← Session 28 — ii′-5e: partial-fraction identity (zero series + F entire)
thinking (51 chars)
Session complete. Summarize for the user concisely.
**Session 028 COMPLETE — ii′-5e-iii done** (commits `3de2a85` + `eca05e7`).
**LiSeries.lean** (~600 lines, 22 decls, all audited: 3 standard axioms only; build GREEN, 8765 jobs):
- **`seriesSummable`** — the zero series `∑' ρ, mult·(1/(s−ρ) + 1/ρ)` is absolutely summable at *every* s (the ii′-1a D-trick: tail bound `2‖s‖/‖ρ‖²` via `1/(s−ρ)+1/ρ = s/(ρ(s−ρ))`, the `(17/12)‖ρ‖²` γ-comparison, Zeta23 `zero_sum_inv_sq` dominator, finite head by compact-finiteness).
- **`seriesFun_analyticOnNhd`** — analytic off the nontrivial zeros, via three-case avoiding balls (ζ≠0 continuity / ζ=0 order normal form / s₀=1 zetaA) + the Weierstrass M-test (`of_norm_le_summable_eventually`) and the banked TendstoLocallyUniformlyOn→differentiableOn→analyticOnNhd pipeline.
- **`nontrivialZero_isolated`** + **`seriesFunSub_analyticAt`** + capstone **`seriesFun_sub_polar_analyticAt`** — the series minus `mult/(s−ρ₀)` is analytic *at* ρ₀. This is exactly the residue-cancellation input 5e-iv needs at the nontrivial zeros.
Archive updated: journal-028 (with 11 new banked Lean gotchas), STATE.md (NEXT ACTION: **ii′-5e-iv**, F entire by principal-part splits, fully scoped with the per-point split plan), NGUYEN.md (checklist + session log), research.db.
Remaining route: 5e-iv → 5e-v (circle bound + ∃ B capstone) → 5f → 3b → 3c → 6 → 7.
raw content_json (database row)
[{"type":"thinking","thinking":"Session complete. Summarize for the user concisely.","signature":""},{"type":"text","text":"**Session 028 COMPLETE — ii′-5e-iii done** (commits `3de2a85` + `eca05e7`).\n\n**LiSeries.lean** (~600 lines, 22 decls, all audited: 3 standard axioms only; build GREEN, 8765 jobs):\n\n- **`seriesSummable`** — the zero series `∑' ρ, mult·(1/(s−ρ) + 1/ρ)` is absolutely summable at *every* s (the ii′-1a D-trick: tail bound `2‖s‖/‖ρ‖²` via `1/(s−ρ)+1/ρ = s/(ρ(s−ρ))`, the `(17/12)‖ρ‖²` γ-comparison, Zeta23 `zero_sum_inv_sq` dominator, finite head by compact-finiteness).\n- **`seriesFun_analyticOnNhd`** — analytic off the nontrivial zeros, via three-case avoiding balls (ζ≠0 continuity / ζ=0 order normal form / s₀=1 zetaA) + the Weierstrass M-test (`of_norm_le_summable_eventually`) and the banked TendstoLocallyUniformlyOn→differentiableOn→analyticOnNhd pipeline.\n- **`nontrivialZero_isolated`** + **`seriesFunSub_analyticAt`** + capstone **`seriesFun_sub_polar_analyticAt`** — the series minus `mult/(s−ρ₀)` is analytic *at* ρ₀. This is exactly the residue-cancellation input 5e-iv needs at the nontrivial zeros.\n\nArchive updated: journal-028 (with 11 new banked Lean gotchas), STATE.md (NEXT ACTION: **ii′-5e-iv**, F entire by principal-part splits, fully scoped with the per-point split plan), NGUYEN.md (checklist + session log), research.db.\n\nRemaining route: 5e-iv → 5e-v (circle bound + ∃ B capstone) → 5f → 3b → 3c → 6 → 7."}]