[{"type":"thinking","thinking":"The LiBasePoints structure: Bn_FAlt_80000_le (upper bound on Bn at 80000, ≤ 329000), Bn_FAlt_80000_ge (lower, ≥ 274000), sInftyLower bounds, then margin = sInfty − Bn + 1: margin(8e4) = sInftyLower(8e4) − Bn + 1: lower via sInfty_lo − Bn_up: 336000 − 329000 = 7000 → ≥ 7001; upper via sInfty_up − Bn_lo: 339000−274000 = 65000. And at 1e5: margin ≥ sInfty_lo(1e5) − Bn_up(1e5): 432500 − 350000 = 82501. Then marginTilde_FAlt_80000_lt_100000 : 65000 < 82501 ✓.\n\nAt c = 1.5, the Bn pieces change: T = 1.5√n+1 (√80000 ≈ 282.8 → T ≈ 425.2; √100000 ≈ 316.2 → T ≈ 475.3), kernel e^{16/9}+1 ≈ 6.917, R = 5.94, 1.5·IIMaj.\n\nThe certification theorems to rebuild:\n- sqrt bounds: √80000 ∈ [282, 283] ✓ exists; then 1.5√80000+1 ∈ [1.5·282+1, 1.5·283+1] = [424, 425.5]. Need log T bounds: log(1.5·√80000+1) ∈ [log 424, log 425.5] ≈ [6.0497, 6.0533] — certify [6.04, 6.06] say. New log lemmas for T = 1.5·sqrt+1 and T+2 and loglog.\n- Bn_FAlt_80000_le/ge at the new Bn.\n- Same at 100000.\n- The final margin theorems.\n\nThis is a substantial rebuild of LiBasePoints' section 3+ (the Bn evaluations). The good news: the true margins are much bigger (111,917 vs 14,682; 181,256 vs 90,653), so the certified bounds can be much coarser — e.g. margin(8e4) ≥ 100000 (11% below true), margin(1e5) ≥ 170000, margin(8e4) ≤ 125000. The coarse slack makes the interval arithmetic comfortable (the old file used ~500-unit coarse bounds like 329000/274000 with similar coarseness).\n\nEstimate Bn(8e4) at c = 1.5: T = 1.5·282.84+1 = 425.26: \n- bridge 4FAlt(T): FAlt(425) ≈ 425·6.052/6.283 + 0.137·6.052 + 0.443·1.80 + 2.463 = 409.2 + 0.829 + 0.797 + 2.463 ≈ 413.3; ×4 = 1653.\n- I_III: 4·425.26·[(2/3)·Z2 + 10.5·log 427.26 + 48]: Z2(425) = 0.6·6.057 + 0.886·1.801 + 2.7 = 3.634+1.596+2.7 = 7.930; (2/3)·7.930 = 5.287; 10.5·6.057 = 63.60; +48 → 116.9; ×1701 = 198,840.\n- 40·1.3352 = 53.4.\n- horizontal: 2·6.917·(7·Ĥ + 5.94): Ĥ = 3·Ĉ₀² + 10.5·6.057+48: Ĉ₀ = 0.433·6.057+0.886·1.801+2.9 = 2.623+1.596+2.9 = 7.119; 3·50.68 = 152.0; +63.60+48 = 263.6; 7·263.6 = 1845.2; +5.94 = 1851.1; ×13.834 = 25,610.\n- 1.5·IIMaj ≤ 1.5·4.56·283.84·2 ≈ 3883 (bound; true much less).\n- +1.\nBn ≈ 1653+198840+53+25610+3883+1 = 230,040. True-ish margin: sInftyLower(8e4) = 0.5·8e4·log(8e4) − 1.41894·8e4 − 4 = 40000·11.2908 − 113515 − 4 = 451,632−113,519 = 338,113. margin ≈ 338,113−230,040+1 = 108,074. Hmm — E-033 said 111,917 (its IIMaj_c is the exact form, and its Bn uses the exact e^{16/9}+1 = 6.92249 and R = 5.93924; also FAlt exact). My hand-arithmetic has ~1-2% error; whatever — the certified bound target: margin(8e4) ≥ 100,000 (comfortable); ≤ 120,000 (for hbase2: 120,000 < margin(1e5)-lower).\n\nBn(1e5) at c = 1.5: T = 475.34:\n- bridge ≈ 4·[475.34·6.1664/6.283 + 0.137·6.1664 + 0.443·1.8195 + 2.463] = 4·[466.5+0.845+0.806+2.463] = 4·470.6 = 1882.\n- I_III: 4·475.34·[(2/3)·8.015 + 10.5·6.1664+48] = 1901.4·[5.343+112.75] = 1901.4·118.09 = 224,560.\n- horizontal: Ĥ = 3·(0.433·6.1664+0.886·1.8195+2.9)² + 10.5·6.1664+48 = 3·(2.670+1.612+2.9)² + 112.75 = 3·7.182²+112.75 = 3·51.58+112.75 = 267.5; 7·267.5 = 1872.5; +5.94 = 1878.4; ×13.834 = 25,986.\n- 1.5·IIMaj ≤ 1.5·4.56·317.23·2 ≈ 4341.\n- total Bn ≈ 1882+224560+53+25986+4341+1 = 256,823.\nsInftyLower(1e5) = 50000·11.5129 − 141894 − 4 = 575,646−141,898 = 433,748. margin ≈ 433,748−256,823+1 = 176,926. E-033 says 181,256 — again my hand-arithmetic is off by ~2% (mostly the I_III coefficient). Anyway: certify margin(1e5) ≥ 165,000 (comfortable) and margin(8e4) ≤ 125,000... wait — hbase2 needs margin(8e4) < margin(1e5): upper bound on margin(8e4) vs lower on margin(1e5): 125,000 < 165,000 ✓ comfortable.\n\nTargets:\n- marginTilde_FAlt_80000_pos : 0 < marginTilde FAlt 80000 — via margin ≥ 100,000.\n- marginTilde_FAlt_80000_lt_100000 : marginTilde FAlt 80000 < marginTilde FAlt 100000 — via ≤ 125,000 < ≥ 165,000.\n\nGiven the machinery, the cleanest: prove Bn_FAlt_80000_le (≤ 237000), Bn_FAlt_80000_ge (≥ 210000), Bn_FAlt_100000_le (≤ 268000), Bn_FAlt_100000_ge (≥ 235000)?? Let me size against the true values (E-033: margin(8e4) = 111917 → Bn(8e4) = 338114 − 111917 + 1 = 226,198 (E-033-exact). margin(1e5) = 181256 → Bn(1e5) = 433,744 − 181,256 + 1 = 252,489.)\n\nSo Bn(8e4) ≈ 226,198; Bn(1e5) ≈ 252,489. Certify:\n- Bn_FAlt_80000_le : Bn ≤ 236,000 (slack ~10k for the IIMaj 1.5×4.56(√n+1)·2 coarse bound ~3,883 + log bounds coarseness ~1-2%)\n- Bn_FAlt_80000_ge : Bn ≥ 213,000\n- margin(8e4) ∈ [338113−236000+1, 339000−213000+1] = [102,114, 126,001] — wait sInftyLower_80000_up is 339000 (existing). margin upper: sInfty_up − Bn_lo + 1 = 339000 − 213000 + 1 = 126,001. Lower: 336000 − 236000 + 1 = 100,001.\n- Bn_FAlt_100000_le : Bn ≤ 262,000; margin(1e5) ≥ 432500 − 262000 + 1 = 170,501.\n- hbase2: 126,001 < 170,501 ✓.\n\nThese slacks are huge (10k+) — the interval arithmetic will be comfortable. \n\nNow — the existing LiBasePoints theorems Bn_FAlt_80000_le/ge etc. reference the OLD Bn (T = √n+1, e⁴+1, 3.3, 1×IIMaj). After the Bn redefinition these theorems' PROOFS break (unfold Bn + arithmetic). I must rebuild them. The file is ~700 lines; the Bn evaluation sections are ~247–650. The rebuild: change T-bounds (1.5·sqrt+1 ∈ [424, 425.5] / [474.5, 476]), the log bounds for T and T+2 (new values: log T ∈ [6.04, 6.06]-ish, log(T+2) similar, loglog ∈ [1.79, 1.82]), the kernel charge e^{16/9}+1 bounds (e^{16/9} = e·e^{7/9}: 16/9 = 1.7778: e^{1.7778} ∈ [5.9, 5.93]: e^{16/9}+1 ∈ [6.9, 6.93]), R = 5.94 exact, IIMaj ×1.5.\n\nLet me now also check: exp (16/9) certified bounds: 16/9 = 1.77777…; e^{1.7778} = e·e^{0.7778}: e^{0.7778} via the (1−x/16)⁻¹⁶ trick upper: (1−0.7778/16)⁻¹⁶ = (0.95139)⁻¹⁶ = 1.05137^{16}... hmm (1/(1−0.04861))^16 = (1.05114)^16 ≈ e^{16·0.04986} = e^{0.7978} ≈ 2.2207. Lower via partial sum Σ₀⁷ 0.7778^i/i! ≈ 2.1764. So e^{0.7778} ∈ [2.1764, 2.2207] — then e^{1.7778} = e·that ∈ [2.71828·2.1764, 2.71828·2.2207] = [5.9152, 6.0356] — too coarse (2% spread). Better: split 16/9 = 1.5 + 0.2778: e^{1.5} ∈ [4.4816, 4.4818] (e·e^{0.5} with e^{0.5} certified tight: Σ partial sums give e^{0.5} to 1e-6 — actually e^{1.5} = e^1·e^{0.5}; e^{0.5} = 1.64872 certified by partial sums to ~1e-7); e^{0.2778}: upper (1−0.2778/16)⁻¹⁶ = (0.98264)⁻¹⁶ = 1.01769^16 = e^{16·0.01753} = e^{0.2806} = 1.3239; lower Σ₀⁶ 0.2778^i/i! = 1.3202. So e^{16/9} ∈ [4.4817·1.3202, 4.4818·1.3239] = [5.9169, 5.9339] → e^{16/9}+1 ∈ [6.9169, 6.9339]. True: 6.92249. Certify e^{16/9}+1 ≤ 6.94 (slack 0.017, 0.25%) and ≥ 6.91. For the Bn upper bound use 6.94·... hmm — the horizontal charge in Bn: 2·(e^{16/9}+1)·(7Ĥ+5.94) ≤ 2·6.94·(7·Ĥ_up+5.94). Fine.\n\nNow — big remaining question: the LiBasePoints theorems are ~400 lines of certified bounds; the rebuild touches the T-dependent bounds. The cleanest approach: rewrite the Bn-evaluation section fresh with the new constants (new log lemmas for the new T values) — most of the exp/log machinery (section 1–2, the e¹¹, e^{0.263}, log80000, sqrt lemmas, sInftyLower bounds) is T-INDEPENDENT and reusable ✓. Only the Bn_FAlt_* theorems + the T-log lemmas need redoing.\n\nAlso — marginTilde_eq_sub and marginTilde_eq_expanded: `ring`-level bookkeeping — the marginTilde/Bn change shape identically (both get T := 1.5√n+1 etc.) so the ring proofs should still close (the blocks were DESIGNED to match the expansion; the mK constant changes on both sides consistently; IIMaj ×1.5 consistent).\n\nWait — actually marginTilde_eq_sub is `simp only [marginTilde, Bn, sInftyLower]` — it closes by rfl after unfolding?? The proof is just `simp only` — so marginTilde = sInftyLower − Bn + 1 holds DEFINITIONALLY (the marginTilde def literally is sInfty-expression − Bn-expression + 1). After my edits both change consistently ✓.\n\nmarginTilde_eq_expanded: funext + simp only [unfold all] + ring — the blocks mFblock etc. must match the marginTilde expansion EXACTLY. I'm changing blocks and marginTilde consistently; ring verifies.\n\nOK. Now the last piece before planning the edit sequence: the hTn in marginTilde_convex (line 2189): Real.sqrt 20 + 1 ≤ Real.sqrt n + 1 — used for hψ (needs ψ[F] negative at T = √n+1) AND the block ψ-negativity lemmas psiOf_*_shift_neg (which are stated at T = √n+1? let me check psiOf_T_mul_log_shift_neg — these \"shift_neg\" lemmas are presumably about psiOf φ (√n+1) < 0 — at the new T = 1.5√n+1 I need psiOf φ (1.5√n+1) < 0 — the lemmas psiOf_*_shift_neg take hTn : √20+1 ≤ √n+1 as hypothesis? From hnn2: psiOf_T_mul_log_shift_neg hTn — let me check its signature. It probably states: ∀ T ≥ something, psiOf (t·log(t+2)) T < 0 — let me grep. If the ψ-negativity lemmas are T-generic (∀ T ≥ T₀), they apply at 1.5√n+1 directly ✓. Let me check.","signature":""},{"type":"text","text":"Checking whether the ψ-negativity lemmas are T-generic (they need to apply at T = 1.5√n+1)."},{"type":"tool_use","id":"Grep_75","name":"Grep","input":{"pattern":"theorem psiOf_\\w*shift_neg|theorem FAlt_psi_neg|theorem contDiffOn_FAlt","path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","output_mode":"content","-n":true,"-A":3},"extra_content":null}]