[{"type":"thinking","thinking":"Committed (f3d7244). ii′-1b is now FULLY COMPLETE (i)+(ii)+(iii).\n\nNow the archive updates per §8.4:\n1. journal/session-017.md\n2. STATE.md update (ii′-1b complete; NEXT ACTION: ii′-2)\n3. NGUYEN.md checklist — mark ii′-1b complete\n4. research.db registration (add-derivation for D-064 follow-up? No — this is a sub-stage completion, not a new derivation. The db has sessions/results tables. Register the session? Session registration via direct sqlite3 insert (journal_file column) — per handoff notes.)\n\nLet me check how the db registration was done for recent sessions — handoff says: \"Session registration via direct sqlite3 insert (journal_file column)\". Let me write the journal first, then STATE, then NGUYEN checklist, then db.\n\nJournal content — session 017 work narrative:\n- ii′-1b(iii) complete: LiSides.lean (~460 lines), rectInt'_Jlag proved.\n- Design decisions: supplied-Sz form (like vendored rectangleIntegral'_mul_logDeriv), zeroMult symbolic at trivial zeros (simplicity deferred), zetaA as the entire vehicle for finiteness/order.\n- Lean notes (hard-won this session) additions:\n  - analyticOrderAt_ne_top_of_isPreconnected is AnalyticOnNhd.-namespaced.\n  - mul_div_cancel_left₀ (b) (ha : a ≠ 0) — explicit arg is b.\n  - Tendsto.congr'/EventuallyEq: ascribe hev as (f =ᶠ[l] g) — the raw ∀ᶠ form has no .symm.\n  - mem_nhdsWithin_of_mem_nhds produces raw set-membership — coerce to EventuallyEq by type ascription before .deriv_eq/.self_of_nhds.\n  - analyticOrderAt_mul rw needs (f :=)(g :=) named to fix the Pi.mul pattern; analyticAt_id.sub analyticAt_const elaborates to (id − const) — lambda mismatch on rw; show-bridge for defeq.\n  - field_simp closes the cleared goal by itself sometimes (trailing ring errors \"no goals\").\n  - zetaA 1 = 1 ≠ 0 avoids riemannZeta 1 junk-value issues entirely.\n- Full build green 8745 jobs; axiom audit 19 decls, 3 standard axioms.\n- STATE: ii′-1b COMPLETE → next ii′-2.\n\nNGUYEN.md checklist: the ii′-1b entry currently says \"→ (i)+(ii) COMPLETE (session 016)... Remaining: (iii) rectangle residue theorem for J...\" — update to full completion.\n\nSTATE.md: update header + NEXT ACTION → ii′-2 (R2(i)/(ii) elementary swaps).\n\nLet me write these now. Also register session in research.db.\n\nFirst the journal.","signature":""},{"type":"text","text":"Committed f3d7244. Now the archive closure: journal, STATE, NGUYEN checklist, db."},{"type":"tool_use","id":"Write_60","name":"Write","input":{"file_path":"D:\\Python\\Quantum\\research\\journal\\session-017.md","content":"# Session 017 — 2026-08-21 — Tier ii′ sub-stage ii′-1b(iii) COMPLETE (ii′-1b now fully done)\n\n## Directive\nContinue tier ii′ per D-064: finish ii′-1b — (iii) the rectangle residue\ntheorem for the Lagarias-kernel integrand J (STATE.md NEXT ACTION).\n\n## Work done\n**ii′-1b(iii) COMPLETE (commit f3d7244)** — LiSides.lean (~460 lines, 19\nnew declarations, single-file check clean, full `lake build` GREEN 8745\njobs — first build covering LiResidue/LiPrincipal, which session 016 had\nleft out of the package root; root NguyenRH1.lean now imports all three):\n\n* `kFun n w = (w/(w−1))ⁿ−1`, `Jlag n w = kFun n w·(−ζ′/ζ(w))`,\n  `Rfun n = Jlag n − ppart n` + differentiability/continuity kits\n  (`kFun_differentiableAt`, `ppart_differentiableAt`,\n  `Jlag_differentiableAt`, `Rfun_differentiableAt`, `ppart_continuousAt`).\n* `zeta_order_ne_top_of_ne` : analyticOrderAt ζ ρ ≠ ⊤ for all ρ ≠ 1 —\n  identity principle through the ENTIRE zetaA (avoids punctured-plane\n  connectedness): analyticOrderAt_ne_top_of_isPreconnected on univ with\n  zetaA 1 = 1 ≠ 0, then order(zetaA) = order((w−1)·ζ) = 0 + order(ζ)\n  via `analyticOrderAt_congr` + `analyticOrderAt_mul`. NOTE: zetaA's\n  value-1-at-1 also makes `riemannZeta 1`'s junk value irrelevant\n  everywhere — no `ζ(1) ≠ 0` lemma needed.\n* `rectangle_mem_nhds_of_mem_not_border` — interior from border avoidance\n  (extracted from the vendored hZint argument).\n* **Removable singularity at 1** (`Rfun_bounded_near_one`): on 𝓝[≠] 1,\n  R = (Ffun n − TaylP n)/(w−1)^{n+1} (`Rfun_eq_sub_div_near_one`, from\n  `Jlag_mul_pow` + `ppart_mul_pow`); `Ffun_sub_TaylP_order`: order ≥ n+1\n  via `natCast_le_analyticOrderAt_iff_iteratedDeriv_eq_zero` —\n  `iteratedDeriv_TaylP` (the i-th deriv of the Taylor poly matches Ffun's:\n  descFactorial/0^{j−i} split, `Finset.sum_eq_single`); then\n  `Rfun_tendsto_one` by cases on order = ⊤ (R → 0) or factorization\n  g = (w−1)^m·h with m ≥ n+1 (R → 0^{m−n−1}·h(1)). Residue at 1 is 0\n  for R — exactly what `residueTheorem_finset` needs with 1 ∈ S.\n* **Residue at a zero** (`Rfun_residue_at`): for ANY ρ ≠ 1,\n  R − mult(ρ)·(1−(ρ/(ρ−1))ⁿ)/(w−ρ) is bounded on 𝓝[≠] ρ — logDeriv normal\n  form ζ = (w−ρ)^m·h, h(ρ) ≠ 0 (from `zeta_order_ne_top_of_ne` +\n  `AnalyticAt.analyticOrderAt_ne_top`), so −ζ′/ζ = −m/(w−ρ) − h′/h; the\n  kFun slope → kFun′(ρ), h′/h and ppart continuous. Multiple zeros OK\n  (logDeriv poles are simple). Sign check: residue = −m·kFun n ρ\n  = m·(1−(ρ/(ρ−1))ⁿ) ✓ matches D-064.\n* **`rectInt'_Jlag`** (the deliverable): rectangle with 1 interior, no\n  zeros of ζ on the border, Sz any finset with\n  `s ∈ Sz ↔ s ∈ Rectangle ∧ zetaA s = 0` (supplied form, like the\n  vendored `rectangleIntegral'_mul_logDeriv`; downstream produces Sz from\n  `finite_zeros_rectangle` — zetaA is entire so it applies across w = 1):\n    RectangleIntegral' (Jlag n) z w\n      = ResW1 n + Σ_{ρ ∈ Sz} (zeroMult ρ)·(1 − (ρ/(ρ−1))ⁿ).\n  Proof: residueTheorem_finset on R with S = insert 1 Sz, A 1 = 0,\n  A ρ = mult·(1−(ρ/(ρ−1))ⁿ); then Jlag = Rfun + ppart,\n  `RectangleIntegral'_add`, `rectInt'_ppart` (session 016).\n\nAxiom audit (PrintAxioms.lean): all 19 declarations — propext,\nClassical.choice, Quot.sound ONLY. No sorryAx.\n\n**Deferred (documented in file header):** zeroMult(−2) = 1 (trivial-zero\nsimplicity — not in Mathlib/Zeta23; would need the functional-equation\nderiv computation). The sum keeps zeroMult symbolic; the\nSz = {nontrivial} ∪ {−2} split on the D-064 rectangle is ii′-4 territory.\n\n## Lean notes (hard-won this session, additions to the handoff list)\n- `analyticOrderAt_ne_top_of_isPreconnected` is **AnalyticOnNhd-namespaced**\n  (`AnalyticOnNhd.analyticOrderAt_ne_top_of_isPreconnected`).\n- `mul_div_cancel_left₀ (b) (ha : a ≠ 0) : a * b / a = b` — the EXPLICIT\n  arg is b (the kept factor); pass `_`.\n- `Tendsto.congr'` needs an `EventuallyEq` — a raw `∀ᶠ s, f s = c` has no\n  `.symm`; ascribe the have as `f =ᶠ[l] g` explicitly.\n- `mem_nhdsWithin_of_mem_nhds` yields the raw set-membership form\n  ({x | p x} ∈ 𝓝 s); coerce to `f =ᶠ[𝓝 s] g` by type ascription before\n  `.deriv_eq` / `.self_of_nhds` dot-projection works.\n- `analyticOrderAt_mul` in a `rw` needs `(f := ...) (g := ...)` named —\n  `analyticAt_id.sub analyticAt_const` elaborates to `id − const`, not\n  the `fun w => w − 1` lambda; `show`-bridge for the defeq.\n- `field_simp` sometimes closes the cleared goal itself — a trailing\n  `ring` then errors \"no goals\". Remove on recompile error.\n- `Nat.cast_analyticOrderNatAt (h : order ≠ ⊤) : ↑orderNat = order` —\n  rewrite direction `← ... at h` to turn an order bound into a nat bound\n  (`ENat.natCast_le_natCast.mp`).\n- `riemannZeta 1 = (γ − log 4π)/2` (Mathlib ZetaAsymp) — junk value but\n  nonzero; routing everything through zetaA avoids needing it.\n\n## State at stop\nii′-1b (R1 residue machinery) FULLY COMPLETE: (i) ResW1_eq_sum, (ii)\nrectInt'_ppart, (iii) rectInt'_Jlag. The sides identity\nsides(T) = ResW1 n + Σ_{ρ ∈ Sz} mult·(1−(ρ/(ρ−1))ⁿ) is now a Lean theorem.\nNext: **ii′-2** — R2(i)/(ii) elementary swaps: λ_n = Σ_j (−1)^{j+1}C(n,j)σ_j\n(binomial + paired swap) and S_∞ = Σ_j C(n,j)τ_{j−1} (D-064 §4 R2).\n"},"extra_content":null}]