[{"type":"thinking","thinking":"My earlier \"Edit\" to restructure §6 accidentally inserted my planning text INTO the file (lines 417-430 contain my thinking notes as Lean comments/text — that's why the edit \"succeeded\": it replaced the §6 header with a new §6 that contained my draft code plus stray commentary). I need to rewrite the whole §6 block (lines 199–430, up to the \"Pieces 4/5\" comment) with the final factored version. Let me replace that whole span.\n\nThe cleanest approach: replace from `/-! ## 6. Piece 1: the Dirichlet bound at the reference point -/` through the end of my planning text (line 430) with the final factored code. The old_string is huge; instead, let me do it with a targeted approach: read the exact span boundaries and use Write... no — Write rewrites the whole file. Better: find a unique anchor at the start and end. Start anchor: `/-! ## 6. Piece 1: the Dirichlet bound at the reference point -/` (line 199). End anchor: the line `/-- **Pieces 4/5**` (line 432). I'll use Edit with old_string = the §6 header line + everything up to... — too long. Alternative: use a bash sed to delete lines 199–431 then Edit to insert. Hmm — instructions say prefer Edit over sed. But Edit needs the exact old_string. The block is ~230 lines. Let me instead reconstruct: I'll read the full file and rewrite via Write (the file is ~508 lines; I have its full content in context from my writes + reads). Actually simplest: Edit with old_string being from the §6 header comment start through the planning-text end — I can reconstruct it exactly from what I wrote... The risk is mismatch. Let me instead do two edits:\n1. Delete the stray planning text (lines 417–431): the block starting with \"Hmm — the integrability should just be reused...\" — wait, that's part of my inserted comment `/--` ... the draft had `exact IntegrableOn.of_finite_on... ` at line 416 then ``` then my notes as PLAIN TEXT (lines 418-430 are not Lean comments — they'd be parse errors!). Lines 418-430: \"Hmm — the integrability...\" — these are NOT in a comment! They follow the ``` that closed... wait no — looking at the text: line 416 `exact IntegrableOn.of_finite_on... ` line 417 ``` — hmm, my Edit's new_string contained the draft Lean code fenced in ``` — NO. Let me look: my Edit new_string was the entire §6 draft INCLUDING the markdown-fenced code and my commentary as raw text. Since the Edit's new_string is inserted VERBATIM into the .lean file, the file now contains raw English text and markdown fences — completely broken Lean.\n\nRight — my \"Edit\" was actually a mistake: I pasted my planning notes (including ``` fences and commentary) into the file. The file is currently broken from line 199 to ~430.\n\nRecovery: rewrite the entire §6 section properly. Let me find the exact span: from line 199 (`/-! ## 6. Piece 1: ...`) to line 430 (the line before `/-- **Pieces 4/5**`). I'll use bash sed to delete those lines, then Edit to insert the final code before the Pieces 4/5 comment. Actually — to be safe and precise, let me use Python or sed to extract lines 1-198 and 432-508, and reassemble with the correct §6. Cleanest: use bash to truncate the file at line 198, then append the new §6 + then the pieces 4/5 section (lines 432+). \n\nPlan:\n1. `head -198 LiTightR.lean > tmp` — keeps through line 198 (the end of §5, before the §6 header).\n\nWait line 199 is `/-! ## 6. Piece 1...` — so keep lines 1–198.\n\n2. `tail -n +432` gives from `/-- **Pieces 4/5**` onward — but I need the `/-! ## 6. Pieces 4/5 -/` header... the original §6 header was `/-! ## 6. Pieces 4/5: the reciprocal differences -/` which my edit REPLACED with the piece-1 stuff. Line 432 starts with the `/-- **Pieces 4/5** ...` docstring. So the section header `/-! ## 6. Pieces 4/5 ... -/` is gone — I'll re-add it in the new text.\n\n3. Assemble: head-198 + new §6 (piece 1) + `/-! ## 7. Pieces 4/5 -/` header + tail-from-432.\n\nLet me do this with bash (file surgery with exact line numbers is the safe route here), writing the new §6 to a separate file first with Write, then concatenating.\n\nActually — cleaner: I'll write the whole §6 block to a temp file via Write, then use bash to splice. Let me first finalize the §6 code. Factored structure:\n\n```lean\n/-! ## 6. Piece 1: the Dirichlet bound at the reference point -/\n\n/-- The Dirichlet term norm at 2. -/\nlemma dirichletTerm_norm_eq (n : ℕ) : ... (KEEP the existing one — it's fine)\n```\nWait — dirichletTerm_norm_eq (lines 201-221) is fine and compiles? It's syntactically fine (untested). The broken part is from line 223 onward. Let me keep 1–221 (through dirichletTerm_norm_eq) and rewrite the rest of §6.\n\nNew §6 (after dirichletTerm_norm_eq):\n\n```lean\n/-- The antiderivative derivative fact: d/dx [−(log x + 1)/x] = log x/x². -/\nlemma hasDerivAt_neg_log_add_one_div {x : ℝ} (hx0 : x ≠ 0) :\n    HasDerivAt (fun y => -(Real.log y + 1) / y) (Real.log x / x ^ 2) x := by\n  have h1 := ((Real.hasDerivAt_log hx0).add_const 1).div (hasDerivAt_id' x) hx0\n  have h2 := h1.neg\n  have hval : -((x⁻¹ * x - (Real.log x + 1) * 1) / x ^ 2) = Real.log x / x ^ 2 := by\n    field_simp\n    ring\n  have hfn : (fun y => -((Real.log y + 1) / y)) = (fun y => -(Real.log y + 1) / y) := by\n    ext y\n    rw [neg_div]\n  rw [← hfn, hval] at *\n  exact h2\n```\nHmm — `rw [← hfn, hval] at *` — the h2's function is (fun y => -((log y + 1)/y)) and value −((x⁻¹·x − (log x+1)·1)/x²). rw [← hfn] at h2: replaces (fun y => -(log y+1)/y) with (fun y => −((log y+1)/y))?? — ← direction rewrites RHS→LHS: occurrences of (fun y => −(log y+1)/y) get replaced by (fun y => −((log y+1)/y)) — but h2's function IS −((log y+1)/y) — no occurrence of the −(log y+1)/y form. Confused. Let me restructure: I want HasDerivAt F (log x/x²) x with F = fun y => −(log y + 1)/y. h2 : HasDerivAt (fun y => −((log y+1)/y)) (−((x⁻¹·x − (log x+1)·1)/x²)) x. The function in h2 vs my F: −((log y+1)/y) vs −(log y+1)/y — defeq by neg_div but not syntactic. And the value. Cleanest: use `convert h2 using 1`:\n```lean\nlemma hasDerivAt_neg_log_add_one_div {x : ℝ} (hx0 : x ≠ 0) :\n    HasDerivAt (fun y => -(Real.log y + 1) / y) (Real.log x / x ^ 2) x := by\n  have h1 := ((Real.hasDerivAt_log hx0).add_const 1).div (hasDerivAt_id' x) hx0\n  have h2 := h1.neg\n  convert h2 using 1\n  · ext y\n    rw [neg_div]\n  · field_simp\n    ring\n```\nHmm — convert ... using 1 on HasDerivAt F f' x = HasDerivAt G g' x gives goals F = G and f' = g' (and x = x — closed). F = G: (fun y => −(log y+1)/y) = (fun y => −((log y+1)/y)): ext y; neg_div ✓ wait — which side? h2's function after .neg: HasDerivAt.neg (h : HasDerivAt f f' x) : HasDerivAt (fun y => −f y) (−f') x — so h2's function is fun y => −((log y+1)/y)·... hold on — h1's function from .div: fun y => c y / d y with c = fun y => log y + 1, d = id — so fun y => (log y + 1) / (id y) — the id y! `(fun y => (Real.log y + 1) / id y)` — id y vs y — defeq but maybe simp only [id_eq] needed. And my F has /y. The convert goal F = G: ext y: −(log y + 1)/y = −((log y + 1)/(id y))·... simp [id_eq, neg_div]? Let me write `· ext y; simp [neg_div]`. And the value goal: log x/x² = −((x⁻¹·x − (log x + 1)·1)/(id x)²) — field_simp; ring — hmm the (id x): simp only [id_eq] first. Let me write:\n```lean\n  convert h2 using 1\n  · ext y\n    simp [neg_div]\n  · simp only [id_eq]\n    field_simp\n    ring\n```\nThe value: −((1/x·x − (log x+1)·1)/x²) vs log x/x²: after simp only [id_eq]: goal (from convert) Real.log x / x^2 = -((x⁻¹ * x - (Real.log x + 1) * 1) / x ^ 2)? or flipped. field_simp clears x⁻¹, x²: should reduce to a ring identity ✓. Might need `Real.log`... fine, iterate at build.\n\nThen:\n```lean\n/-- Tendsto (fun y => −(log y + 1)/y) atTop (𝓝 0). -/\nlemma tendsto_neg_log_add_one_div :\n    Tendsto (fun y => -(Real.log y + 1) / y) atTop (𝓝 0) := by\n  have hA : Tendsto (fun x => Real.log x / x) atTop (𝓝 0) := by\n    have h := tendsto_pow_log_div_mul_add_atTop 1 0 1 one_ne_zero\n    simpa using h\n  have hB : Tendsto (fun x => (1 : ℝ) / x) atTop (𝓝 0) :=\n    tendsto_one_div_atTop_nhds_zero\n  have hC : Tendsto (fun x => Real.log x / x + 1 / x) atTop (𝓝 (0 + 0)) :=\n    hA.add hB\n  have heq : (fun y => -(Real.log y + 1) / y) = (fun x => Real.log x / x + 1 / x) := by\n    ext x\n    rw [neg_div, add_div]\n  rw [heq]\n  simpa using hC\n```\nWait — heq: −(log y + 1)/y = log y/y + 1/y: rw [neg_div]: −((log y+1)/y); then add_div: (log y + 1)/y = log y/y + 1/y — so −(log y+1)/y = −(log y/y + 1/y) — that's NOT log y/y + 1/y (sign flip!). OOPS: −(log y + 1)/y = −(log y/y + 1/y) ≠ log y/y + 1/y. My target function is −(log y+1)/y which tends to 0 since (log y+1)/y → 0. So heq should be: (fun y => −(log y + 1)/y) = (fun x => −(log x/x + 1/x)):\n```lean\n  have heq : (fun y => -(Real.log y + 1) / y) = (fun x => -(Real.log x / x + 1 / x)) := by\n    ext x\n    rw [neg_div, add_div]\n  rw [heq]\n  exact (by simpa using hC).neg\n```\n(log y+1)/y = log y/y + 1/y (add_div ✓); −((log y+1)/y) = −(log y/y + 1/y) ✓ via neg_div then add_div ✓. Then Tendsto (fun x => −(...)) via hC.neg : Tendsto (fun x => −(log x/x + 1/x)) (𝓝 (−(0+0))) = 𝓝 0 — simpa using hC.neg? hC : Tendsto ... (𝓝 (0+0)); hC.neg : Tendsto (fun x => −(log x/x+1/x)) atTop (𝓝 (−(0+0))) — −(0+0) = 0: `by simpa using hC.neg` ✓.\n\nThen:\n```lean\n/-- IntegrableOn (log x/x²) (Ioi 3). -/\nlemma integrableOn_log_div_sq :\n    IntegrableOn (fun x => Real.log x / x ^ 2) (Set.Ioi (3 : ℝ)) volume := by\n  apply integrableOn_Ioi_deriv_of_nonneg' (fun x hx => hasDerivAt_neg_log_add_one_div (by\n    have := (Set.mem_Ici.mp hx); linarith))\n  · intro x hx\n    have hx3 : (3 : ℝ) < x := hx\n    have hlog : (0 : ℝ) ≤ Real.log x := Real.log_nonneg (by linarith)\n    positivity\n  · exact tendsto_neg_log_add_one_div\n```\nWait — the signature: integrableOn_Ioi_deriv_of_nonneg' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x) (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a). The g is implicit — unified from hderiv: g := fun y => −(log y+1)/y, g' := fun x => log x/x² ✓ (a := 3 from hx : Set.Ici 3). The l := 0 ✓. The Ioi membership hx : x ∈ Set.Ioi 3 — `have hx3 : (3:ℝ) < x := hx` ✓ (Ioi membership is defeq to <). ✓.\n\n```lean\n/-- The integral: ∫_{Ioi 3} log x/x² = (log 3 + 1)/3. -/\nlemma integral_log_div_sq_Ioi_three :\n    ∫ x in Set.Ioi (3 : ℝ), Real.log x / x ^ 2 = (Real.log 3 + 1) / 3 := by\n  have h := integral_Ioi_of_hasDerivAt_of_tendsto'\n    (fun x hx => hasDerivAt_neg_log_add_one_div (by\n      have := (Set.mem_Ici.mp hx); linarith))\n    integrableOn_log_div_sq tendsto_neg_log_add_one_div\n  rw [h]\n  field_simp\n  ring\n```\nh : ∫ x in Ioi 3, log x/x² = 0 − (−(log 3 + 1)/3) — the m − f a form: m = 0, f a = −(log 3 + 1)/3: 0 − (−(log 3+1)/3) = (log 3+1)/3 ✓. After rw [h], goal: 0 − (−(log 3 + 1)/3) = (log 3 + 1)/3: `ring` alone? 0 − (−X) = X ✓ ring closes (division by 3 as atoms — (log 3+1)/3 is an atom: 0 − (−X) = X ✓ ring). Just `ring`. Hmm — the exact form of f 3: −(log 3 + 1)/3 vs −((log 3+1)/3) — ring normalizes ✓. So:\n```lean\n  rw [h]\n  ring\n```\nHmm — but rw [h] rewrites the integral to `0 − (fun y => -(log y+1)/y) 3`? — the f a application is a beta-redex: (fun y => −(log y+1)/y) 3 — ring might not reduce the redex. Use `simp only []` to beta-reduce, or `show`. Safer:\n```lean\n  rw [h]\n  show (0 : ℝ) - (-(Real.log 3 + 1) / 3) = (Real.log 3 + 1) / 3\n  ring\n```\nHmm — the `show` must be defeq to the post-rw goal `0 − (fun y => −(log y+1)/y) 3 = ...` — beta-redex defeq ✓.\n\nThen the antitone:\n```lean\n/-- AntitoneOn (log x/x²) on Ici 3 (derivative (1 − 2 log x)/x³ ≤ 0). -/\nlemma antitoneOn_log_div_sq :\n    AntitoneOn (fun x => Real.log x / x ^ 2) (Set.Ici (3 : ℝ)) := by\n  apply antitoneOn_of_deriv_nonpos (convex_Ici 3)\n  · intro x hx\n    have hx0 : x ≠ 0 := by have := (Set.mem_Ici.mp hx); linarith\n    exact (Real.continuousAt_log hx0).continuousWithinAt.div\n      (continuousWithinAt_pow _ _ _) (pow_ne_zero 2 hx0)\n  · intro x hx\n    have hx0 : x ≠ 0 := by\n      have : (3 : ℝ) < x := Set.mem_Ioi.mp (by rwa [Set.interior_Ici] at hx)\n      linarith\n    exact ((Real.differentiableAt_log hx0).div (differentiableAt_pow _ _)\n      (pow_ne_zero 2 hx0)).differentiableWithinAt\n  · intro x hx\n    have hx3 : (3 : ℝ) < x := Set.mem_Ioi.mp (by rwa [Set.interior_Ici] at hx)\n    have hx0 : x ≠ 0 := by linarith\n    have hlog3 : (1 : ℝ) ≤ Real.log 3 := by\n      have h := Real.log_le_log (Real.exp_pos 1) (by linarith [Real.exp_one_lt_d9.le] :\n        Real.exp 1 ≤ 3)\n      rwa [Real.log_exp] at h\n    have hlogx : (1 : ℝ) ≤ Real.log x :=\n      hlog3.trans (Real.log_le_log (by positivity : (0 : ℝ) < 3) hx3.le)\n    have hderiv : deriv (fun y => Real.log y / y ^ 2) x\n        = (1 - 2 * Real.log x) / x ^ 3 := by\n      have h := (Real.hasDerivAt_log hx0).div (hasDerivAt_pow 2 x) (pow_ne_zero 2 hx0)\n      rw [h.deriv]\n      congr 1\n      field_simp\n      ring\n    rw [hderiv]\n    exact div_nonpos_of_nonpos_of_nonneg (by linarith [hlogx]) (by positivity)\n```\nHmm — antitoneOn_of_deriv_nonpos's argument structure: (hD : Convex ℝ D) (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D)) (h : ∀ x ∈ interior D, deriv f x ≤ 0) : AntitoneOn f D. My bullets: the `apply` leaves three goals (hf, hf', h) — each `intro x hx` then prove ✓. Note `Set.interior_Ici : interior (Ici a) = Ioi a` ✓. The `by rwa [Set.interior_Ici] at hx` — hx : x ∈ interior (Ici 3) → after rw: x ∈ Ioi 3 → Set.mem_Ioi.mp hx : 3 < x ✓.\n\nThe deriv computation: `(Real.hasDerivAt_log hx0).div (hasDerivAt_pow 2 x) (pow_ne_zero 2 hx0)`: HasDerivAt log (1/x) x div HasDerivAt (fun y => y^2) (2*x^(2-1)) x — hasDerivAt_pow (n) (x) : HasDerivAt (fun y => y^n) (n*x^(n-1)) x ✓. The div value: ((1/x)·x² − log x·(2x))/(x²)². h.deriv : deriv (fun y => log y/y^2) x = that value. Then congr 1 + field_simp + ring to show ((1/x)·x² − log x·(2x))/(x²)² = (1 − 2 log x)/x³: field_simp [hx0] clears; ring... the identity: ((x²/x) − 2x log x)/x⁴ = (x − 2x log x)/x⁴ = (1 − 2 log x)/x³ ✓ (divide num/den by x). field_simp then ring should manage (x ≠ 0 in context). Potential fragility — iterate at build.\n\nAlso continuity: `Real.continuousAt_log hx0 : ContinuousAt log x` — exists? Real.continuousAt_log (hx : x ≠ 0) : ContinuousAt Real.log x ✓ I believe. continuousWithinAt_pow: `ContinuousWithinAt.pow`? The form `(continuousWithinAt_pow _ _ _)` — hmm risky. Alternative: (continuousWithinAt_id.pow 2)? Let me use: `(ContinuousWithinAt.pow (continuousWithinAt_id) 2)`? hmm. Or continuousWithinAt_const... Actually simpler: `ContinuousWithinAt.div (ContinuousWithinAt.log ...)`... Let me use differentiability-based: the function is differentiable at x (x ≠ 0) ⟹ continuousWithinAt: `(Real.differentiableAt_log hx0).div (differentiableAt_pow _ _) (pow_ne_zero 2 hx0)` gives DifferentiableAt ⟹ .continuousAt.continuousWithinAt ✓. So bullet 1: intro x hx; exact ((Real.differentiableAt_log hx0).div (differentiableAt_pow _ _) (pow_ne_zero 2 hx0)).continuousAt.continuousWithinAt — cleaner (no separate continuity lemmas). ✓\n\nReal.differentiableAt_log (hx : x ≠ 0) : DifferentiableAt ℝ Real.log x ✓ exists.\ndifferentiableAt_pow (n : ℕ)? : DifferentiableAt ℝ (fun x => x^n) x? signature: `differentiableAt_pow (n : ℕ) (x : ℝ)`? or just (n)? Check at build — risky name. There's `differentiableAt_pow (n : ℕ) (x : 𝕜) : DifferentiableAt 𝕜 (·^n) x`? hmm the exact form varies. Fallback: `DifferentiableAt.pow differentiableAt_id` hmm... or `differentiableAt_pow : DifferentiableAt 𝕜 (fun x => x ^ n) x`? Let me just try and fix at build.\n\nThen the tail bound + summability:\n```lean\n/-- Summability of the log-tail. -/\nlemma summable_log_div_sq_tail :\n    Summable (fun n : ℕ => Real.log (n + 4) / (n + 4 : ℝ) ^ 2) := by\n  have h := AntitoneOn.summable_of_integrableOn_Ioi antitoneOn_log_div_sq\n    integrableOn_log_div_sq (fun t ht => by\n      have ht3 : (3 : ℝ) < t := ht\n      have hlog : (0 : ℝ) ≤ Real.log t := Real.log_nonneg (by linarith)\n      positivity)\n  -- h : Summable (fun n ↦ (fun x => log x/x²) (↑n + 3 + 1))-ish\n  ...\n```\nHmm — I need the exact signature of AntitoneOn.summable_of_integrableOn_Ioi. From line 215-218 usage: `summable_of_integrableOn_Ioi (N := 0) (mod_cast anti) (mod_cast integrable) (mod_cast nonneg)` — so it takes N as a named/implicit ℕ arg, then anti : AntitoneOn f (Ici (N:ℝ)), integrable : IntegrableOn f (Ioi (N:ℝ)), nonneg : ∀ t ∈ Ioi N, 0 ≤ f t, giving Summable (fun n ↦ f (↑(n + N + 1)))? or f (↑n + ↑N + 1)? The mod_cast suggestions the casts. Let me grep its exact statement.","signature":""},{"type":"text","text":"The file got corrupted by a bad edit (my planning notes were pasted in). Let me check the exact summability lemma signature, then do surgical repair."},{"type":"tool_use","id":"Bash_167","name":"Bash","input":{"command":"M=D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib; grep -n \"theorem AntitoneOn.summable_of_integrableOn_Ioi\\b\" -A 5 $M/Analysis/SumIntegralComparisons.lean | head -8","description":"Get exact AntitoneOn.summable_of_integrableOn_Ioi signature"},"extra_content":null}]