[{"type":"thinking","thinking":"The ψ-negativity lemmas are T-generic: they take T with hypotheses like `Real.sqrt 20 + 1 ≤ T` or `1 < T`. At T = 1.5√n+1 (n ≥ 20): 1.5√n+1 ≥ 1.5√20+1 ≥ √20+1 ✓ (since 1.5√20 ≥ √20) — so `√20+1 ≤ 1.5·√n+1` follows from √n ≥ √20 by linarith. ✓ All reusable.\n\nFAlt_psi_neg (2607): `deriv (deriv FAlt) T − deriv FAlt T \\ (T−1) < 0` — wait, there's a backslash `\\` at line 2608: `deriv FAlt T \\ (T - 1)` — that's set-difference notation?? No wait, that's probably a display artifact of the grep — it's `/` presumably... Actually the grep output shows `\\` — could be a typo in the file?? No — marginTilde_convex's hψ hypothesis uses `deriv (deriv F) T - deriv F T / (T - 1) < 0` ✓ — FAlt_psi_neg must match that form. The grep -A3 probably mangled it. Fine.\n\nAlso block_range_bounds — T-generic with hT : √20+1 ≤ T ✓.\n\nEverything checks out. Now the full edit plan for the ii′-6-a redo (c = 1.5):\n\n**LiSkeleton.lean edits:**\n\n1. **New k-shift infrastructure** (add near 1715–1753):\n   - `contDiffOn_ksqrt_shift (k : ℝ) (hk : 0 < k) : ContDiffOn ℝ 2 (fun m => k * Real.sqrt m + 1) (Set.Ioi 20)`\n   - `mapsTo_ksqrt_shift (k) (hk : 0 < k) : MapsTo (fun m => k*√m+1) (Ioi 20) (Ioi (-1))`\n   - `mapsTo_ksqrt_shift_Ioi1 (k) (hk : 0 < k) : MapsTo ... (Ioi 1)`\n   - `mapsTo_ksqrt_shift_Ici_Ioi1 (k) (hk : 0 < k) : MapsTo (Ici 20) (Ioi 1)`\n   - `psiOf_sqrt_krel (φ) (k n) : psiOf φ (k*√n+1) = deriv (deriv φ) (k*√n+1) − deriv φ (k*√n+1)/(k*√n)` — hmm wait, psiOf has /(T−1) and T−1 = k√n ✓ so the statement is psiOf φ (k√n+1) = φ″ − φ′/(k√n) ✓.\n   - `chain_rule_sqrt_kshift_on (φ) {k : ℝ} (hk : 0 < k) {c : ℝ} (hc : c ≤ 1) (hφ : ContDiffOn ℝ 2 φ (Set.Ioi c)) (n : ℝ) (hn : 0 < n) : deriv (deriv fun n => φ (k * Real.sqrt n + 1)) n = (k^2) * (deriv (deriv φ) (k*√n+1) − deriv φ (k*√n+1)/(k*√n)) / (4*n)`.\n\n2. **IIMaj strengthening**:\n   - iimaj_bracket_bound: `< 2` → `< 4 / 3` (last line norm_num; the certified product 4.56·1.2237·0.2124·1.1099 = 1.3155 < 1.3333 ✓).\n   - IIMaj_deriv2_lt: keep (harmless) + add `IIMaj_deriv2_lt_three : deriv (deriv IIMaj) n < 1/(3*n)` (bracket < 4/3 ⟹ /(4n) = 1/(3n)).\n\n3. **marginTilde** (766): T := 1.5·√n+1; (Real.exp (16/9) + 1); 5.94; 1.5·IIMaj. Update the doc comment (D-073 correction pointer, protocol step 2).\n\n4. **marginExpanded blocks** (1815–1832): inner fun → 1.5√m+1; e⁴+1 → e^{16/9}+1; mIIM → (−1.5)·IIMaj; mK → new constant. Update the section comments.\n\n5. **marginTilde_eq_expanded**: simp list unchanged (same def names); ring re-checks.\n\n6. **contDiffOn_m*** lemmas: update to the k-shift infrastructure (replace contDiffOn_sqrt_shift/mapsTo_sqrt_shift with the k-versions at k = 1.5). The unfold of each block matches (blocks now have 1.5·√m+1).\n\n7. **continuousOn_marginExpanded**: hT becomes 1.5√m+1 (hsqrt.const_mul + add const); hmaps via mapsTo_ksqrt_shift_Ici_Ioi1.\n\n8. **marginTilde_convex**: \n   - hCDFc/hCDTL1c/... composition bundles via the k-shift mapsTo.\n   - hTn : √20+1 ≤ 1.5√n+1.\n   - v-blocks: chain_rule_sqrt_kshift_on + ← psiOf_sqrt_krel; values become (−w)·(1.5^2·(psiOf φ T/(4n))) — wait: chain_rule gives 1.5²·(φ″−φ′/(1.5√n))/(4n); psiOf_sqrt_krel rewrites (φ″−φ′/(1.5√n)) = psiOf φ T... direction: psiOf_sqrt_krel : psiOf φ (k√n+1) = φ″ − φ′/(k√n) — so `← psiOf_sqrt_krel` rewrites (φ″ − φ′/(k√n)) → psiOf ✓ same pattern.\n   - hnn* nonneg lemmas: adapt with the 1.5² factor (mul_nonneg with the extra positive 1.5² — I'll write them as: 0 ≤ (−w)·(1.5^2·(psiOf φ T/(4n))) — prove: psiOf/(4n) < 0 ⟹ 1.5²·(that) < 0 (mul_neg_of_pos_of_neg (by norm_num) ...) ⟹ (−w)·neg ≥ 0.\n   - vIIM: (−1.5)·IIMaj″; final: linarith [hIIM3, hnn*] with hIIM3 : IIMaj″ < 1/(3n).\n   - The hψ hypothesis form: unchanged (∀ T ≥ √20+1) ✓ — applied at 1.5√n+1: hψ _ (show √20+1 ≤ 1.5√n+1 by ...) ✓.\n\n9. **Bn** (2431): same changes as marginTilde + doc comment (protocol step 2: D-073 correction comment).\n\n10. **SfBoundProp** (2456): endpoint `(3 * 10 ^ 12 - 1 : ℝ) ^ 2` → `((2 / 3) * (3 * 10 ^ 12 - 1 : ℝ)) ^ 2` — the new N₁ = (2/3)²·(3e12−1)² = 4×10²⁴. Update doc comment.\n\n11. **lambda_pos_finite_height** (2766): endpoint + doc comment update.\n\n**LiBasePoints.lean edits:**\n12. Bn_FAlt_80000_le / Bn_FAlt_80000_ge / Bn_FAlt_100000_le / Bn_FAlt_100000_ge — rebuild at the new Bn with new T-log lemmas.\n13. marginTilde_FAlt_80000_pos / marginTilde_FAlt_80000_lt_100000 — reassemble.\n14. lambda_pos_finite_height_verified (655): endpoint change.\n\nLet me read the rest of LiBasePoints (247–720) to see the full Bn-evaluation section I need to rebuild, and 600–700 for the margin theorems + verified.\n\nThis is a big session. Let me organize into sub-stages:\n- 6-a-redo-i: LiSkeleton infra (k-shift lemmas + bracket strengthening + IIMaj_lt_three) — build check.\n- 6-a-redo-ii: Bn/marginTilde/blocks/mK redefinition + eq_expanded/eq_sub + contDiffOn/continuousOn + convexity re-proof — build check.\n- 6-a-redo-iii: SfBoundProp/lambda endpoints + LiBasePoints rebuild + verified — build + audit + commit.\n\nLet me read the remaining LiBasePoints sections now.","signature":""},{"type":"text","text":"All ψ-lemmas are T-generic — reusable. Now reading the full LiBasePoints Bn-evaluation section to scope its rebuild."},{"type":"tool_use","id":"Read_76","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean","offset":183,"limit":220},"extra_content":null}]