[{"type":"thinking","thinking":"5f-0 committed. Now 5f-i: the reference-point difference identity + the elementary pieces. Let me start a new Lean file LiTightR.lean.\n\n**5f-i content:**\n1. The difference identity. From partialFraction_identity at s and s₀:\n   logDeriv ζ s = B + seriesFun s − 1/s − 1/(s−1) + c − ψ(s/2)/2  (c := (log π)/2)\n   logDeriv ζ s₀ = B + seriesFun s₀ − 1/s₀ − 1/(s₀−1) + c − ψ(s₀/2)/2\n   \n   Subtract: logDeriv ζ s = logDeriv ζ s₀ + (seriesFun s − seriesFun s₀) − (1/s − 1/s₀) − (1/(s−1) − 1/(s₀−1)) − (ψ(s/2) − ψ(s₀/2))/2.\n\n2. seriesFun s − seriesFun s₀ = Σ_ρ m·(1/(s−ρ) − 1/(s₀−ρ)): via tsum_sub (both summable: seriesSummable) + termwise algebra:\n   seriesTerm s ρ − seriesTerm s₀ ρ = m·(1/(s−ρ) + 1/ρ) − m·(1/(s₀−ρ) + 1/ρ) = m·(1/(s−ρ) − 1/(s₀−ρ)).\n   So: seriesFun s − seriesFun s₀ = ∑' ρ, m·(1/(s−ρ) − 1/(s₀−ρ)) := diffSeries s s₀.\n   \n3. Split the diff series into near (nearZeros t) and far: ∑' = Σ_near + Σ_far via the tsum_split_finset machinery. nearZeros t is a Finset ℂ in LiDisc — let me check its exact def (line 970): `def nearZeros (t : ℝ) : Finset ℂ := (zeroDiscFinset t).filter (|ρ.im − t| < 1)`-ish. The series is over the subtype {ρ // IsNontrivialZero ρ}. The split needs the subtype version: {ρ : subtype | ρ ∈ nearZeros t} as a Finset via Finset.subtype or filter. LiCircleBound's tsum_split_finset — let me check its exact form (5e-v-b chunk 1: \"tsum_split_finset (tsum = finset sum + if-zeroed tail via tsum_eq_sum)\").\n\n4. Then R(s) := logDeriv ζ s − Σ_near m/(s−ρ):\n   R(s) = logDeriv ζ s₀ − Σ_near m/(s₀−ρ) + Σ_far m·(1/(s−ρ) − 1/(s₀−ρ)) − (1/s − 1/s₀) − (1/(s−1) − 1/(s₀−1)) − (ψ(s/2) − ψ(s₀/2))/2.\n   With 1/(s−ρ) − 1/(s₀−ρ) = (s₀−s)/((s−ρ)(s₀−ρ)) termwise.\n\n5. Piece bounds:\n   - P1: ‖logDeriv ζ s₀‖ ≤ dirichletC2 (re s₀ = 2 ✓) ≤ 0.995 — the numeric bound on dirichletC2 (tsum of LSeries term norms). Hmm — dirichletC2 := ∑' n, ‖LSeries.term Λ 2 n‖. LSeries.term f a n = f n / n^a (for n ≥ 1; 0 for n = 0?). ‖term‖ = Λ(n)/n² (for n ≥ 1; n=0: term = 0). Bound: Λ(n) ≤ log n (Mathlib: ArithmeticFunction.vonMangoldt_le_log : Λ(n) ≤ log n ✓ exists I believe). So dirichletC2 ≤ ∑'_{n≥2} log n / n². The tsum ≤ 0.995 via: first terms exact (n=2: log 2/4, n=3: log 3/9) + tail Σ_{n≥4} log n/n² ≤ ∫₃^∞ log x/x² dx = (log 3 + 1)/3. The integral comparison for a decreasing function: need the lemma. Mathlib: `Real...`? Hmm — for the tail: Σ_{n≥4} log n/n² where f(x) = log x/x² decreasing for x ≥ e^{1/2} ≈ 1.65 ✓. The comparison Σ_{n≥N+1} f(n) ≤ ∫_N^∞ f: I recall Mathlib has something in `Mathlib.Analysis.SumOverResidueClass`... or `integral_test`? There's `MeasureTheory...`? Let me search for \"tsum_le_integral\" or build it: for each n ≥ 3: f(n+1) ≤ ∫_n^{n+1} f (f decreasing ⟹ f(n+1) ≤ f(x) on [n, n+1]) — `IntervalIntegral...` / `intervalIntegral.integral_mono_on`. Then tsum of f(n+1) ≤ tsum of ∫_n^{n+1} f = ∫₃^∞ f (the tsum of consecutive integrals = the big integral: `intervalIntegral...` hmm — needs the \"sum of integrals = integral over union\" for a sequence of consecutive intervals: `MeasureTheory.integral_iUnion...`? There's a simpler path: Mathlib has the integral test: `Real.summable_...`? hmm. Alternatively bound the tail by a computable series: for n ≥ 4, log n ≤ n^{1/2}·(something)... log n/n² ≤ n^{−3/2} for n ≥ ... (log n ≤ √n ⟺ n ≥ ~... log n ≤ √n for all n ≥ 1? log n ≤ n/2 ≤ ... hmm log n ≤ n^{1/2} fails for small n? log 1 = 0 ≤ 1 ✓, log 2 = 0.69 ≤ 1.41 ✓ — log x ≤ √x for x ≥ 0? At x = 1: 0 ≤ 1 ✓; derivative comparison... max of log x/√x at x = e²: 2/e ≈ 0.736 < 1 ✓ so log n ≤ 0.74·√n ⟹ log n/n² ≤ 0.74/n^{3/2}; Σ_{n≥4} n^{−3/2} ≤ ∫₃^∞ x^{−3/2} = 2/√3 ≈ 1.155 — too big (budget for the tail: 0.995 − 0.1733 − 0.1221 = 0.6996). 0.74·1.155 = 0.855 > 0.6996. Not enough. So need the actual integral ∫₃^∞ log x/x² = (log 3+1)/3 = 0.6995 — knife-edge vs budget 0.6996!! OUCH — 0.9949 vs budget 1.01 — fine actually: total piece-1 bound = 0.1733 + 0.1221 + 0.6995 = 0.9949 ≤ 1.01 budget ✓ (I confused myself — the tail budget is 0.6996 and the integral is 0.6995 ✓ fits, but TOO tight — need the integral bound done RIGHT with a hair of room). Hmm — actually the exact tail Σ_{n≥4} log n/n²: true value ≈ 0.9375 − 0.1733 − 0.1221 = 0.6421 (since Σ_{n≥2} log n/n² = −ζ′(2) ≈ 0.93755). The integral bound gives 0.6995 (lossy by 0.057). Total: 0.9949 vs budget 1.01 ✓ 0.015 room. OK — tight but fine. Alternatively include n=4,5 exactly: log4/16 = 0.0866, log5/25 = 0.0644; tail from 6: ∫₅^∞ = (log 5+1)/5 = 0.5219: total 0.1733+0.1221+0.0866+0.0644+0.5219 = 0.9683 — more room (0.04). Take more exact terms if needed. The integral comparison lemma is the work item. Let me search Mathlib for it at build time: candidates: `MeasureTheory.integral_...`? In practice: `summable_nat_add_iff`, and the termwise ∫_{n}^{n+1} f ≥ f(n+1): then Σ_{n=3}^{N} f(n+1) ≤ ∫₃^{N+1} f; take N → ∞: tsum ≤ lim of integrals = ∫₃^∞ (improper). The improper integral value (log 3+1)/3 via FTC on the antiderivative −(log x + 1)/x. Both halves need some API work; ~100 lines. OK.\n   \n   Hmm wait — actually, maybe simpler: bound dirichletC2 against a KNOWN Mathlib value: is there `Real... -ζ'(2)`? No. Or use Λ(n) ≤ log 2·(number-of-prime-power...) no. The integral route it is.\n\n   - P2: near sum at s₀: ‖Σ_near m/(s₀−ρ)‖ ≤ Σ_near m·‖1/(s₀−ρ)‖ ≤ Σ_near m·1 (|s₀−ρ| ≥ 1: ‖s₀−ρ‖ ≥ |re(s₀−ρ)| = 2 − re ρ ≥ 1 since re ρ < 1... wait re ρ < 1 gives 2 − re ρ > 1 — strictly. ≥ 1 suffices for ‖1/(s₀−ρ)‖ ≤ 1: norm one-div: ‖1/w‖ = 1/‖w‖ ≤ 1/1 = 1 ✓). Then Σ_near m ≤ P1-count ≤ 0.6L + 0.886LL + 2.7 — the near sum Σ over nearZeros t... but P1's window is |γ−t| ≤ 1 while nearZeros t is |γ−t| < 1 — ⊂ ✓ so the nearZeros mult sum ≤ the abs-window mult sum ≤ P1 bound. Need the finset inclusion bridge: nearZeros t ⊆ zeroAbsWindow_finset t (as finsets of ℂ: nearZeros t's elements are nontrivial zeros with |im − t| < 1 ⊂ |·| ≤ 1 ✓). Then sum_mono (nonneg terms).\n   \n   Wait — is nearZeros t EXACTLY the nontrivial zeros with |im − t| < 1? From LiDisc: nearZeros t := (zeroDiscFinset t).filter (|ρ.im − t| < 1) — zeroDiscFinset t is zeros in a DISC around 2+it — the disc might not contain ALL |γ−t|<1 zeros! Let me check zeroDiscFinset's radius: from LiDisc, the disc was B(2+it, 5/2)-ish. A nontrivial zero with |γ−t| < 1 has re ∈ (0,1), so |ρ − (2+it)|² = (2−re)² + (γ−t)² ≤ 4 + 1 < (5/2)² = 6.25 ✓ — yes 4+1 = 5 < 6.25 so all near zeros are in the disc ✓. But is zeroDiscFinset t exactly the nontrivial zeros in that disc? From LiDisc: zeroDiscFinset + zeroDisc_isNontrivial (re ≥ 1 via one_le_re nonvanishing; re ≤ 0 via ΓR-pole/FE) — the disc zeros are all nontrivial ✓. So nearZeros t = {nontrivial zeros with |γ−t| < 1} exactly (as a finset of ℂ). Good — the P1 bridge is direct.\n\n   - P4/P5: ‖1/s − 1/s₀‖ = ‖(s₀−s)/(s·s₀)‖ = |s₀−s|/(‖s‖‖s₀‖) ≤ 4.5/(|t|·|t|) ≤ 0.045 (|s₀−s| = |2−σ| ≤ 4.5 since σ ∈ [−5/2, 3]: |2−σ| ≤ max(|2−(−5/2)|, |2−3|) = 4.5 ✓; ‖s‖ ≥ |im s| = |t| ≥ 10; ‖s₀‖ ≥ |t|). Similarly 1/(s−1) − 1/(s₀−1): |s₀−s|/(‖s−1‖‖s₀−1‖) ≤ 4.5/(|t|−0)²... ‖s−1‖ ≥ |im(s−1)| = |t| ✓, ‖s₀−1‖ = |1+it| ≥ |t| ✓. ≤ 0.045 ✓.\n\n6. The identity assembly (the norm chain): ‖R(s)‖ ≤ P1 + P2 + P3 + P4 + P5 + P6.\n\n7. P6 (5f-ii): the ψ-difference.\n8. P3 (5f-ii): the far-sum machine.\n9. The pinch lemma (5f-iii): the final real-variable fold.\n\nLet me start LiTightR.lean with the identity + P1/P2/P4/P5 (5f-i). Actually — the tsum_split_finset signature: let me check LiCircleBound's.","signature":""},{"type":"text","text":"5f-0 committed (`f850d9a`). Starting 5f-i — the difference identity + elementary pieces. Checking the split machinery signatures first."},{"type":"tool_use","id":"Bash_130","name":"Bash","input":{"command":"cd D:/Python/Quantum/research/lean/NguyenRH1/NguyenRH1; grep -n \"theorem tsum_split_finset\\|lemma tsum_split_finset\" -A 6 LiCircleBound.lean | head -12; echo ===NEARZ===; sed -n '965,985p' LiDisc.lean; echo ===VML===; grep -rn \"vonMangoldt_le_log\\|vonMangoldt_le\" D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/NumberTheory/VonMangoldt.lean 2>/dev/null | head -4","description":"Check tsum_split_finset, nearZeros def, vonMangoldt_le_log"},"extra_content":null}]