[{"type":"thinking","thinking":"Session 035: ii′-5f — the tight R-bound from the partial-fraction identity. Per STATE.md NEXT ACTION:\n\n> Session 035: **ii′-5f** — the tight R-bound from the identity: D-056's six-piece analysis on `partialFraction_identity` at s vs the reference point s₀ = 2+it ⟹ |R(s)| ≤ 13.73·C₀(t) + 7.97 ≤ 10.5·log(|t|+2) + 48 EXACTLY (the baked LiSkeleton constants; D-068 §6 preview: near sum ≤ C₀(t) via the perimeter ZeroCountProp, far difference ≤ 12.73·C₀ + 7.06 via Σ 1/(k(k+1)) = 1, Z₂ ≤ 0.65 certified, ψ-difference via digamma_stirling; B := Gentire 0 cancels in the two-point difference — no evaluation needed). ~1 session.\n\nFirst, per the session START protocol: read STATE.md (done implicitly — I just updated it, but let me follow protocol properly; I know the state), skim latest journal (session-034, just written). \n\nThe key input now is **D-056** — the reference-point R-bound proof with A = 10.5, B₀ = 48. I need to read D-056 carefully to understand the six-piece analysis, and D-068 §6 (the preview of what 5f consumes). Also look at what \"the R-bound\" means in the LiSkeleton context — SfBoundProp is the target eventually, and the R-bound is |ζ′/ζ(s) − Σ_near...| ≤ A log|t| + B₀ on the contour? Let me recall from NGUYEN.md:\n\n- D-056 (session 009): \"reference-point R-bound PROVED (A = 10.5, B₀ = 48, honest C₀ incl. log log term — D-036 §0 erratum) + digamma remainder |r| ≤ 1.2/|w|² PROVED + I_I housekeeping.\"\n- D-036: explicit |ξ′/ξ| / |L′/L| contour bounds.\n\nThe R-bound in the D-036/D-037/R-004 context: R(s) = ζ′/ζ(s) − Σ_{|γ−t|<1} m_ρ/(s−ρ) is the \"R\" in the R-bound |R(s)| ≤ A log(|t|+2) + B₀ used for the contour estimates (I_II/I_III integrals in the Lagarias Thm 6.1 bridge). Hmm — actually let me not guess. Let me read D-056 and D-068 §6 first.\n\nAlso need to understand what the Lean deliverable for 5f should be. The SfBoundProp proof (ii′-6) needs: |Sf n| ≤ Bn FAlt n on [10⁵, N₁], via the contour assembly: |Sf| ≤ |λ_n(T)| + |sides| + |triv| where the sides integrals need the R-bound with the tight constants 10.5/48 (the constants baked into LiSkeleton's Hhat/marginTilde: Hhat T = 3·C0hat² + 10.5·log(T+2) + 48).\n\nSo 5f's deliverable: a Lean theorem stating the tight R-bound, presumably:\n‖logDeriv ζ s − Σ_{ρ ∈ nearZeros t} m_ρ/(s−ρ)‖ ≤ 10.5·log(|t|+2) + 48 for s on the relevant contour (σ ∈ [−1, 2]? |t| ≥ T₀-something), proved from partialFraction_identity + ZeroCountProp.\n\nLet me read D-056 first — it's the analytic content. Also D-068 §6 preview. And check what LiSkeleton expects (where does 10.5/48 appear — Hhat).\n\nPlan:\n1. Read D-056 (the proof to formalize).\n2. Read D-068 §6 (already read — the preview says: near sum ≤ C₀(t) (perimeter ZeroCountProp), far difference ≤ 12.73 C₀ + 7.06 (Σ 1/(k(k+1)) = 1), Z₂ ≤ 0.65 certified, ψ-difference via digamma_stirling ⟹ |R| ≤ 13.73·C₀(t) + 7.97 ≤ 10.5·log(|t|+2) + 48 EXACTLY).\n3. Check what nearZeros is in Lean (LiDisc/LiReflect define nearZeros t as a finset).\n4. Design the 5f statement + proof decomposition.\n5. Build it.\n\nLet me start reading. D-056 is in research/derivations/.","signature":""},{"type":"text","text":"Starting session 035 — ii′-5f. Per protocol, reading the design inputs first: D-056 (the tight-constant proof to formalize) and the relevant Lean context (nearZeros, Hhat)."},{"type":"tool_use","id":"TodoWrite_68","name":"TodoWrite","input":{"todos":[{"content":"Read D-056 (six-piece analysis) + D-068 §6 + nearZeros/Hhat Lean context","status":"in_progress","activeForm":"Reading D-056 and Lean context"},{"content":"Design 5f statement + proof decomposition (D-069)","status":"pending","activeForm":"Designing 5f"},{"content":"Build 5f pieces in Lean","status":"pending","activeForm":"Building 5f"},{"content":"Build green + axiom audit + commit + journal/STATE/NGUYEN/db","status":"pending","activeForm":"Closing session 035"}]},"extra_content":null}]