[{"type":"thinking","thinking":"Session 027 fully closed. Let me give the final summary.","signature":""},{"type":"text","text":"**Session 027 COMPLETE** — commits `e28eff6` (5e-i) + `0e9336e`/`ec13029` (5e-ii) + `31303a2` (closure).\n\n## What landed\n\n**ii′-5e-i** (`LiGapCircles.lean`): `exists_gapCircle_radius` — for each k ≥ 1, a radius R ∈ [4k+5/2, 4k+3] with **both** clearances the identity's Liouville step needs: dist(R/2, ℤ) ≥ 1/4 (trivial zeros) and every nontrivial zero ordinate either ≥ 1 away or at clearance ≥ 1/(16·C0hat(4k+9/2)). Built with zero new pigeonhole code via the ×2-rescale trick on ii′-4's midpoint lemma.\n\n**ii′-5e-ii** (`LiDigammaLeft.lean`, 15 decls): the four digamma pieces from D-068's correction C2, all with explicit constants:\n- **(B)** `norm_digamma_le_of_two_le_re`: ‖ψ(w)‖ ≤ 20·log(2+‖w‖) for re w ≥ 2 — Weierstrass series split at ⌈re w⌉₊ (harmonic head via Mathlib's `harmonic_le_one_add_log`, tail via a new telescoping-tsum lemma; positive integers via a proved ψ(n) = −γ + H_{n−1}).\n- **(C)** `digamma_recip_sum_le`: Σ‖(w+j)⁻¹‖ ≤ 2/c + 2 + 2·log m under dist(w,−ℕ) ≥ c — the V-split, the heart of the near-axis case (O(log m) is essential for the o(R) squeeze).\n- **(A)** `norm_digamma_le_of_dist_nat`: the near-axis left-plane bound for re w ≤ 2 — recurrence to re ∈ [3,4) + (C) + StirlingVert/(B) at the landing point.\n- **(G)** `logDeriv_Gammaℝ_eq`: logDeriv ΓR s = −(log π)/2 + ½ψ(s/2) for **all** im s ≠ 0 (the m-fold shift — no longer confined to re > 0 or the +4 window).\n\nFull build GREEN (8764 jobs); 17 decls axiom-audited: 3 standard axioms only.\n\n## Next (session 028)\nii′-5e-iii — the zero series `Σ' m_ρ(1/(s−ρ) + 1/ρ)`: summability (ii′-1a dominator + `zero_sum_inv_sq`), analyticity off the zeros (locally-uniform limits), and isolation at each zero. Then 5e-iv (F entire) and 5e-v (circle bound + squeeze ⟹ ∃ B identity). STATE.md has the detailed breakdown."}]