[{"type":"thinking","thinking":"The user wants to continue with session 040 — the main line executing repair (R) at c = 1.5. This is Lean work: the ii′-6-a redo per D-073.\n\nThe 8-step protocol re-run at c = 1.5:\n1. Bn := Bn_c(n) at c = 1.5 (changes: T: √n+1 → 1.5√n+1; kernel charge e⁴+1 → e^{16/9}+1; right-tail 3.3 → 5.94; IIMaj ×1.5; N₁ → 4×10²⁴)\n2. Fresh documented definition + correction comment\n3. Re-prove marginTilde_convex\n4. Re-certify base points (8×10⁴, 10⁵)\n5. Re-run endpoint margins\n6. Resume ii′-6-c with the fixed stepping (later sub-stage)\n7. ii′-7 paper note (later)\n8. Audit history (done)\n\nSo session 040's work = ii′-6-a-redo (steps 1–5) in Lean. This is a significant Lean session. Let me plan:\n\nFirst, understand what session 038's ii′-6-a did (the Bn redefinition at the session-038 constants). The files involved:\n- LiSkeleton.lean: Bn (line ~2413), marginTilde (~754), marginExpanded + mK constant (~1819), marginTilde_eq_expanded, marginTilde_convex, marginTilde_eq_sub\n- LiBasePoints.lean: marginTilde_FAlt_80000_pos, marginTilde_FAlt_80000_lt_100000\n- lambda_pos_finite_height_verified (statement has N₁?)\n\nWait — also SfBoundProp's range: n ≤ (3×10¹²−1)² currently. With c = 1.5, N₁ = ((3×10¹²−1)/1.5)² = (2×10¹²−2/3)². Hmm: (3e12−1)/1.5 = 2e12 − 0.6667 — not an integer-friendly form. How is N₁ expressed in Lean? Probably as (3·10¹² − 1)² : ℝ or similar. With c = 1.5: N₁ = ((3·10¹²−1)/1.5)² = (2/3)²·(3·10¹²−1)² = (4/9)·(3e12−1)². That's still a clean real expression: (4/9)·(3·10¹²−1)² ≈ 4×10²⁴. Good — the Lean statement can use (4/9)·(3·10¹²−1)².\n\nAlso, the bridge needs T = 1.5√n+1 ≤ T₀ = 3×10¹²: n ≤ N₁ ⟹ √n ≤ (2/3)(3e12−1) = 2e12−0.667 ⟹ 1.5√n ≤ 3e12−1 ⟹ 1.5√n+1 ≤ 3e12 ✓.\n\nBut wait — actually, I need to think about what T appears in each Bn piece and how the Lean proofs use it:\n- Bridge 4·FAlt(T): needs every zero in Sz (|γ| ≤ T) on the line — hRH: RHUpTo (3×10¹²) — the bridge proof shows T ≤ 3×10¹² from n ≤ N₁.\n- Wide pigeonhole exists_zero_avoiding_T_wide: window [a, a+1] with a = 1.5√n: needs a ≥ 3 (✓ n ≥ 10⁵) and gives T ∈ [a, a+1] with clearance 1/(9·C0hat(a+2)). But in the Lean assembly, T is CHOSEN by the pigeonhole, so T ∈ [1.5√n, 1.5√n+1] — the Bn formula uses T = 1.5√n+1 as the worst case. Hmm — but the session-038 Bn used T = √n+1; the assembly must show the chosen T ≤ √n+1... fine, same pattern with 1.5.\n- The kernel bounds in Lean (ii′-4): kFun_shift_norm_le_exp_four_add_one (horizontals, re s ≤ 3, n ≤ im²) and kFun_shift_norm_le_exp_sqrt_half_add_one (right vertical, 2√n ≤ re s). The horizontal kernel bound at T: |k_n| ≤ e⁴+1 needs n ≤ im² (T² ≥ n). At T = 1.5√n: T² = 2.25n ≥ n ✓ the OLD lemma still applies! But the IMPROVED charge e^{16/9}+1 needs a new lemma: |k_n(s)| ≤ e^{4n/T²}+1 with T² = 2.25n gives e^{4/2.25}+1 = e^{16/9}+1. The ii′-4 lemma kFun_shift_norm_le_exp_four_add_one was proved for n ≤ im²; the generalized form e^{4n/im²}+1 or a new specialized lemma for 2.25n ≤ im² giving e^{16/9}+1.\n\nWait, hmm. Let me re-derive the kernel charge carefully. The (6.11) bound: |1+1/s| ≤ 1+(σ+1)/(σ²+T²) for s = σ+iT, σ ∈ [−3,3]. Then |k_n(s)| ≤ (1+4/(9+T²))ⁿ+1 (max at σ=3) ≤ e^{4n/(9+T²)}+1. At T² = n (old): ≤ e^{4n/(9+n)}+1 ≤ e⁴+1. At T² = 2.25n (new): ≤ e^{4n/(9+2.25n)}+1 ≤ e^{4/2.25}+1 = e^{16/9}+1 ✓ (since 4n/(9+2.25n) ≤ 4/2.25 = 16/9).\n\nSo the new kernel lemma: for 2.25·n ≤ (im s)², re s ∈ [−3,3] (or the kFun shift form): ‖kFun n s‖ ≤ e^{16/9}+1. Or more reusable: a general lemma giving e^{4n/im²}+1 and then specialize. The ii′-4 proof structure can be cloned.\n\nAlso the right-tail stepping lemma (Lemma 4.1 of D-073) will be needed in ii′-6-c — not in 6-a. Session 040 = steps 1–5 (Bn redefinition + convexity + base points + endpoint margins), per the sub-stage map. That's already a full Lean session (~like session 038's ii′-6-a).\n\nWhere are things in the Lean package? Let me look at the current definitions:\n- Bn in LiSkeleton.lean (~2413)\n- marginTilde (~754)\n- marginExpanded + mK (~1819)\n- marginTilde_eq_expanded\n- marginTilde_convex\n- marginTilde_eq_sub\n- Z2hat (added session 038)\n- LiBasePoints.lean: marginTilde_FAlt_80000_pos / _lt_100000\n- SfBoundProp (range endpoint (3×10¹²−1)²)\n- lambda_pos_finite_height_verified (uses N₁?)\n\nLet me check the current Lean files for these definitions. This is session 040 now. Actually — should I first write journal-039? The session boundary is fuzzy: the user said \"let's continue\" — session 039 continues into the main-line work, or do we call this session 040? STATE.md says \"Session 040: MAIN LINE EXECUTES REPAIR (R)\". I'll treat the continuation as session 040 and write journal-039 covering the ii′-6-R thread (it ended with the thread completion + user decision to continue). Actually the cleanest: journal-039 = the ii′-6-R side thread (through de1a805). Session 040 starts now with the ii′-6-a redo. I'll write journal-039 first (quick), then start Lean work.\n\nPlan for session 040 (ii′-6-a redo at c = 1.5):\n1. Read the relevant Lean definitions (Bn, marginTilde, marginExpanded/mK, Z2hat, convexity proof, base points, SfBoundProp range, N₁ usage).\n2. Redefine Bn at c = 1.5 with the D-073 correction comment.\n3. Lockstep marginTilde/marginExpanded/mK.\n4. Re-prove marginTilde_convex.\n5. Re-certify base points (new values: margins improve — E-033 says margin(8e4) = 111,917, margin(1e5) = 181,256 at c = 1.5 — wait, but those are vs the OLD margin values at session 038 which were... the session-038 certified values were ≥ 7,001/82,501 with true 14,682/90,653. At c = 1.5 the TRUE margins are 111,917/181,256 — much bigger! Because Bn shrinks near the base points (smaller horizontal charge). Wait, is that right? Let me sanity check: session-038 Bn at n = 1e5: Bn = 4F + 4T[...] + 40·1.3352 + 2(e⁴+1)(7Ĥ+3.3) + IIMaj + 1 with T = 317.2: I_III = 1268·[(2/3)·7.63 + 10.5·5.77+48] = 1268·[5.09+108.6] = 1268·113.7 = 144,200; horizontal = 2·55.6·(7·(3·51.1+108.6)+3.3) = 111.2·(7·261.9+3.3) = 111.2·1837 = 204,300; Bn ≈ 144,200+1,700+53+204,300+1 = 350,300 ✓ (matches Bn_FAlt_100000_le 350,000 from session 038!). margin = sInftyLower(1e5) − Bn + 1 = 433,744−350,000 = 83,744 ✓ (≈ the 82,501/90,653 ballpark ✓).\n\nAt c = 1.5: I_III = 1900·118.1 = 224,400 (BIGGER — T larger); horizontal = 2·6.92·(7·267.6+5.94) = 13.84·1879 = 26,000 (much smaller). Bn ≈ 224,400+1,880+53+26,000+1 = 252,300. margin = 433,744−252,300 = 181,400 ✓ matches E-033's 181,256.\n\nSo the horizontal shrink (204,300 → 26,000) beats the I_III growth (144,200 → 224,400). Margins improve hugely at the base: 83,744 → 181,256 at 10⁵; at 8×10⁴: session-038 margin was 14,682 (true) — at c = 1.5: 111,917 (E-033). The base-point certifications get much easier (bigger margins).\n\nBUT — the margin is NOT monotonically better everywhere: at N₁ the margin is 1.08e26 (c=1.5) vs 2.458e26 (old N₁ = 9e24) — smaller but the range is bigger. Fine.\n\n6. Re-run endpoint margins.\n7. Full build + axiom audit.\n\nAlso — the range endpoint change: SfBoundProp's `n ≤ (3·10¹²−1)²` becomes `n ≤ (4/9)·(3·10¹²−1)²` — wait, need to check how it's stated. And N₁ in lambda_pos_finite_height(_verified): the theorem says λ_n > 0 for n ≤ N₁. Let me read the Lean files.\n\nAlso — hmm, one more consideration: does anything else depend on T = √n+1? The LiAssembly.lean (sides identity + bridge) from ii′-6-b: the bridge zeroSum_norm_le_four_FAlt_add_one uses RHUpTo + T ≤ 3×10¹² — where does T come from there? The wide pigeonhole exists_zero_avoiding_T_wide gives T ∈ [a, a+1] for any a ≥ 3; the assembly instantiates a = √n presumably. At c = 1.5: a = 1.5√n. The assembly proofs may have the window baked — need to check LiAssembly.lean and LiContour.lean for how T is produced/consumed. The bridge proof: T ≤ 3×10¹² from n ≤ N₁: at the new N₁, 1.5√n+1 ≤ 3×10¹² needs n ≤ (4/9)(3e12−1)² ✓ consistent.\n\nSo the freeze-protocol exception scope: Bn redefinition (approved in the 8-step protocol) + the downstream margin proofs + (for 6-b/6-c adjustments, later) the assembly's pigeonhole instantiation. For session 040 (6-a redo): Bn, marginTilde, marginExpanded/mK, convexity, base points, SfBoundProp range, lambda_pos theorems' N₁ — the statement-level changes ripple to lambda_pos_finite_height_verified's statement (N₁ value) — hmm, that's a theorem statement change; the protocol covers it (the range change is part of the repair; D-073 consequence audit documents N₁ → 4e24).\n\nLet me start by reading the Lean files to find all the pieces. The Lean package is at research/lean/NguyenRH1/NguyenRH1/*.lean. Let me locate the definitions.\n\nThis will be a long session of careful Lean work. Let me set up todos and start reading.\n\nFirst: write journal-039 quickly (session 039 closure), then dive into session 040.\n\nActually — the journal: per §8.4 the journal is written at session END. Session 039 hasn't formally ended (user said continue). I'll write journal-039 when we stop or when session 040 ends. But the cleaner practice: since STATE.md says \"Session 040\", I'll write journal-039 now as closure of the side-thread session, and journal-040 at the end. Hmm — the user said \"let's continue\" — treating it as one continuous session is also fine, but STATE.md explicitly labels the next work \"Session 040\". I'll write journal-039 now (short — the side thread), commit, then proceed with session 040 work.\n\nLet me do that.","signature":""},{"type":"text","text":"Continuing into session 040 — main-line repair (R) at c = 1.5. First, closing out session 039's journal (the side-thread session), then starting the ii′-6-a redo."},{"type":"tool_use","id":"TodoWrite_54","name":"TodoWrite","input":{"todos":[{"content":"Write journal-039 (ii′-6-R side thread closure) + commit","status":"in_progress","activeForm":"Writing journal-039"},{"content":"Session 040: read Lean state (Bn, marginTilde, marginExpanded/mK, Z2hat, convexity, base points, SfBoundProp range, assembly T-instantiation)","status":"pending","activeForm":"Reading current Lean state"},{"content":"ii′-6-a redo step 1–2: Bn redefinition at c = 1.5 + D-073 correction comment; lockstep marginTilde/marginExpanded/mK; N₁ = (4/9)(3·10¹²−1)²","status":"pending","activeForm":"Redefining Bn at c = 1.5"},{"content":"ii′-6-a redo step 3: re-prove marginTilde_convex at the new constants","status":"pending","activeForm":"Re-proving margin convexity"},{"content":"ii′-6-a redo steps 4–5: re-certify base points (8×10⁴, 10⁵) + endpoint margins at N₁ = 4×10²⁴","status":"pending","activeForm":"Re-certifying base points"},{"content":"Full build green + axiom audit + journal-040 + commit ii′-6-a-redo","status":"pending","activeForm":"Building and auditing"}]},"extra_content":null}]