[{"type":"thinking","thinking":"Clean again. Now the final sections: 9 (DinfC iterated values at 1/2 + digamma transfer) and 10 (reindexing + affine comp + τ_j assembly). Let me write Section 9 now — carefully, applying the lessons (beta-redexes, function-level forms, cast ascriptions).\n\nSection 9:\n```lean\n/-! ## 9. The polygamma values of DinfC and Complex.digamma at 1/2 -/\n\n/-- The k-th iterated derivative of DinfC at 1/2 (k ≥ 1). -/\nlemma iteratedDeriv_DinfC_half (k : ℕ) (hk : 1 ≤ k) :\n    iteratedDeriv k DinfC ((1 : ℂ) / 2) =\n      -((-1 : ℂ)) ^ k * (k ! : ℂ) * ∑' m : ℕ, ((1 / 2 : ℂ) + ↑m) ^ (-(k : ℤ) - 1) := by\n  -- DinfC = A + B as functions\n  have hsplit : DinfC = (fun s : ℂ => -(Real.eulerMascheroniConstant : ℂ) - 1 / s) +\n      (fun s => ∑' m : ℕ, dpsiTermC s m) := by\n    funext s\n    rw [DinfC]\n    simp [Pi.add_apply]\n  rw [hsplit]\n  have hcfA : ContDiffAt ℂ k (fun s : ℂ => -(Real.eulerMascheroniConstant : ℂ) - 1 / s)\n      ((1 : ℂ) / 2) := by\n    have h1 : AnalyticAt ℂ (fun s : ℂ => 1 / s) ((1 : ℂ) / 2) :=\n      analyticAt_const.div analyticAt_id (by norm_num)\n    exact (analyticAt_const.sub h1).contDiffAt\n  have hcfB : ContDiffAt ℂ k (fun s => ∑' m : ℕ, dpsiTermC s m) ((1 : ℂ) / 2) := by\n    have h := dpsiTermC_tsum_differentiableOn\n    -- DifferentiableOn on the open ball ⟹ AnalyticAt at 1/2 ⟹ ContDiffAt\n    have h2 : AnalyticOnNhd ℂ (fun s => ∑' m : ℕ, dpsiTermC s m) dpsiDomain :=\n      h.analyticOnNhd dpsiDomain_isOpen\n    exact (h2 _ dpsiDomain_half).contDiffAt\n  rw [iteratedDeriv_add hcfA hcfB]\n  -- the A part\n  have hA : iteratedDeriv k (fun s : ℂ => -(Real.eulerMascheroniConstant : ℂ) - 1 / s)\n      ((1 : ℂ) / 2) = -((-1 : ℂ)) ^ k * (k ! : ℂ) * (((1 : ℂ) / 2)) ^ (-(k : ℤ) - 1) := by\n    have heq : (fun s : ℂ => -(Real.eulerMascheroniConstant : ℂ) - 1 / s) =\n        (fun _ : ℂ => -(Real.eulerMascheroniConstant : ℂ)) + fun s => -(1 / (s + 0)) := by\n      funext s\n      simp [Pi.add_apply]\n      ring\n    rw [heq]\n    have hcf1 : ContDiffAt ℂ k (fun _ : ℂ => -(Real.eulerMascheroniConstant : ℂ)) _ :=\n      contDiff_const.contDiffAt\n    have hcf2 : ContDiffAt ℂ k (fun s : ℂ => -(1 / (s + 0))) ((1 : ℂ) / 2) := by\n      have h2 : AnalyticAt ℂ (fun s : ℂ => s + 0) ((1 : ℂ) / 2) :=\n        analyticAt_id.add analyticAt_const\n      have h3 : AnalyticAt ℂ (fun s : ℂ => 1 / (s + 0)) ((1 : ℂ) / 2) :=\n        analyticAt_const.div h2 (by simpa using (by norm_num : ((1 : ℂ) / 2) ≠ 0))\n      exact h3.neg.contDiffAt\n    rw [iteratedDeriv_add hcf1 hcf2]\n    rw [iteratedDeriv_const, if_neg (by omega : k ≠ 0)]\n    rw [iteratedDeriv_neg_one_div_add_const k 0 (by simp; norm_num : ((1:ℂ)/2) + 0 ≠ 0)]\n    rw [add_zero, zero_add]\n  -- the B part: from iteratedDerivWithin → iteratedDeriv (open domain)\n  have hB : iteratedDeriv k (fun s => ∑' m : ℕ, dpsiTermC s m) ((1 : ℂ) / 2) =\n      -((-1 : ℂ)) ^ k * (k ! : ℂ) * ∑' m : ℕ, ((1 / 2 : ℂ) + ↑m + 1) ^ (-(k : ℤ) - 1) := by\n    rw [← iteratedDerivWithin_of_isOpen dpsiDomain_isOpen dpsiDomain_half]\n    rw [iteratedDerivWithin_tsum_dpsiTermC k hk]\n    rw [tsum_mul_left]  -- pull −(−1)^k·k! out of the tsum — wait the tsum's terms have the factor — tsum_mul_left : ∑' b, a * f b = a * ∑' b, f b — the per-term: −(−1)^k·↑k!·X_m — as a * f m with a := −(−1)^k·↑k!... — the term is written `-(−1)^k * ↑k! * X` — that's (−(−1)^k * ↑k!) * X ✓ so a := −(−1)^k * ↑k! — hmm — wait the iteratedDerivWithin_tsum_dpsiTermC's conclusion: ∑' m, −(−1)^k * ↑k! * ((1/2:ℂ)+↑m+1)^(−↑k−1) — so the tsum's function is fun m => (−(−1)^k * ↑k!) * X_m — tsum_mul_left gives (−(−1)^k*↑k!) * ∑' X_m ✓\n  rw [hA, hB]\n  -- combine the two series: (1/2)^{−(k+1)} + Σ_{m≥0}(3/2+m)^{−(k+1)} = Σ_{m≥0}(1/2+m)^{−(k+1)}\n  rw [← tsum_mul_left]  -- fold the factor back? hmm — I have C·X₀-form + C·ΣY — combine: C·(X₀ + ΣY) via ← mul_add\n  rw [← mul_add]\n  congr 2  -- the −(−1)^k·k! factor\n  -- (1/2)^{−(k+1)} + Σ_{m≥0}(1/2+↑m+1)^{−(k+1)} = Σ_{m≥0}(1/2+↑m)^{−(k+1)}\n  ...\n```\nThe combine step: after rw [hA, hB], the goal:\nC·(1/2)^{−(k+1)} + C·Σ_{m≥0}(1/2+↑m+1)^{−(k+1)} = C·Σ_{m≥0}(1/2+↑m)^{−(k+1)} where C := −(−1)^k·↑k!.\nrw [← mul_add]: LHS → C·((1/2)^{−(k+1)} + Σ (1/2+↑m+1)^{...}) ✓ then congr 1: C·X = C·Y → X = Y:\n(1/2)^{−(k+1)} + Σ_{m≥0}(1/2+↑m+1)^{−(k+1)} = Σ_{m≥0}(1/2+↑m)^{−(k+1)}\nRHS via tsum_eq_zero_add: Σ_{m≥0} f m = f 0 + Σ_{m≥0} f (m+1): `tsum_eq_zero_add (hf : Summable f) : ∑' n, f n = f 0 + ∑' n, f (n+1)` ✓ — with f m := (1/2+↑m)^{−(k+1)}: f 0 = (1/2)^{−(k+1)} ✓ and f (m+1) = (1/2 + ↑(m+1))^{−(k+1)} = (1/2 + ↑m + 1)^{−(k+1)} ✓ (↑(m+1) = ↑m + 1 push_cast ✓). The Summable f: the (1/2+↑m)^{−(k+1)} series — summable (norm ~ 1/m^{k+1} ✓ via the M-test dominator — need a small summability lemma: Summable (fun m => ((1/2:ℂ)+↑m)^{−(k+1):ℤ}) — via of_norm_bounded with 1/(m+1)²-family... — hmm — need it. |（1/2+↑m)^{−(k+1)}| = (1/2+m)^{−(k+1)} ≤ 1/(m+1/2)^{k+1} ≤ 2^{k+1}/(2m+1)^{k+1}... — let me bound: (1/2 + m) ≥ (m+1)/2 ⟹ (1/2+m)^{k+1} ≥ (m+1)^{k+1}/2^{k+1} ⟹ the inverse ≤ 2^{k+1}/(m+1)^{k+1} ✓ summable ✓. So:\n```lean\n  have hsum0 : Summable (fun m : ℕ => ((1 / 2 : ℂ) + ↑m) ^ (-(k : ℤ) - 1)) := by\n    apply Summable.of_norm_bounded (summable_div_add_one_pow (2^(k+1)) (k+1) (by omega))\n    intro m\n    ... ‖(1/2+↑m)^{−(k+1)}‖ ≤ 2^{k+1}/(m+1)^{k+1}\n```\nThis needs the norm chains again — ~25 lines. OK.\n\nThe final algebra: −(−1)^k = (−1)^{k+1}: pow_succ: (−1)^{k+1} = (−1)^k·(−1) = −(−1)^k ✓ — so the C factor −(−1)^k·↑k! = (−1)^{k+1}·↑k! via pow_succ + ring.\n\nHmm — but wait — my target statement in iteratedDeriv_DinfC_half: I wrote the RHS as −(−1)^k·k!·Σ — let me keep that form (avoid the sign flip here; do it at the assembly). Actually for the ASSEMBLY I want (−1)^{j+1}·j!·lagariasA(j+1) — so the −(−1)^j = (−1)^{j+1} flip happens there. Let me state iteratedDeriv_DinfC_half with the form `((-1:ℂ))^(k+1) * (k!:ℂ) * ∑' m, ((1/2:ℂ)+↑m)^(−(k:ℤ)−1)` — absorbing the sign. So hA/hB with the (−1)^{k+1} forms: −(−1)^k = (−1)^{k+1} via:\n```lean\n  have hsign : -((-1 : ℂ)) ^ k = (-1 : ℂ) ^ (k + 1) := by rw [pow_succ]; ring\n```\n✓.\n\nOK this section is getting really long and it's past midnight. Let me write Sections 9+10 carefully in one append and grind through the compile errors. Given the number of fiddly steps, let me be extra careful with forms. Actually — let me simplify the structure of the final assembly: skip the Within-machinery form and directly state the DinfC value lemma with the (1/2+↑m) series, then the digamma transfer, then the reindexing as a separate lemma, then the final τ-assembly as the capstone in LiArchimedean.lean.","signature":""},{"type":"text","text":"Clean. Now the final push — Sections 9–10: the values at 1/2, the digamma transfer, and the reindexing to lagariasA."},{"type":"tool_use","id":"Bash_469","name":"Bash","input":{"command":"cd \"D:/Python/Quantum\" && cat >> research/lean/NguyenRH1/NguyenRH1/LiDigamma.lean << 'EOF'\n\n/-! ## 9. The values at 1/2 and the transfer to digamma -/\n\n/-- The (1/2 + m)-power series is summable (norm ≤ 2^{k+1}/(m+1)^{k+1}). -/\nlemma summable_zpow_half_add_nat (k : ℕ) (hk : 1 ≤ k) :\n    Summable (fun m : ℕ => ((1 / 2 : ℂ) + ↑m) ^ (-(k : ℤ) - 1)) := by\n  apply Summable.of_norm_bounded (summable_div_add_one_pow (2 ^ (k + 1)) (k + 1) (by omega))\n  intro m\n  have hpos : (0 : ℝ) < (m : ℝ) + 1 / 2 := by positivity\n  have h1 : ‖((1 / 2 : ℂ) + ↑m)‖ = (m : ℝ) + 1 / 2 := by\n    have e : ((1 / 2 : ℂ) + ↑m) = ↑((m : ℝ) + 1 / 2) := by push_cast; ring\n    rw [e, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg hpos.le]\n  rw [norm_zpow, h1]\n  rw [show (-(k : ℤ) - 1 : ℤ) = -(((k + 1 : ℕ) : ℤ)) by push_cast; ring, zpow_neg, zpow_natCast]\n  rw [div_eq_mul_inv, ← one_div, ← one_div]\n  apply one_div_le_one_div_of_le (by positivity)\n  have h2 : ((m : ℝ) + 1) ^ (k + 1) ≤ 2 ^ (k + 1) * ((m + 1 / 2 : ℝ)) ^ (k + 1) := by\n    rw [show (2 : ℝ) ^ (k + 1) * ((m + 1 / 2 : ℝ)) ^ (k + 1) =\n      (2 * (m + 1 / 2 : ℝ)) ^ (k + 1) by rw [mul_pow]]\n    apply pow_le_pow_left₀ (by positivity : (0 : ℝ) ≤ (m : ℝ) + 1)\n    · ring_nf\n      linarith [Nat.cast_nonneg (α := ℝ) m]\n  exact h2\n\n/-- The polygamma values of DinfC at 1/2 (j ≥ 1). -/\nlemma iteratedDeriv_DinfC_half (k : ℕ) (hk : 1 ≤ k) :\n    iteratedDeriv k DinfC ((1 : ℂ) / 2) =\n      ((-1 : ℂ)) ^ (k + 1) * (k ! : ℂ) * ∑' m : ℕ, ((1 / 2 : ℂ) + ↑m) ^ (-(k : ℤ) - 1) := by\n  have hsign : ∀ k : ℕ, -((-1 : ℂ)) ^ k = (-1 : ℂ) ^ (k + 1) := fun k => by\n    rw [pow_succ]; ring\n  have hsplit : DinfC = (fun s : ℂ => -(Real.eulerMascheroniConstant : ℂ) - 1 / s) +\n      (fun s => ∑' m : ℕ, dpsiTermC s m) := by\n    funext s\n    rw [DinfC]\n    simp [Pi.add_apply]\n  rw [hsplit]\n  have hcfA : ContDiffAt ℂ k (fun s : ℂ => -(Real.eulerMascheroniConstant : ℂ) - 1 / s)\n      ((1 : ℂ) / 2) := by\n    have h1 : AnalyticAt ℂ (fun s : ℂ => 1 / s) ((1 : ℂ) / 2) :=\n      analyticAt_const.div analyticAt_id (by norm_num)\n    exact (analyticAt_const.sub h1).contDiffAt\n  have hcfB : ContDiffAt ℂ k (fun s => ∑' m : ℕ, dpsiTermC s m) ((1 : ℂ) / 2) := by\n    have h2 : AnalyticOnNhd ℂ (fun s => ∑' m : ℕ, dpsiTermC s m) dpsiDomain :=\n      dpsiTermC_tsum_differentiableOn.analyticOnNhd dpsiDomain_isOpen\n    exact (h2 _ dpsiDomain_half).contDiffAt\n  rw [iteratedDeriv_add hcfA hcfB]\n  have hA : iteratedDeriv k (fun s : ℂ => -(Real.eulerMascheroniConstant : ℂ) - 1 / s)\n      ((1 : ℂ) / 2) = ((-1 : ℂ)) ^ (k + 1) * (k ! : ℂ) * ((1 : ℂ) / 2) ^ (-(k : ℤ) - 1) := by\n    have heq : (fun s : ℂ => -(Real.eulerMascheroniConstant : ℂ) - 1 / s) =\n        (fun _ : ℂ => -(Real.eulerMascheroniConstant : ℂ)) + fun s => -(1 / (s + 0)) := by\n      funext s\n      simp [Pi.add_apply]\n      ring\n    rw [heq]\n    have hcf1 : ContDiffAt ℂ k (fun _ : ℂ => -(Real.eulerMascheroniConstant : ℂ)) ((1 : ℂ) / 2) :=\n      contDiff_const.contDiffAt\n    have hcf2 : ContDiffAt ℂ k (fun s : ℂ => -(1 / (s + 0))) ((1 : ℂ) / 2) := by\n      have h2 : AnalyticAt ℂ (fun s : ℂ => s + 0) ((1 : ℂ) / 2) :=\n        analyticAt_id.add analyticAt_const\n      have h3 : AnalyticAt ℂ (fun s : ℂ => 1 / (s + 0)) ((1 : ℂ) / 2) :=\n        analyticAt_const.div h2 (by simpa using (by norm_num : ((1 : ℂ) / 2) ≠ 0))\n      exact h3.neg.contDiffAt\n    rw [iteratedDeriv_add hcf1 hcf2]\n    rw [iteratedDeriv_const, if_neg (by omega : k ≠ 0)]\n    rw [iteratedDeriv_neg_one_div_add_const k 0 (by simp)]\n    rw [add_zero, zero_add, hsign]\n  have hB : iteratedDeriv k (fun s => ∑' m : ℕ, dpsiTermC s m) ((1 : ℂ) / 2) =\n      ((-1 : ℂ)) ^ (k + 1) * (k ! : ℂ) * ∑' m : ℕ, ((1 / 2 : ℂ) + ↑m + 1) ^ (-(k : ℤ) - 1) := by\n    rw [← iteratedDerivWithin_of_isOpen dpsiDomain_isOpen dpsiDomain_half]\n    rw [iteratedDerivWithin_tsum_dpsiTermC k hk]\n    rw [tsum_mul_left]\n    rw [hsign]\n  rw [hA, hB]\n  rw [← mul_add]\n  congr 2\n  have hsum := summable_zpow_half_add_nat k hk\n  rw [tsum_eq_zero_add hsum]\n  apply Finset.sum_congr rfl |>.trans' |>.symm  -- placeholder, will fix\n  sorry\nEOF\ncd research/lean/NguyenRH1 && export PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiDigamma.lean 2>&1 | grep -v \"^$\" | head -20","description":"Append section 9 skeleton and compile","timeout":600000},"extra_content":null}]