[{"type":"thinking","thinking":"The user wants to start ii′-5e-v-d — the Schwarz squeeze ⟹ the ∃ B capstone `partialFraction_identity`. Per STATE.md, the design is:\n\n1. **Every circle point is good** (¬badPoint s: not nontrivial (3a), ≠ 0/1 (norms), ≠ −2n (R/2 ∉ ℤ by hgapInt)) ⟹ Gentire s = Fexpr s (5e-iv's gentire_eq_Fexpr) ⟹ ‖Gentire‖ ≤ the same bound on the sphere.\n2. **MapsTo**: Gentire differentiable on ball 0 R (from gentire_analyticAt_and_eq.1) + Zeta23's AnalyticOn.norm_le_of_norm_le_on_sphere ⟹ MapsTo (ball 0 R) into closedBall (Gentire 0) (R₂(R) + 1) with R₂(R) := 2·lowZeroConst + dirichletC2 + 400000·A₀·Ĉ₀(R+2)·(log R)² + ‖Gentire 0‖.\n3. **Schwarz**: Complex.dist_le_div_mul_dist_of_mapsTo_ball (ball-form; bridge via ball ⊆ closedBall and R₂ < R₂+1): dist (Gentire z₀) (Gentire 0) ≤ (R₂(R)+1)/R·‖z₀‖ for ‖z₀‖ < R.\n4. **The ratio → 0**: Ĉ₀(R+2) ≤ 5·log R (log(R+4) ≤ 2·log R for R ≥ 3; loglog ≤ 1+log; L ≥ 9) ⟹ R₂(R)+1 ≤ C₀' + 2.01·10⁶·A₀·(log R)³ (symbolic C₀') ⟹ (R₂+1)/R ≤ (C₀'+64·2.01·10⁶·A₀)·R^{−1/4} via log R ≤ 4·R^{1/4} (Real.log_le_rpow — CHECK exists: 1 ≤ x, 0 < r ⟹ log x ≤ r⁻¹·x^r) ⟹ (log R)³ ≤ 64·R^{3/4} (rpow arithmetic). For fixed z₀: pick k with 4k+5/2 > max(‖z₀‖, (C·‖z₀‖/ε)⁴) ⟹ dist < ε ⟹ dist (Gentire z₀) (Gentire 0) = 0 ⟹ **Gentire ≡ Gentire 0 =: B**.\n5. **Capstone** (D-068 §5): `partialFraction_identity (hZC : ZeroCountProp) : ∃ B : ℂ, ∀ s, riemannZeta s ≠ 0 → s ≠ 0 → s ≠ 1 → (∀ n ≥ 1, s ≠ −2n) → logDeriv riemannZeta s = B + seriesFun s − 1/s − 1/(s−1) + (Real.log Real.pi)/2 − Complex.digamma (s/2)/2` — from Gentire s = Fexpr s at good points (5e-iv) + Gentire ≡ B: Fexpr unfolds to exactly this.\n\nPer session protocol:\n1. First read journal-033 for the gotchas + exact statement of Fexpr_circle_le, and journal-030/032 as referenced.\n2. Look at the current LiCircleBound.lean structure to see where to add things (§13 or new file LiSqueeze.lean).\n3. Check key Mathlib/Zeta23 lemmas exist: AnalyticOn.norm_le_of_norm_le_on_sphere, Complex.dist_le_div_mul_dist_of_mapsTo_ball, Real.log_le_rpow.\n\nLet me set up a task list and start. First reads:\n- research/journal/session-033.md (gotchas + state)\n- The end of LiCircleBound.lean (Fexpr_circle_le statement + what's around it)\n- LiFEntire.lean capstone statement (gentire_analyticAt_and_eq)\n\nLet me plan the work:\n\n**Sub-stage plan for 5e-v-d:**\n1. Read journals (033 for latest gotchas), D-068 §5 (capstone design), tail of LiCircleBound.lean, relevant parts of LiFEntire.lean.\n2. Verify Mathlib lemma names: `Complex.dist_le_div_mul_dist_of_mapsTo_ball` (the Schwarz lemma for ball maps), `Real.log_le_rpow` or similar (log x ≤ r⁻¹ x^r).\n3. Build the steps:\n   - Step 1: `gentire_circle_le`: on the gap circle, ‖Gentire s‖ ≤ R₂bound (from Fexpr_circle_le + gentire_eq_Fexpr + every circle point good).\n   - Step 2: MapsTo from norm_le_of_norm_le_on_sphere (need to check the exact Zeta23 name — it might be `DifferentiableOn.norm_le_of_norm_le_on_sphere` or in Zeta23).\n   - Step 3: Schwarz bound.\n   - Step 4: ratio → 0, constancy via picking k large.\n   - Step 5: capstone partialFraction_identity.\n4. Build, audit, commit at clean boundaries.\n\nNote the ZeroCountProp hypothesis: the gap circles exist under ZeroCountProp (perimeter). The capstone takes (hZC : ZeroCountProp).\n\nAlso note: Gentire 0 — what's its value? At s=0, Gentire is the punctured limit. We don't need to evaluate it; B := Gentire 0. Wait — actually in the capstone statement, the identity is logDeriv ζ s = B + seriesFun s − 1/s − 1/(s−1) + (log π)/2 − ψ(s/2)/2. Fexpr s = logDeriv ζ s − seriesFun s + 1/s + 1/(s−1) − (log π)/2 + ψ(s/2)/2. So Fexpr s = B ⟹ logDeriv ζ s = B + seriesFun s − 1/s − 1/(s−1) + (log π)/2 − ψ(s/2)/2. Yes exactly. Good.\n\nNote B is known to be γ + (log 4π)/2... wait, actually the constant in the partial fraction identity: ζ′/ζ(s) = B + Σ_ρ (1/(s−ρ) + 1/ρ) − 1/s − 1/(s−1) + (log π)/2 − (1/2)ψ(s/2) where B = −1 − γ/2 + log(4π)/2... Actually in Lagarias' convention, ζ′/ζ(s) = log 2π − 1 − γ/2 − 1/(s−1) ... let me not worry; the capstone just says ∃ B. The tight-constant stage (5f) might need B's value, but that's for later. Actually for the moment identity ii′-3b we may need B = −1 − γ/2 + log(4π)... wait no — the D-068 design says capstone is ∃ B. Fine.\n\nActually hold on — one nuance: is it known that Gentire 0 is the right B? The identity at good points: Fexpr s = Gentire s = Gentire 0 = B for all good s. So B := Gentire 0. Fine, no evaluation needed.\n\nLet me start. First, read the journals and files. I'll read:\n- research/journal/session-033.md\n- research/derivations/D-068-*.md (find the file)\n- The tail of research/lean/NguyenRH1/NguyenRH1/LiCircleBound.lean\n- LiFEntire.lean capstone region\n\nLet me find files first. LiCircleBound.lean is at research/lean/NguyenRH1/NguyenRH1/LiCircleBound.lean presumably (package NguyenRH1 with root NguyenRH1). Let me glob.\n\nAlso create the task list with TaskCreate... wait, the tools available: TaskCreate is deferred. I have TodoWrite available directly. Actually looking at the tools: TodoWrite is available. TaskCreate is deferred. I'll use TodoWrite.\n\nLet me start by reading key files in parallel.","signature":""},{"type":"text","text":"Starting ii′-5e-v-d. Per STATE.md, all inputs are in hand; the design has 5 steps. Let me set up the task list and read the latest state first."},{"type":"tool_use","id":"TodoWrite_1","name":"TodoWrite","input":{"todos":[{"content":"Read journal-033 + D-068 §5 + tails of LiCircleBound/LiFEntire","status":"in_progress","activeForm":"Reading journals and Lean file state"},{"content":"Verify Mathlib/Zeta23 lemma names (Schwarz ball-form, log_rpow, norm_le_on_sphere)","status":"pending","activeForm":"Verifying lemma names"},{"content":"Step 1: gentire_circle_le (circle points good ⟹ Gentire = Fexpr bound)","status":"pending","activeForm":"Building gentire_circle_le"},{"content":"Step 2: MapsTo (ball 0 R) into closedBall (Gentire 0) (R₂+1)","status":"pending","activeForm":"Building MapsTo step"},{"content":"Step 3: Schwarz dist bound for fixed z₀","status":"pending","activeForm":"Building Schwarz step"},{"content":"Step 4: ratio → 0 ⟹ Gentire ≡ Gentire 0 =: B","status":"pending","activeForm":"Building constancy argument"},{"content":"Step 5: capstone partialFraction_identity (hZC : ZeroCountProp)","status":"pending","activeForm":"Building the capstone"},{"content":"Build green + axiom audit + commit + session 034 journal","status":"pending","activeForm":"Building, auditing, committing"}]},"extra_content":null}]