[{"type":"thinking","thinking":"Progress — only one error left visible at 3738: the calc step `_ ≤ 4.5 * ((0.433 * 0.833 + 0.886 * 0.3475 + 2.9) * 0.1 + 0.433 * 0.3303 + 0.886 * 0.1216) := by linarith [hle, heval]` — hle : tsum(farGRightMaj(i+11)) ≤ tsum(farGRightMajCombo(i+11)) and heval : tsum(combo-form) ≤ 4.5·(...). But hle's RHS is tsum(farGRightMajCombo (i+11)) — and heval's LHS is tsum(4.5·(...)) — DIFFERENT functions: farGRightMajCombo (i+11) vs 4.5·((c0)·(1/((i:ℝ)+11)²) + 0.433·(log/²) + 0.886·(loglog/²)). The bridge between them is the Summable.congr in farGRightMajCombo_shift_summable — but heval's LHS is the combo-form directly. So I need: tsum(farGRightMajCombo (i+11)) = tsum(4.5·(combo-form i)) — the congr from farGRightMajCombo_shift_summable's proof showed pointwise farGRightMajCombo (k+11) = 4.5·(combo-form k) — as a TSUM equality: `tsum_congr`:\n```\n  have hcombo_eq : (∑' k : ℕ, farGRightMajCombo (k + 11))\n      = ∑' k : ℕ, 4.5 * ((0.433 * 0.833 + 0.886 * 0.3475 + 2.9) * (1 / ((k : ℝ) + 11) ^ 2)\n          + 0.433 * (Real.log ((k : ℝ) + 11) / ((k : ℝ) + 11) ^ 2)\n          + 0.886 * (Real.log (Real.log ((k : ℝ) + 11)) / ((k : ℝ) + 11) ^ 2)) := by\n    apply tsum_congr\n    intro k\n    rw [farGRightMajCombo]\n    have hcast : ((k + 11 : ℕ) : ℝ) = (k : ℝ) + 11 := by push_cast; ring\n    rw [hcast]\n    ring\n```\nThen hle.trans: tsum(farGRightMaj(i+11)) ≤ tsum(farGRightMajCombo(i+11)) = tsum(combo-form) ≤ 4.5·(...) (heval). Assemble in the calc:\n```\n  have hle2 : (∑' k : ℕ, farGRightMaj t (k + 11))\n      ≤ 4.5 * ((0.433 * 0.833 + 0.886 * 0.3475 + 2.9) * 0.1\n          + 0.433 * 0.3303 + 0.886 * 0.1216) := by\n    rw [hcombo_eq] at hle\n    exact hle.trans heval\n```\nthen the calc step uses hle2.","signature":""},{"type":"text","text":"One missing link: the tsum-level combo equality. Adding it."},{"type":"tool_use","id":"Bash_249","name":"Bash","input":{"command":"cd D:/Python/Quantum/research/lean/NguyenRH1 && python - <<'EOF'\np = 'NguyenRH1/LiTightR.lean'\nsrc = open(p, encoding='utf-8').read()\n\nold = \"\"\"  calc (∑' k : ℕ, farGRightMaj t k)\n      = (∑ k ∈ Finset.range 11, farGRightMaj t k) + ∑' k : ℕ, farGRightMaj t (k + 11) := hsplit\n    _ = 0 + ∑' k : ℕ, farGRightMaj t (k + 11) := by rw [hzero]\n    _ ≤ 4.5 * ((0.433 * 0.833 + 0.886 * 0.3475 + 2.9) * 0.1\n        + 0.433 * 0.3303 + 0.886 * 0.1216) := by linarith [hle, heval]\n    _ = 4.5 * (0.433 * (0.833 * 0.1 + 0.3303)\n        + 0.886 * (0.3475 * 0.1 + 0.1216) + 2.9 * 0.1) := by ring\"\"\"\nnew = \"\"\"  have hcombo_eq : (∑' k : ℕ, farGRightMajCombo (k + 11))\n      = ∑' k : ℕ, 4.5 * ((0.433 * 0.833 + 0.886 * 0.3475 + 2.9) * (1 / ((k : ℝ) + 11) ^ 2)\n          + 0.433 * (Real.log ((k : ℝ) + 11) / ((k : ℝ) + 11) ^ 2)\n          + 0.886 * (Real.log (Real.log ((k : ℝ) + 11)) / ((k : ℝ) + 11) ^ 2)) := by\n    apply tsum_congr\n    intro k\n    rw [farGRightMajCombo]\n    have hcast : ((k + 11 : ℕ) : ℝ) = (k : ℝ) + 11 := by push_cast; ring\n    rw [hcast]\n    ring\n  have hle2 : (∑' k : ℕ, farGRightMaj t (k + 11))\n      ≤ 4.5 * ((0.433 * 0.833 + 0.886 * 0.3475 + 2.9) * 0.1\n          + 0.433 * 0.3303 + 0.886 * 0.1216) := by\n    rw [hcombo_eq] at hle\n    exact hle.trans heval\n  calc (∑' k : ℕ, farGRightMaj t k)\n      = (∑ k ∈ Finset.range 11, farGRightMaj t k) + ∑' k : ℕ, farGRightMaj t (k + 11) := hsplit\n    _ = 0 + ∑' k : ℕ, farGRightMaj t (k + 11) := by rw [hzero]\n    _ ≤ 4.5 * ((0.433 * 0.833 + 0.886 * 0.3475 + 2.9) * 0.1\n        + 0.433 * 0.3303 + 0.886 * 0.1216) := hle2\n    _ = 4.5 * (0.433 * (0.833 * 0.1 + 0.3303)\n        + 0.886 * (0.3475 * 0.1 + 0.1216) + 2.9 * 0.1) := by ring\"\"\"\nassert old in src\nsrc = src.replace(old, new)\nopen(p, 'w', encoding='utf-8').write(src)\nprint(\"done\")\nEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiTightR.lean 2>&1 | head -20","description":"Add combo tsum equality and recompile","timeout":420000},"extra_content":null}]