[{"type":"thinking","thinking":"Wait — before splicing, let me double-check a few risky spots in the mid file:\n\n1. `Real.sum_le_exp_of_nonneg (show (0:ℝ) ≤ 1.1 by norm_num) 6` — signature: Real.sum_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : ∑ k ∈ range n, x^k/k! ≤ exp x? — let me check: from D-062 §5 notes \"exp lower bounds by partial sums (Real.sum_le_exp_of_nonneg)\". Signature guess: `Real.sum_le_exp_of_nonneg (hx : 0 ≤ x) (n : ℕ) : ∑ i ∈ Finset.range n, x ^ i / i ! ≤ Real.exp x`. So h : ∑ k ∈ range 6, 1.1^k/k! ≤ exp 1.1. Then expanding the sum and norm_num: ∑ = 1 + 1.1 + 1.21/2 + 1.331/6 + 1.4641/24 + 1.61051/120 = 1 + 1.1 + 0.605 + 0.221833 + 0.061004 + 0.013421 = 3.001258. norm_num at h should evaluate the LHS (rational!) — 1.1 is a decimal = rational ✓ k! values ✓. h : 3.001258... ≤ exp 1.1 — then linarith [h] : 3 ≤ exp 1.1 — hmm norm_num at h evaluates the sum to a numeral: h : (3.00126... : ℝ) ≤ exp 1.1; linarith needs 3 ≤ that numeral: norm_num should prove 3 ≤ 3.00126 within linarith's reach? linarith [h] with h : C ≤ exp 1.1 and goal 3 ≤ exp 1.1 — needs 3 ≤ C as a fact: linarith can verify 3 ≤ C if C is a numeral? linarith handles numeric comparisons of literals ✓ (3 ≤ 3.001258 is a linear fact about constants ✓).\n\nHmm — but `Real.sum_le_exp_of_nonneg` signature — is the sum over `range n` with x^k/k!? Let me not guess: D-062 §5 says \"exp lower bounds by partial sums (Real.sum_le_exp_of_nonneg)\" — so it exists with some form. The exact form might be `Real.sum_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : (∑ i ∈ Finset.range n, x ^ i / i !) ≤ Real.exp x`. I'll adjust at build if needed.\n\n2. `(summable_nat_add_iff 4).mpr (f := fun n : ℕ => Real.log (n:ℝ)/(n:ℝ)^2) h` — summable_nat_add_iff : Summable (fun n => f (n + k)) ↔ Summable f. .mpr : Summable f → Summable (fun n => f (n + 4)). My h : Summable (fun n : ℕ => (fun x => log x/x²) ↑n) — is that Summable f with f := fun n => log ↑n/↑n²? h from AntitoneOn.summable_of_integrableOn_Ioi gives Summable (fun n : ℕ ↦ f ↑n) where f := (fun x => log x/x²) — so h : Summable (fun n => log ↑n/↑n²) ✓ matches f := fun n : ℕ => Real.log (n:ℝ)/(n:ℝ)^2 ✓. Then .mpr gives Summable (fun n => f (n + 4)) = Summable (fun n => log (↑(n+4))/(↑(n+4))²). Then h2.congr (fun n => ?_) — the congr target: Summable (fun n => log (↑n+4)/(↑n+4)²): pointwise: log (↑(n+4))/(↑(n+4))² = log (↑n+4)/(↑n+4)²: push_cast + ring_nf — hmm: ↑(n+4) = ↑n + ↑4 = ↑n + 4: push_cast rewrites ✓ then the goal becomes log (↑n+4)/(↑n+4)² = log (↑n+4)/(↑n+4)² — rfl — ring_nf closes ✓ hopefully.\n\nWait — actually h2.congr : Summable.congr (hf : Summable f) (h : ∀ i, f i = g i) : Summable g — so `h2.congr (fun n => ?_)` where the pointwise h : f n = g n. My `refine h2.congr fun n => ?_; push_cast; ring_nf` — the goal: (fun n => log ↑(n+4)/↑(n+4)²) n = (fun n => log (↑n+4)/(↑n+4)²) n — beta-redexes both sides — push_cast then ring_nf — hmm ring_nf on log-atoms: the goal after beta is log ↑(n+4)/↑(n+4)² = log (↑n+4)/(↑n+4)²: push_cast turns ↑(n+4) into ↑n + 4 ✓ then both sides identical — ring_nf or rfl. OK.\n\nHmm — but does `(summable_nat_add_iff 4).mpr` accept the (f := ...) named arg? f is implicit in summable_nat_add_iff {f : ℕ → M} (k : ℕ) — pin via (f := ...) ✓.\n\n3. `sum_add_tsum_nat_add` — exact name/direction: `sum_add_tsum_nat_add (k : ℕ) (hf : Summable f) : (∑ i ∈ range k, f i) + ∑' i, f (i + k) = ∑' i, f i`? or the reverse. My hsplit : ∑' n, h n = (∑ range 4) + ∑' (h (n+4)): if the lemma is (∑ range k) + tsum-shifted = tsum, then hsplit = (sum_add_tsum_nat_add 4 hsumh).symm ✓ as written. If it's the reverse direction, drop .symm. Check at build. The name might also be `Summable.sum_add_tsum_nat_add`. Risk — adjust at build.\n\n4. The hhead computation: `Finset.sum_range_succ` chain from range 4 down: ∑ range 4 = ∑ range 3 + f 3 — applying sum_range_succ four times then sum_range_zero gives f 0 + f 1 + f 2 + f 3 + 0... The sum_range_succ : ∑ i ∈ range (n+1), f i = ∑ i ∈ range n, f i + f n. So ∑ range 4 h = ((∑ range 3 h) + h 3) = (((∑ range 2 h) + h 2) + h 3) = ((((∑ range 1 h) + h 1) + h 2) + h 3) = (((((∑ range 0 h) + h 0) + h 1) + h 2) + h 3) — after 4× sum_range_succ + sum_range_zero: ((((0 + h 0) + h 1) + h 2) + h 3). Then my rw chain: `simp only [...]` then the `show h 1 = ...` rws. Hmm — the `simp only [Nat.zero_eq, if_pos, ...]` — for h 0: h 0 = if 0 = 0 then 0 else ... = 0 via if_pos rfl — but h 0 is `h 0` — with hh : h = fun n => if n = 0 then... — after rw [hh], h 0 becomes (fun n => if n = 0 then (0:ℝ) else ...) 0 — a beta-redex — simp only [if_pos] with the beta... The `rw [hh]` inside the show-haves: `show h 1 = (0:ℝ) from by rw [hh]; simp` — after rw [hh]: (fun n => ...) 1 = 0 — simp handles the beta + if_neg (1 ≠ 0) + log 1 = 0 + 0/1 = 0 ✓ probably. This head computation is fiddly — iterate at build.\n\nActually — a cleaner head computation: use `Finset.sum_range_four`? There's `Finset.sum_range_one/two/three/four`? Hmm — `Finset.sum_range_succ` chain is the standard. Or convert to explicit: `decide`? no (real values). Let me keep the chain and fix at build.\n\n5. hle's form in tsum_log_div_sq_tail_le: antitoneOn_log_div_sq.tsum_comp_add_le_integral 3 integrableOn_log_div_sq nonneg — tsum_comp_add_le_integral (N : ℕ) (anti : AntitoneOn f (Ici (N:ℝ))) (integrable ...) (nonneg ...) : ∑' (n : ℕ), f (↑(n + N + 1)) ≤ ∫ x in Ioi N, f x. With N := 3: ∑' n, f (↑(n+3+1)) — note: ↑(n + 3 + 1) with the ℕ add then cast. My target: ∑' n, log (↑n+4)/(↑n+4)². The congr: f (↑(n+3+1)) = log (↑(n+3+1))/(↑(n+3+1))² vs log (↑n+4)/(↑n+4)²: ↑(n+3+1) = ↑n + 4 via push_cast ✓. My code:\n```lean\n  rw [integral_log_div_sq_Ioi_three] at hle\n  refine hle.trans_eq' ?_\n  refine tsum_congr fun n => ?_\n  push_cast\n  ring_nf\n```\nhle : ∑' n, (fun x => log x/x²) ↑(n+3+1) ≤ (log 3+1)/3 (after the rw — wait, rw [integral_log_div_sq_Ioi_three] at hle rewrites the integral to the value ✓). Then hle.trans_eq' (h : b = c)? : a ≤ b → b = c → a ≤ c — trans_eq' : a ≤ b → b = c → a ≤ c ✓. The tsum_congr gives equality of the tsums... my goal after refine hle.trans_eq' ?_ : ∑' n, (fun x => log x/x²) ↑(n+3+1) = ∑' n, log (↑n+4)/(↑n+4)² — via tsum_congr + push_cast + ring_nf ✓ plausible.\n\n6. In dirichletC2_le_one: the final `linarith [hmain, hlog2, hlog3, hlog2nn, hlog3nn]`: goal dirichletC2 ≤ 1 — wait I did `rw [dirichletC2]` at the start — dirichletC2 IS the def: rw [dirichletC2] unfolds to the tsum ✓ (def-rw). Then goal: ∑' n, ‖term‖ ≤ 1. hmain : ∑' n, ‖term‖ ≤ log 2/4 + log 3/9 + (log 3+1)/3. linarith with log 2 ≤ 0.6932, log 3 ≤ 1.1: log2/4 + log3/9 + (log3+1)/3 ≤ 0.6932/4 + 1.1/9 + 2.1/3 = 0.1733 + 0.12222 + 0.7 = 0.9955 ≤ 1 ✓ linarith (nonneg logs for the monotonicity — linarith does linear arithmetic: log2/4 ≤ 0.6932/4 needs log2 ≤ 0.6932 ✓ linear ✓; the combination: linarith handles division-by-constant ✓).\n\n7. `Real.log_two_lt_d9 : Real.log 2 < 0.6931471808` ✓ exists (banked). `.le.trans (by norm_num)` : log 2 ≤ 0.6932 ✓.\n\n8. The summable_nat_add_iff direction: `summable_nat_add_iff (k : ℕ) : Summable (fun n => f (n + k)) ↔ Summable f` — so .mp : Summable (fun n => h (n+4)) → Summable h ✓ as written in hsumh. ✓\n\n9. In hsumh's h1: `apply summable_log_div_sq_tail.congr; intro n; rw [hh]; show (if (n+4:ℕ) = 0 then ...) = ...` — Summable.congr (hf : Summable f) (h : ∀ i, f i = g i) : Summable g — apply ... .congr: goal Summable (fun n => h (n+4)): congr with f := the tail's function: needs ∀ n, log(↑n+4)/(↑n+4)² = h (n+4)?? — congr's h : f i = g i where f = tail-fn, g = the goal's fn: log (↑n+4)/(↑n+4)² = h (n+4): my show has them flipped: `show (if ...) = log (↑n+4)/...` — that's g n = f n. Order matters for congr — Summable.congr (hf : Summable f) (h : ∀ i, f i = g i) : Summable g — so h : f i = g i = (tail-fn) i = h(n+4)-form... Let me just symmetrize at build if it fails.\n\nLet me now splice and build.","signature":""},{"type":"text","text":"Now splice: head + new §6 + tail."},{"type":"tool_use","id":"Bash_170","name":"Bash","input":{"command":"cd D:/Python/Quantum/research/lean/NguyenRH1/NguyenRH1 && cat /tmp/litightr_head.txt litightr_mid.tmp /tmp/litightr_tail.txt > LiTightR.lean && rm litightr_mid.tmp && wc -l LiTightR.lean && export PATH=\"$HOME/.elan/bin:$PATH\" && cd D:/Python/Quantum/research/lean/NguyenRH1 && lake env lean NguyenRH1/LiTightR.lean 2>&1 | head -40","description":"Splice LiTightR and compile","timeout":300000},"extra_content":null}]