← Session 28 — ii′-5e: partial-fraction identity (zero series + F entire)
thinking (125 chars)
fiber_mult_count compiles clean now (no output). Now the main tail lemma. Writing zero_tail_inv_sq_le with the full assembly.
Clean. Now the main tail lemma assembly.
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean",
"old_string": " have hpow : (2 : \u211d) ^ (j + 1) * X = 2 * ((2 : \u211d) ^ j * X) := by rw [pow_succ']; ring\n rw [hpow] at h1' h2' \u22a2\n linarith [h1', h2', hA0, hlogpos]\n\nend NguyenRH1\n\nend",
"new_string": " have hpow : (2 : \u211d) ^ (j + 1) * X = 2 * ((2 : \u211d) ^ j * X) := by rw [pow_succ']; ring\n rw [hpow] at h1' h2' \u22a2\n linarith [h1', h2', hA0, hlogpos]\n\n/-- The per-fiber tail bound: \u03a3 over the dyadic fiber of tailTerm X \u2264\n(1/(4^j\u00b7X\u00b2))\u00b7(fiber count bound). -/\nlemma fiber_tailTerm_sum_le {X : \u211d} (hX : 200 \u2264 X) (j : \u2115) :\n \u2211' b : {\u03c1 : {\u03c1 // IsNontrivialZero \u03c1} // dyadClass X \u03c1 = j},\n ENNReal.ofReal (tailTerm X (b : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}))\n \u2264 ENNReal.ofReal ((1 / ((2 : \u211d) ^ j * X) ^ 2) *\n (A0count * ((2 : \u211d) ^ (j + 1) * X + 3) * Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5))) := by\n classical\n have hX0 : (0 : \u211d) < X := by linarith\n -- the fiber subtype's tsum = the finset sum (finiteness)\n haveI := (dyadClass_fiber_finite hX0 j).fintype\n rw [show ({\u03c1 : {\u03c1 // IsNontrivialZero \u03c1} // dyadClass X \u03c1 = j} : Type)\n =\u21a5({\u03c1 : {\u03c1 // IsNontrivialZero \u03c1} | dyadClass X \u03c1 = j}) from rfl]\n rw [tsum_fintype]\n -- hmm: tsum_fintype needs the Fintype instance on the subtype set\n sorry\n\n/-- **The far-tail bound**: for X \u2265 200, the zeros with |\u03b3| > X\ncontribute \u03a3 m/(1+\u03b3\u00b2) \u2264 16\u00b7A\u2080\u00b7log X / X. -/\ntheorem zero_tail_inv_sq_le {X : \u211d} (hX : 200 \u2264 X) :\n \u2211' \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}, tailTerm X \u03c1\n \u2264 16 * A0count * Real.log X / X := by\n sorry\n\nend NguyenRH1\n\nend"
}
raw content_json (database row)
[{"type":"thinking","thinking":"fiber_mult_count compiles clean now (no output). Now the main tail lemma. Writing zero_tail_inv_sq_le with the full assembly.","signature":""},{"type":"text","text":"Clean. Now the main tail lemma assembly."},{"type":"tool_use","id":"Edit_416","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean","old_string":" have hpow : (2 : ℝ) ^ (j + 1) * X = 2 * ((2 : ℝ) ^ j * X) := by rw [pow_succ']; ring\n rw [hpow] at h1' h2' ⊢\n linarith [h1', h2', hA0, hlogpos]\n\nend NguyenRH1\n\nend","new_string":" have hpow : (2 : ℝ) ^ (j + 1) * X = 2 * ((2 : ℝ) ^ j * X) := by rw [pow_succ']; ring\n rw [hpow] at h1' h2' ⊢\n linarith [h1', h2', hA0, hlogpos]\n\n/-- The per-fiber tail bound: Σ over the dyadic fiber of tailTerm X ≤\n(1/(4^j·X²))·(fiber count bound). -/\nlemma fiber_tailTerm_sum_le {X : ℝ} (hX : 200 ≤ X) (j : ℕ) :\n ∑' b : {ρ : {ρ // IsNontrivialZero ρ} // dyadClass X ρ = j},\n ENNReal.ofReal (tailTerm X (b : {ρ : ℂ // IsNontrivialZero ρ}))\n ≤ ENNReal.ofReal ((1 / ((2 : ℝ) ^ j * X) ^ 2) *\n (A0count * ((2 : ℝ) ^ (j + 1) * X + 3) * Real.log (3 * (2 : ℝ) ^ (j + 1) * X + 5))) := by\n classical\n have hX0 : (0 : ℝ) < X := by linarith\n -- the fiber subtype's tsum = the finset sum (finiteness)\n haveI := (dyadClass_fiber_finite hX0 j).fintype\n rw [show ({ρ : {ρ // IsNontrivialZero ρ} // dyadClass X ρ = j} : Type)\n =↥({ρ : {ρ // IsNontrivialZero ρ} | dyadClass X ρ = j}) from rfl]\n rw [tsum_fintype]\n -- hmm: tsum_fintype needs the Fintype instance on the subtype set\n sorry\n\n/-- **The far-tail bound**: for X ≥ 200, the zeros with |γ| > X\ncontribute Σ m/(1+γ²) ≤ 16·A₀·log X / X. -/\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 sorry\n\nend NguyenRH1\n\nend"},"extra_content":null}]