[{"type":"thinking","thinking":"The user wants to work on ii′-4. Let me recall what ii′-4 is:\n\nFrom STATE.md NEXT ACTION:\n> Session 021: **ii′-4 — L3 zero-avoiding T + L1 kernel bounds** (D-064 §4 R1/R3 support; elementary, ~1 session). Content:\n> - L3: for each n in the range, a zero-avoiding T ∈ [√n, √n+1] — the pigeonhole: ≤ C₀(√n+1) zeros with |γ−t| < 1 ⟹ a gap center with clearance ≥ 1/(3C₀) (D-036 §2); the zero count from the perimeter (Trudgian N(T)/S(T) declared, D-062 §4).\n> - L1: kernel bounds |k_n| ≤ 2 (on Re ≥ ...), e⁴+1, e^{√n/2}+1 (D-036); the trivial-zero residue value 1−(2/3)ⁿ at −2 — needs zeroMult(−2) = 1 (trivial-zero simplicity, deferred from ii′-1b(iii): route via the FE + Gammaℝ_eq_zero_iff + ζ(3) ≠ 0, D-066 §6 note).\n\nThis is a Lean formalization task in research/lean/NguyenRH1. This is session 021's work. This is a big, non-trivial task. Per the session protocol:\n\n1. Session START protocol: Read STATE.md (done), skim latest journal entry + results/index.md.\n2. The work itself: prove L3 (zero-avoiding T via pigeonhole) and L1 (kernel bounds) in Lean.\n\nGiven this is a substantial task (estimated ~1 session), I should:\n1. First read the relevant design docs (D-064 §4, D-036 §2, D-066 §6 note) to understand exactly what's needed.\n2. Look at the existing Lean files to understand conventions (LiSkeleton.lean, LiSides.lean, etc.).\n3. Plan the Lean work.\n\nShould I enter plan mode? The task is well-specified by STATE.md/D-064 — the user has already decided what ii′-4 is; the design docs exist. The user said \"Let's work on ii′-4 now.\" The approach is documented. I think I should dig into the details and start. But given the scale (a full session of Lean work), a task list (TodoWrite) is appropriate.\n\nActually, let me think about whether to use EnterPlanMode. The task content is already designed in D-064/D-066 — this is execution of a pre-approved plan, not a new design. The user directive is clear. I'll proceed with implementation directly, but first read the design docs carefully.\n\nSession protocol first: read STATE.md (already read at conversation start — it's current), skim latest journal (session-020). Let me read journal-020 quickly for the Lean gotchas, and the relevant sections of D-064/D-036/D-066.\n\nPlan for ii′-4 content:\n\n**L3: zero-avoiding T.**\nMathematical content: For each n in [10⁵, N₁] (or maybe ∀ n in range), we need T ∈ [√n, √n+1] such that the horizontal line at height T (wait — actually it's a contour at height T) avoids zeros: the rectangle used in rectInt'_Jlag has top edge at height T, and we need no zeros ON the border. The pigeonhole: in the interval [√n, √n+1], zeros with ordinates in that interval number ≤ C₀(√n+1)... Actually from D-036 §2: the count of zeros with |γ − t| < 1 is ≤ C₀ log... hmm, need to read D-036 §2.\n\nActually the standard argument: N(T+1) − N(T−1) ≤ c log T (or an explicit bound like C₀). Partition [√n, √n+1] into 3C₀+... sub-intervals; one sub-interval contains no zero ordinate; pick T at its center with clearance ≥ 1/(3C₀) or similar.\n\nIn Lean, this needs:\n- A bound on the number of zeros in a unit interval — from the perimeter: Trudgian N(T)/S(T) declared bounds. D-062 §4 put \"Trudgian N(T)/S(T)\" inside the trusted perimeter. So there should be a declared hypothesis about zero counts.\n- Pigeonhole: finite set of reals in an interval; if count·clearance < length, there's a gap.\n\n**L1: kernel bounds.**\nFrom D-036: the Lagarias kernel k_n or Jlag on the vertical sides: |k_n| ≤ 2, and bounds like e⁴+1, e^{√n/2}+1. Plus the trivial-zero residue value 1−(2/3)ⁿ at ρ = −2, which needs zeroMult(−2) = 1 (simplicity of the trivial zero at −2), via the functional equation + Gammaℝ_eq_zero_iff + ζ(3) ≠ 0.\n\nLet me read the design docs first: D-064 §4 (R1/R3 support description of ii′-4), D-036 §2 (the pigeonhole + constants), D-066 §6 (the zeroMult(−2) route note). Also the existing Lean files to see what's available: LiSkeleton.lean (perimeter declarations — Trudgian N(T)/S(T)?), LiSides.lean (rectInt'_Jlag, zeroMult), LiSwaps.lean.\n\nThis is a large task. Let me start by creating a todo list, then reading the docs in parallel.\n\nGiven the size, maybe I should also check how the existing Lean package is organized: files in research/lean/NguyenRH1/NguyenRH1/: LiSkeleton.lean, LiResidue.lean, LiPrincipal.lean, LiSides.lean, LiSwaps.lean, LiZetaCoeff.lean, LiDigamma.lean, LiArchimedean.lean, LiBasePoints.lean, NguyenRH1.lean (root). Let me look.\n\nStrategy for the session:\n1. Read D-036 §2, D-064 §4, D-066 §6 for exact mathematical content.\n2. Read LiSkeleton.lean (perimeter: what's declared — Trudgian bound form), LiSides.lean (zeroMult, Jlag definitions).\n3. Design the Lean statements for L3 and L1.\n4. Implement L3 (pigeonhole) — likely a new file LiPigeonhole.lean or add to existing.\n5. Implement L1 kernel bounds + zeroMult(−2) = 1.\n6. Build + axiom audit.\n7. Session end protocol: journal-021, update NGUYEN.md checklist, STATE.md, commit.\n\nThis will be a long session. Let me start. First, todos and reading.\n\nActually wait — should I use plan mode? The instructions say to prefer EnterPlanMode for non-trivial implementation. But this is a research project with a pre-existing approved design (D-064), a session protocol, and the user just said \"let's work on ii′-4\" — the plan is already documented in STATE.md/D-064. Entering plan mode would be redundant; the user expects execution per the established session protocol. I'll proceed directly but carefully, starting with reading the design docs.\n\nLet me kick off with the reads in parallel: D-036 §2 area, D-064 §4, D-066 §6. First find the files' exact names.","signature":""},{"type":"text","text":"Starting session 021 work on ii′-4. Per protocol, let me pull up the design docs for the exact mathematical content (D-036 §2 pigeonhole, D-064 §4, D-066 §6) and the current Lean state."},{"type":"tool_use","id":"TodoWrite_33","name":"TodoWrite","input":{"todos":[{"content":"Read design docs: D-036 §2, D-064 §4, D-066 §6 (zeroMult route)","status":"in_progress","activeForm":"Reading design docs for ii′-4 content"},{"content":"Survey Lean package state (LiSkeleton perimeter, LiSides zeroMult/Jlag)","status":"pending","activeForm":"Surveying Lean package state"},{"content":"Implement L3: zero-avoiding T via pigeonhole","status":"pending","activeForm":"Implementing L3 zero-avoiding T"},{"content":"Implement L1: kernel bounds + trivial-zero residue (zeroMult(−2) = 1)","status":"pending","activeForm":"Implementing L1 kernel bounds"},{"content":"Full build + axiom audit","status":"pending","activeForm":"Building and auditing axioms"},{"content":"Session end protocol: journal-021, NGUYEN.md, STATE.md, commit","status":"pending","activeForm":"Running session end protocol"}]},"extra_content":null}]