[{"type":"thinking","thinking":"Now NGUYEN.md updates. The relevant sections:\n1. §5 Phase 5B header line (ACTIVE → COMPLETE).\n2. The ii′-6 entry: [~] → [x] + completion line.\n3. The ii′-7 entry: [ ] → [x] + closure summary.\n4. §7 Result Ledger #6: append the Lean closure note.\n\nLet me read the exact current text of those blocks in NGUYEN.md (I have it from the system prompt, but let me make precise edits). The ii′-6 entry starts with:\n```\n- [~] ii′-6 L0 assembly: sides identity + bridge |λ_n(T)| ≤ 4·FAlt(T)\n      (hRH, BOTH-signs correction F-A) + |Sf n| ≤ Bn FAlt n on [10⁵, N₁].\n      ⚠ **FINDING F-D (session 038, D-072)**: ...\n```\nAnd ends with the 6-c-i line and \"6-c-ii (I_III) designed, Lean not yet written — recipe in STATE.md.\"\n\nThe ii′-7 entry:\n```\n- [ ] ii′-7 Closure: theorem sfBoundProp : SfBoundProp; final\n      lambda_pos_finite_height_closed (3 perimeter hypotheses only);\n      full axiom audit; paper Appendix B + tag upgrade (D-062 §4\n      endpoint); archive closure + D-037 line-63 erratum pointer.\n```\n\nPhase 5B header:\n```\n### 🔶 PHASE 5B — TIER ii′: close SfBoundProp in Lean (ACTIVE 2026-08-21 — user directive: BEFORE submission; design D-064; ~9–16 sessions)\n```\n\nLedger #6:\n```\n| 6 | Explicit finite-height Li theorem: λ_n > 0 for all n ≤ N₁ = (T₀−1)² ≈ 9.0×10²⁴ (T₀ = 3×10¹² Platt–Trudgian); conservative window N₁* = 2.73×10²¹; first assembled explicit chain (S_∞ elementary lower bound + traced contour constants + margin convexity + finite verification) | [PROVED-CONDITIONAL(computational RH-to-T₀; finite verification)] — R-bound fully traced D-056 (A = 10.5, B₀ = 48, honest C₀; E-028 verified, 26× slack); N₀ = 80 000; margin convexity PROVED D-057 (E-029) | R-004, D-037, D-056, D-057, E-023, E-028 |\n```\n\nLet me make these edits.","signature":""},{"type":"text","text":"Now the NGUYEN.md updates (Phase 5B header, ii′-6, ii′-7, Ledger #6)."},{"type":"tool_use","id":"Edit_242","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\NGUYEN.md","old_string":"### 🔶 PHASE 5B — TIER ii′: close SfBoundProp in Lean (ACTIVE 2026-08-21 — user directive: BEFORE submission; design D-064; ~9–16 sessions)","new_string":"### ✅ PHASE 5B — TIER ii′: close SfBoundProp in Lean (COMPLETE 2026-08-26, session 041 — sfBoundProp PROVED + lambda_pos_finite_height_closed; design D-064)"},"extra_content":null},{"type":"tool_use","id":"Edit_243","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\NGUYEN.md","old_string":"      → **ii′-6-c STARTED (session 040 block B)**: FINDING F-E (D-074\n      §9) — the I_I kernel charge e^{√n/2}+1 is off by one at\n      re w = 2√n near t = 0 (true value e^{√n/2+1/8}; latent in the\n      original D-036 charge); REPAIR ADOPTED: I_I charged honestly as\n      e^{1/3}·IIMaj (uniform kernel ≤ e^{√n/2+1/3}+1, n ≥ 4), the\n      overrun absorbed by the fold's free dropped 2π (e^{1/3} ≤ 1.4 ≤\n      2π) — NO change to Bn/IIMaj/convexity/pinch/base points\n      (fatten-IIMaj rejected: iimaj_bracket_bound's 4.56 is linear).\n      **6-c-i COMPLETE (a8c5b1e)**: LiSideBounds.lean scaffolding\n      (Z2hat_mono, nearZerosSub_mult_sum_le, logDeriv continuity,\n      Jlag_norm_eq, both zero-freeness lemmas, rectangle side\n      decomposition, VIntegral/HIntegral machinery, even symmetry).\n      6-c-ii (I_III) designed, Lean not yet written — recipe in\n      STATE.md.\n- [ ] ii′-7 Closure: theorem sfBoundProp : SfBoundProp; final\n      lambda_pos_finite_height_closed (3 perimeter hypotheses only);\n      full axiom audit; paper Appendix B + tag upgrade (D-062 §4\n      endpoint); archive closure + D-037 line-63 erratum pointer.","new_string":"      → **ii′-6-c STARTED (session 040 block B)**: FINDING F-E (D-074\n      §9) — the I_I kernel charge e^{√n/2}+1 is off by one at\n      re w = 2√n near t = 0 (true value e^{√n/2+1/8}; latent in the\n      original D-036 charge); REPAIR ADOPTED: I_I charged honestly as\n      e^{1/3}·IIMaj (uniform kernel ≤ e^{√n/2+1/3}+1, n ≥ 4), the\n      overrun absorbed by the fold's free dropped 2π (e^{1/3} ≤ 1.4 ≤\n      2π) — NO change to Bn/IIMaj/convexity/pinch/base points\n      (fatten-IIMaj rejected: iimaj_bracket_bound's 4.56 is linear).\n      **6-c-i COMPLETE (a8c5b1e)**: LiSideBounds.lean scaffolding\n      (Z2hat_mono, nearZerosSub_mult_sum_le, logDeriv continuity,\n      Jlag_norm_eq, both zero-freeness lemmas, rectangle side\n      decomposition, VIntegral/HIntegral machinery, even symmetry).\n      **ii′-6-c..6-d COMPLETE (session 041)**: 6-c-ii I_III\n      (vIntegral_Jlag_left_norm_le, 6e5a98f), 6-c-iii I_I F-E form\n      (vIntegral_Jlag_right_le_exp_third_IIMaj, 9bd8ba9), 6-c-iv the\n      mid-horizontal segment machine (arcsinh-FTC from scratch —\n      integral_one_div_sqrt_sq_add_sq + endpoints + per_zero ≤\n      2·log(16.5/δ) by pure log-mono, 1fed1df), 6-c-v the horizontal\n      assembly (mid + DL-5 tails both sides, 5d3dc0a), and **6-d THE\n      FOLD (774fec5): `sfBoundProp : SfBoundProp` PROVED** — |Sf n| ≤\n      Bn FAlt n on [10⁵, N₁] from EXACTLY the six declared perimeter\n      hypotheses (RHUpTo, ZeroCountProp, ZeroCountProp2,\n      ZeroCountGlobalProp, NoLowZeros, MZeroProp); the pinch NI + 5.5R\n      ≤ 7·Ĥ discharged by a fully elementary log-mono chain (no\n      certified log numerics); the bridge cloned at T ≤ 3·10¹² (the\n      range actually used — T ≤ √n+1 holds exactly incl. n = N₁).\n      **ii′-6 COMPLETE.**\n- [x] ii′-7 Closure: theorem sfBoundProp : SfBoundProp; final\n      lambda_pos_finite_height_closed (3 perimeter hypotheses only);\n      full axiom audit; paper Appendix B + tag upgrade (D-062 §4\n      endpoint); archive closure + D-037 line-63 erratum pointer.\n      → COMPLETE (session 041): **lambda_pos_finite_height_closed\n      PROVED** (a0943a6, LiClosed.lean) — λ_n > 0 for ALL 1 ≤ n ≤\n      N₁ = (3×10¹²−1)² from EXACTLY EIGHT declared hypotheses (the two\n      computational certificates + the six analytic perimeter Props);\n      full axiom audit (AuditSession041: 29 decls incl. the closed\n      chain — 3 standard axioms only, 0 sorryAx); Paper A closure\n      (b64a839, compiles clean 9 pp — Appendix B rewritten: second\n      tier COMPLETE; the B(n) display corrected + the F-A/F-B/F-D/F-E\n      correction note footnote in §3 Step 2; abstract/intro to the\n      eight-hypotheses headline; AI Contribution Statement factual\n      refresh); archive closure (D-037 erratum block, R-004 Lean-\n      closure status, NGUYEN.md, STATE.md, journal-041)."},"extra_content":null}]