[{"type":"thinking","thinking":"The convexity proof structure is fully clear:\n- `vF`: mFblock's deriv2 = (−2)·(psiOf F/(4n)) — with mFblock := (−4)·F, this becomes (−4)·(psiOf F/(4n)); hnn1: nonneg because the coefficient (−2 or −4) is nonpos and psiOf F < 0 → the `by norm_num` discharges the coefficient-sign: `mul_nonneg_of_nonpos_of_nonpos (by norm_num : (-2:ℝ) ≤ 0) ...` — with −4, `by norm_num : (-4:ℝ) ≤ 0` works the same ✓ mechanical.\n- `vTL1`: mTL1b's coefficient ((8/3)*0.433+42) → ((8/3)*0.6+42): appears in vTL1's equation (rw [deriv2_const_mul _ ...] — the constant is inferred from the unfolded def) and hnn2's `by norm_num` (the constant is negative either way ✓).\n- `vId`: mIdb's ((8/3)*2.9+192) → ((8/3)*2.7+192): appears in vId's `show (fun m => -(8/3*2.9 + 192) * ...)` — the show must be updated to the new constant; hnn4's norm_num ✓.\n- All other rows unchanged.\n\nSo the convexity proof is almost mechanical: the constant changes propagate via the unfold + deriv2_const_mul, and the sign haves use `by norm_num` which handles any specific numeral. The only text edits: the def constants (mFblock/mTL1b/mIdb), the `show` in vId, and possibly hnn1/hnn2/hnn4's hardcoded constants if they're written literally (hnn1 has `-2 * ...` literal in the statement — update to -4; hnn2 has `(-((8/3)*0.433 + 42))` literal — update; hnn4 has `(-((8/3)*2.9 + 192))` literal — update).\n\nLet me also check marginTilde_eq_expanded (simp only + ring — should survive verbatim since it unfolds the defs) and marginTilde_eq_sub (simp only [marginTilde, Bn, sInftyLower] — survives).\n\nAnd check what `hnn1` looks like after the F change — vF's statement: `deriv (deriv (mFblock F)) n = (-2) * (psiOf F (Real.sqrt n + 1) / (4 * n))` — the (-2) is deriv2_const_mul's constant extracted from the def — after changing mFblock to (-4), the unfold+rw produces (-4). The vF statement text has the literal (-2) — must edit to (-4). Similarly vTL1/vId literals.\n\nThen LiBasePoints: marginTilde_FAlt_80000_pos (0 < marginTilde FAlt 80000) and marginTilde_FAlt_80000_lt_100000 (marginTilde FAlt 80000 < marginTilde FAlt 100000). These are certified by rational interval bounds — the certified values were margiñ(8×10⁴) ≥ 8001 (true 15,764) and margiñ(10⁵) ≥ 85,501. With the corrected Bn:\n- margiñ(8×10⁴)_new = margiñ_old + (Bn_old − Bn_new)(8×10⁴). Bn_old − Bn_new = (2F − 4F) + 4T·(2/3)(Ĉ₀−Z2) = −2F + 4T·(2/3)·(2.9−2.7 + (0.433−0.6)·L) = −2F + 4T·(2/3)·(0.2 − 0.167L). At n = 8×10⁴, T = 283.84, L = log(285.84) = 5.655: = −2·259.2 + 4·283.84·(2/3)·(0.2−0.944) = −518.4 + 757.6·(−0.744) = −518.4 − 563.7 = −1082. So Bn_new(8×10⁴) = Bn_old + 1082 → margiñ_new(8×10⁴) = margiñ_old − 1082. Wait — the corrected Bn is LARGER at 8×10⁴? Let me recompute with E-030's numbers: at n=10⁵: Bn_baked = 341,861.50, Bn_honest = 343,096.52 — the corrected Bn is LARGER by 1,235 at 10⁵! But margin_honestBn(10⁵) = 90,652.75 vs margin_bakedBn(10⁵) = 91,887.77 — the honest margin is SMALLER by 1,235. Hmm! Earlier I said \"honest B is smaller near 10⁵\" — that was the honest ASSEMBLY (224k) vs baked Bn (342k). The corrected Bn (which is what margiñ uses) is BIGGER than baked Bn by the killers (+2F + I_III correction) minus nothing. So margiñ with corrected Bn DROPS by ≈ 1,082 at 8×10⁴ and ≈ 1,235 at 10⁵. The certified bounds were ≥ 8001 (true 15,764) and ≥ 85,501 — after the drop: true values ≈ 14,682 and ≈ 84,266 — both still comfortably above the certified thresholds. Good — the re-certification with the same thresholds should work (maybe even reuse the same certified constants since the interval bounds had slack — but the value changed so the computation must be redone; the slack was large).\n\nWait — E-030's margin columns: at n=10⁵: margin_honestBn = 90,652.75, margin_bakedBn = 91,887.77. The certified margiñ(10⁵) ≥ 85,501 — new true margin 90,652 > 85,501 ✓ still certifiable with the same threshold. At 8×10⁴: not in E-030's grid; compute: sInftyLower(8×10⁴) = 0.5·80000·log(80000) − 1.41894·80000 − 4 = 40000·11.2898 − 113,515 − 4 = 451,592 − 113,519 = 338,073. margiñ_old true = 15,764 → Bn_old(8×10⁴) = 338,073 + 1 − 15,764 = 322,310. Bn_new = 322,310 + 1,082 = 323,392. margiñ_new = 338,074 − 323,392 = 14,682 ≥ 8001 ✓.\n\nOK. Now execute. Order:\n1. Edit LiSkeleton: Z2hat def + Bn + marginTilde + mFblock/mTL1b/mIdb + vF/vTL1/vId literals + hnn1/hnn2/hnn4 literals.\n2. Build LiSkeleton — check marginTilde_eq_expanded/eq_sub/convex.\n3. LiBasePoints: check if the certified bounds still go through (they compute the margin value from marginTilde's def — the def changed so the arithmetic changes; the proofs use interval arithmetic with specific rational bounds — need to check how robust: marginTilde_FAlt_80000_pos proves 0 < marginTilde FAlt 80000 via lower-bounding each component; the changed components (4F, (2/3)Z2) enter with their interval bounds — the proofs will break where the specific old coefficients are used. Let me look at LiBasePoints' structure for those proofs after the LiSkeleton edit goes through.)\n\nLet me start the edits.","signature":""},{"type":"text","text":"The structure is fully mapped. Executing the edits: Z2hat def + Bn + marginTilde + the three changed blocks + their ψ-row literals."},{"type":"tool_use","id":"Edit_201","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","old_string":"/-- The honest closed-form unit-interval zero-count majorant (D-056 §2):\nĈ₀(t) = 0.433 log(t+2) + 0.886 log log(t+2) + 2.9. -/\ndef C0hat (t : ℝ) : ℝ :=\n  0.433 * Real.log (t + 2) + 0.886 * Real.log (Real.log (t + 2)) + 2.9\n\n/-- H majorant with A = 10.5, B₀ = 48 (D-056). -/\ndef Hhat (T : ℝ) : ℝ :=\n  3 * C0hat T ^ 2 + 10.5 * Real.log (T + 2) + 48","new_string":"/-- The honest closed-form unit-interval zero-count majorant (D-056 §2):\nĈ₀(t) = 0.433 log(t+2) + 0.886 log log(t+2) + 2.9. -/\ndef C0hat (t : ℝ) : ℝ :=\n  0.433 * Real.log (t + 2) + 0.886 * Real.log (Real.log (t + 2)) + 2.9\n\n/-- The two-unit-window zero-count majorant Z₂(t) = 0.6 log(t+2) +\n0.886 log log(t+2) + 2.7 (the `ZeroCountProp2` bound form, D-069 §2).\nNamed for use in the corrected `Bn` (D-071 F-B): the I_III near-sum's\nwindow |γ−t| < 1 is a two-unit window, honestly counted by Z₂, not Ĉ₀.\nDefinitionally the `ZeroCountProp2` closed form (that Prop's statement\nis untouched). -/\ndef Z2hat (t : ℝ) : ℝ :=\n  0.6 * Real.log (t + 2) + 0.886 * Real.log (Real.log (t + 2)) + 2.7\n\n/-- H majorant with A = 10.5, B₀ = 48 (D-056). -/\ndef Hhat (T : ℝ) : ℝ :=\n  3 * C0hat T ^ 2 + 10.5 * Real.log (T + 2) + 48"},"extra_content":null}]