← Session 22 — ii′-1: liCoeff pairing fix + residue machinery
thinking (254 chars)
Session 020 fully closed. Let me give the user the summary. It's ~01:00 AM local. This was a big session — piece (A) complete, which was the heaviest rock of ii′-3a. Let me summarize the state and the road ahead, and note the hour for the break decision.
**Session 020 complete — ii′-3a is now FULLY DONE** (both pieces A and B). Commits through 722eded.
## The capstone (piece A)
**`iteratedDeriv_logDeriv_Gammaℝ`**: for all j ≥ 0,
```
iteratedDeriv j (fun s => logDeriv Gammaℝ (1+s)) 0 = (−1)^{j+1}·j!·lagariasA(j+1)
```
The chain (~1200 new lines across LiArchimedean.lean + LiDigamma.lean):
- The Gammaℝ logDeriv split + **τ₀ = −a₁** via Mathlib's `digamma_one_half`
- The **digamma partial-fraction series built from scratch** (not in Mathlib): `logGammaSeq` → real identity via `hasDerivAt_of_tendstoLocallyUniformlyOn` (the deriv-limit interchange + Weierstrass M-test) → complex series analytic on the ball via the Weierstrass convergence theorem → identity principle through the real segment
- **Polygamma values at 1/2** via `iteratedDerivWithin_tsum` (termwise iterated derivation)
- The odd-m reindexing (bespoke `oddGeThreeEquiv : ℕ ≃ {m // Odd m ∧ 3 ≤ m}`) + the 2^{j+1} scaling
Full build GREEN (8749 jobs); 18 key declarations audited — **3 standard axioms only**.
Two reusable elaboration gotchas banked in the journal: bare `π` auto-binds as a spurious implicit with these opens (always `Real.pi`), and bare `↑(...)` with no expected ℂ type silently drops the coercion (always ascribe `(↑(...) : ℂ)`).
## Phase 5B map
ii′-0 ✅ → 1a ✅ → 1b ✅ → 2 ✅ → 3-design ✅ → **3a ✅** → 4 (next: L3 zero-avoiding T + L1 kernel bounds — elementary) → 5 (R-bound) → 3b (moments) → 3c (assembly → crux) → 6 → 7
Piece (A) was the heaviest self-contained rock in the tier. What remains: ii′-4 (elementary, ~1 session), ii′-5 (the R-bound, 2–3 sessions, the other big analytic piece), then the moment identities (ii′-3b) that combine everything, and the final assembly.
It's just past 1 AM — a clean sub-stage boundary and a natural place to break. Say the word if you want ii′-4 now, or we can pick it up fresh next session.
raw content_json (database row)
[{"type":"thinking","thinking":"Session 020 fully closed. Let me give the user the summary. It's ~01:00 AM local. This was a big session — piece (A) complete, which was the heaviest rock of ii′-3a. Let me summarize the state and the road ahead, and note the hour for the break decision.","signature":""},{"type":"text","text":"**Session 020 complete — ii′-3a is now FULLY DONE** (both pieces A and B). Commits through 722eded.\n\n## The capstone (piece A)\n\n**`iteratedDeriv_logDeriv_Gammaℝ`**: for all j ≥ 0,\n```\niteratedDeriv j (fun s => logDeriv Gammaℝ (1+s)) 0 = (−1)^{j+1}·j!·lagariasA(j+1)\n```\n\nThe chain (~1200 new lines across LiArchimedean.lean + LiDigamma.lean):\n- The Gammaℝ logDeriv split + **τ₀ = −a₁** via Mathlib's `digamma_one_half`\n- The **digamma partial-fraction series built from scratch** (not in Mathlib): `logGammaSeq` → real identity via `hasDerivAt_of_tendstoLocallyUniformlyOn` (the deriv-limit interchange + Weierstrass M-test) → complex series analytic on the ball via the Weierstrass convergence theorem → identity principle through the real segment\n- **Polygamma values at 1/2** via `iteratedDerivWithin_tsum` (termwise iterated derivation)\n- The odd-m reindexing (bespoke `oddGeThreeEquiv : ℕ ≃ {m // Odd m ∧ 3 ≤ m}`) + the 2^{j+1} scaling\n\nFull build GREEN (8749 jobs); 18 key declarations audited — **3 standard axioms only**.\n\nTwo reusable elaboration gotchas banked in the journal: bare `π` auto-binds as a spurious implicit with these opens (always `Real.pi`), and bare `↑(...)` with no expected ℂ type silently drops the coercion (always ascribe `(↑(...) : ℂ)`).\n\n## Phase 5B map\n\nii′-0 ✅ → 1a ✅ → 1b ✅ → 2 ✅ → 3-design ✅ → **3a ✅** → 4 (next: L3 zero-avoiding T + L1 kernel bounds — elementary) → 5 (R-bound) → 3b (moments) → 3c (assembly → crux) → 6 → 7\n\nPiece (A) was the heaviest self-contained rock in the tier. What remains: ii′-4 (elementary, ~1 session), ii′-5 (the R-bound, 2–3 sessions, the other big analytic piece), then the moment identities (ii′-3b) that combine everything, and the final assembly.\n\nIt's just past 1 AM — a clean sub-stage boundary and a natural place to break. Say the word if you want ii′-4 now, or we can pick it up fresh next session."}]