← Session 22 — ii′-1: liCoeff pairing fix + residue machinery
thinking (582 chars)
Session 017 complete. Let me summarize for the user.
Sub-stage ii′-1b is now FULLY COMPLETE. This is a natural sub-stage boundary — clean, committed, resumable state per §8.5. The session accomplished exactly what was planned. I should report and note the next action (ii′-2). Per the pacing rules, I don't need to ask about breaks unless approaching 5 PM — I don't know the current time. Let me check the time... actually I shouldn't assume; the BREAK PROMPT says if approaching 5:00 PM local. I don't know the time — let me just report the state; the user can decide to continue.
**Session 017 complete — ii′-1b is now FULLY DONE** (R1 residue machinery, sessions 016–017).
## What was proved (LiSides.lean, ~460 lines, 19 declarations)
**`rectInt'_Jlag`** — the rectangle residue theorem for the Lagarias-kernel integrand, the ii′-1b deliverable:
```
RectangleIntegral' (Jlag n) z w = ResW1 n + Σ_{ρ∈Sz} (zeroMult ρ)·(1 − (ρ/(ρ−1))ⁿ)
```
for any rectangle with 1 interior, ζ ≠ 0 on the border, and Sz capturing the zeros inside (supplied form — downstream gets it from `finite_zeros_rectangle`).
Key pieces:
- **Removable singularity at 1**: `R = J − ppart` bounded on 𝓝[≠] 1, via `(w−1)ⁿ⁺¹·R = F − TaylₙF` vanishing to order ≥ n+1 (`natCast_le_analyticOrderAt_iff_iteratedDeriv_eq_zero`)
- **Residues at zeros** via logDeriv normal form — multiple zeros handled (logDeriv poles are always simple)
- **`zeta_order_ne_top_of_ne`** via the entire `zetaA` (identity principle on univ — neatly avoids punctured-plane connectedness, and `zetaA 1 = 1` makes ζ(1)'s junk value irrelevant)
## Verification
- Full `lake build` GREEN (8745 jobs) — root now imports LiResidue/LiPrincipal/LiSides (session 016 had left them out of the root closure)
- Axiom audit: **3 standard axioms only** on all 19 declarations, no sorryAx
- Deferred with documentation: `zeroMult(−2) = 1` (trivial-zero simplicity) — the sum keeps multiplicities symbolic; the Sz = {nontrivial} ∪ {−2} split belongs to ii′-4
## Archive
Commits f3d7244 (work) + 7ec1ceb (closure): journal-017, STATE.md (NEXT ACTION → **ii′-2**: the elementary swaps λ_n = Σⱼ(−1)ʲ⁺¹C(n,j)σⱼ and S_∞ = ΣⱼC(n,j)τ_{j−1}), NGUYEN.md checklist + session log, research.db session 17.
Phase 5B map: ii′-0 ✅ → 1a ✅ → 1b ✅ → **2 (next)** → 3 (long pole) → 4 → 5 → 6 → 7. Continue with ii′-2?
raw content_json (database row)
[{"type":"thinking","thinking":"Session 017 complete. Let me summarize for the user.\n\nSub-stage ii′-1b is now FULLY COMPLETE. This is a natural sub-stage boundary — clean, committed, resumable state per §8.5. The session accomplished exactly what was planned. I should report and note the next action (ii′-2). Per the pacing rules, I don't need to ask about breaks unless approaching 5 PM — I don't know the current time. Let me check the time... actually I shouldn't assume; the BREAK PROMPT says if approaching 5:00 PM local. I don't know the time — let me just report the state; the user can decide to continue.","signature":""},{"type":"text","text":"**Session 017 complete — ii′-1b is now FULLY DONE** (R1 residue machinery, sessions 016–017).\n\n## What was proved (LiSides.lean, ~460 lines, 19 declarations)\n\n**`rectInt'_Jlag`** — the rectangle residue theorem for the Lagarias-kernel integrand, the ii′-1b deliverable:\n```\nRectangleIntegral' (Jlag n) z w = ResW1 n + Σ_{ρ∈Sz} (zeroMult ρ)·(1 − (ρ/(ρ−1))ⁿ)\n```\nfor any rectangle with 1 interior, ζ ≠ 0 on the border, and Sz capturing the zeros inside (supplied form — downstream gets it from `finite_zeros_rectangle`).\n\nKey pieces:\n- **Removable singularity at 1**: `R = J − ppart` bounded on 𝓝[≠] 1, via `(w−1)ⁿ⁺¹·R = F − TaylₙF` vanishing to order ≥ n+1 (`natCast_le_analyticOrderAt_iff_iteratedDeriv_eq_zero`)\n- **Residues at zeros** via logDeriv normal form — multiple zeros handled (logDeriv poles are always simple)\n- **`zeta_order_ne_top_of_ne`** via the entire `zetaA` (identity principle on univ — neatly avoids punctured-plane connectedness, and `zetaA 1 = 1` makes ζ(1)'s junk value irrelevant)\n\n## Verification\n- Full `lake build` GREEN (8745 jobs) — root now imports LiResidue/LiPrincipal/LiSides (session 016 had left them out of the root closure)\n- Axiom audit: **3 standard axioms only** on all 19 declarations, no sorryAx\n- Deferred with documentation: `zeroMult(−2) = 1` (trivial-zero simplicity) — the sum keeps multiplicities symbolic; the Sz = {nontrivial} ∪ {−2} split belongs to ii′-4\n\n## Archive\nCommits f3d7244 (work) + 7ec1ceb (closure): journal-017, STATE.md (NEXT ACTION → **ii′-2**: the elementary swaps λ_n = Σⱼ(−1)ʲ⁺¹C(n,j)σⱼ and S_∞ = ΣⱼC(n,j)τ_{j−1}), NGUYEN.md checklist + session log, research.db session 17.\n\nPhase 5B map: ii′-0 ✅ → 1a ✅ → 1b ✅ → **2 (next)** → 3 (long pole) → 4 → 5 → 6 → 7. Continue with ii′-2?"}]