12.1284 ✓).\n\nNow — the farG_tail_eq formalization is the fiddly part. Let me also confirm `tsum_split_finset` is the LiCircleBound ℝ version (exists ✓ line 249).\n\nFor the mainFinset's mem in the neg-branch: `mem_mainFinset (by linarith : (0:ℝ) ≤ t)` ✓.\n\nAlso `Nat.floor_add_natCast` — verify it exists: grep.","signature":""},{"type":"text","text":"§18 compiles. Committing, then writing the piece-3 assembly (§19)."},{"type":"tool_use","id":"Bash_251","name":"Bash","input":{"command":"cd D:/Python/Quantum && git add research/lean/NguyenRH1/NguyenRH1/LiTightR.lean && git -c user.name=\"NGUYEN-RH-1\" -c user.email=\"research@local\" commit -m \"$(cat <<'EOF'\nSession 036 (5f-ii piece 3e-ii): the right-tail machine\n\nLiTightR.lean section 18 (~430 lines): farGRight (farG restricted to\ntail buckets k >= floor(t)+1 on the right side), farGRight_term_le\n(via farG_le_farW_mul), farGRight_fiber_finite/count (zeroWindow(t+k)\n+ C0hat(t+k+1) <= C0hat(2k+1)), farGRight_fiber_le, loglog_ge_neg_point37\n+ farGRightBridge_nonneg, farGRightMaj(+combo, nonnegs, le_combo, the\nshifted brick summables), farGRightMaj_series_le\n(4.5[0.433(0.833*0.1+0.3303)+0.886(0.3475*0.1+0.1216)+2.9*0.1]), and\nfarGRight_le (the ENNReal fiberwise assembly). Full file elaborates\nclean, zero warnings.\nEOF\n)\"","description":"Commit piece-3e-ii sub-stage"},"extra_content":null}]