[{"type":"thinking","thinking":"The mathematician's notes say:\n\n1. **Consistency item**: margin number has three values in the trail: 16,020 (E-028, original constants) → 15,764 (post-FAlt) → >9,960 (certified). The paper's §3.1 still says 16,020. Make the main-text number match whichever bound the Lean theorem actually consumes, with the trail in a footnote. — Action: check draft.tex §3.1 for the 16,020 number and fix with footnote trail.\n\n2. **hbase/hbase2 discharge** (~400–600 lines, pre-verified, low risk): when done, five hypotheses become three, all perimeter-class. That's the moment the headline claim upgrades. \"Consider holding the paper's formal-status paragraph until then; it's close.\" — So: DO the hbase/hbase2 discharge NOW (that's the clear advice: \"Discharge hbase/hbase2 next (cheap, pre-verified)\").\n\n3. Standing objections to Part 2 (Suzuki/valley): Theorem 3's \"modulo standard tempered-FT bookkeeping\" and o(eᵃ) in its statement, L3/L4 open, authorship policy. Not touched by this session. (Not actionable now — noted.)\n\n4. Bottom line: discharge hbase/hbase2 next, reconcile the margin-number trail in the main text, and Paper A's formal-verification story is complete. Tier ii′ (rectangle contour) is post-publication work — don't let it delay the paper.\n\nSo the clear directives:\n- **A. Discharge hbase/hbase2** (tier (ii) step 2) — the main task now.\n- **B. Fix the margin-number trail in draft.tex §3.1** (16,020 → 15,764 with footnote trail, or match the Lean-consumed bound).\n- C. Note the Part-2 objections as standing (no action).\n- D. Tier ii′ is post-publication — deprioritize.\n\nLet me start. First B (quick paper fix), then A (the interval grind — the big one).\n\nFor B: find where §3.1 says 16,020. Let me grep draft.tex for 16 020 / 16020 / 16,020.\n\nFor A: the plan from D-062 §5. Let me structure the Lean work:\n\nGoal 1: `theorem marginTilde_FAlt_80000_pos : 0 < marginTilde FAlt 80000`\nGoal 2: `theorem marginTilde_FAlt_80000_lt_100000 : marginTilde FAlt 80000 < marginTilde FAlt 100000`\nThen rewrite lambda_pos_finite_height to use them (drop hbase/hbase2 hypotheses — the theorem then depends only on hRH/hfinite/hSf).\n\nThe bounds needed (from D-062 §5, all pre-verified):\n\nFor margin(80000) > 0: slower(80000) > Bn(80000) — via slower ≥ 336998 and Bn ≤ 327038.\n- slower(80000) = (1/2)(80000) log 80000 + (C₁−γ/2)·80000 − 4 where (C₁−γ/2) = −(1+log 2+log π)/2 exactly (γ cancels).\n  Lower: log 80000 ≥ 11.263; upper for the subtracted coefficient: (1+log2+logπ)/2 ≤ (1+0.6931472+1.1448)/2 = 1.4189736.\n  slower ≥ 40000·11.263 − 1.4189736·80000 − 4 = 450520 − 113517.888 − 4 = 336998.1.\n- Bn(80000) upper: T = √80000+1 ≤ 284 (√80000 ≤ 283, 283² = 80089).\n  · FAlt(T) ≤ 0.1591551·(284·5.7) + 0.137·5.7 + 0.443·1.75 + 2.463 ≤ 261.66 → 2F ≤ 523.32\n    Need 1/(2π) ≤ 0.1591551: 2π ≥ 6.283185307 ⟹ 1/(2π) ≤ 0.159155. Check: 1/6.283185307 = 0.15915494...; 0.1591551 ≥ 0.15915494 ✓ margin 0.0001% — exact rational, fine. Actually let me use 0.15916 for comfort: 1/(2π) ≤ 0.1592? 1/6.283185307 = 0.1591549431. Use 0.1592 (margin 0.03%). FAlt ≤ 0.1592·1618.8 + ... = 257.71+... recompute: 0.1592·284·5.7 = 0.1592·1618.8 = 257.71; +0.7809+0.77525+2.463 = 261.73; 2F ≤ 523.46.\n  · Ĉ₀(T) ≤ 0.433·5.7 + 0.886·1.75 + 2.9 = 6.9186\n  · 4T[...] ≤ 1136·[(2/3)(6.9186) + 10.5·5.7 + 48] = 1136·112.4624 = 127757.3\n  · Ĥ ≤ 3·(6.9186)² + 10.5·5.7 + 48 = 143.60 + 107.85 = 251.45\n  · 2(e⁴+1)(7Ĥ+3.3) ≤ 2·55.6015·(7·251.45+3.3) = 111.203·1763.45 = 196103.6 — hmm need e⁴+1 ≤ 55.6015: e⁴ ≤ 54.6015 ✓.\n  · 40·1.3352 = 53.408\n  · IIMaj(80000) ≤ 4.56·285·2 = 2599.2 — wait: IIMaj n = psiTilde(√n) = 4.56·(s+1)·(e^{−as}+e^{−bs}) ≤ 4.56·284·2·... s = √80000 ≤ 283: s+1 ≤ 284; e terms ≤ 1 each. ≤ 4.56·284·2 = 2589.5.\n  · +1\n  Total ≤ 523.46 + 127757.3 + 53.408 + 196103.6 + 2589.5 + 1 = 327028.3 ✓ ≤ 336998.1 − ... margin ≥ 336998.1 − 327028.3 + 1 = 9970.8 > 0 ✓.\n\nFor margin(80000) < margin(100000):\n- margin(80000) upper: slower(80000) ≤ 40000·11.3 − 1.4165736·80000 − 4 (lower coefficient: (1+log2+logπ)/2 ≥ (1+0.6931471+1.14)/2 = 1.41657355) = 452000 − 113325.884 − 4 = 338670.1.\n  Bn(80000) lower: T ≥ 283 (T = 283.84; √80000 ≥ 282 ⟸ 282² = 79524 ≤ 80000 ✓ so T ≥ 283):\n  · 4T[...] ≥ 4·283·[(2/3)(0.433·5.6 + 0 + 2.9) + 10.5·5.6 + 48]: log(T+2) ≥ log 285 ≥ 5.6; Ĉ₀ ≥ 0.433·5.6 + 0.886·(log log ≥ 0 — careful: log log(T+2) ≥ 0 ⟸ log(T+2) ≥ 1 ✓ so Ĉ₀ ≥ 0.433·5.6 + 0 + 2.9 = 5.3248) ⟹ ≥ 1132·[3.5499 + 58.8 + 48] = 1132·110.3499 = 124916.3\n  · 2(e⁴+1)(7Ĥ+3.3) ≥ 2·55.5978·(7·191.87+3.3): Ĥ ≥ 3·(5.3248)² + 10.5·5.6 + 48 = 85.07+58.8+48 = 191.87; 7Ĥ+3.3 ≥ 1346.39; ≥ 111.1956·1346.39 = 149715.6\n  · other terms ≥ 0: 2FAlt(T) ≥ 0? FAlt(T) = (1/2π)T log T + 0.137 log T + 0.443 log log T + 2.463 ≥ 0 for T ≥ 1 ✓ (each term ≥ 0: T log T ≥ 0, log T ≥ 0, log log T ≥ 0 ⟸ T ≥ e... T ≥ 283 ✓ log T ≥ 5.6 > 1 ⟹ log log T > 0 ✓; const 2.463 > 0). So 2F ≥ 0. IIMaj ≥ 0 (all factors nonneg: 4.56 > 0, s+1 > 0, exp > 0) ✓. 40·1.3352 ≥ 53... include for comfort: ≥ 53.4.\n  Bn(80000) ≥ 124916.3 + 149715.6 + 53.4 + 0 + 0 (+1? the +1 is added: Bn = ... + IIMaj + 1: ≥ 124916.3+149715.6+53.4+0+1 = 274686.3).\n  margin(80000) ≤ 338670.1 − 274686.3 + 1 = 63984.8.\n- margin(100000) lower: slower(100000) ≥ 50000·11.5 − 1.4189736·100000 − 4 = 575000 − 141897.36 − 4 = 433098.6.\n  Bn(100000) ≤ : T ≤ 318 (√100000 ≤ 317, 317² = 100489 ✓).\n  · log T ≤ 5.8, log log ≤ 1.77, log(T+2) ≤ 5.8 (T+2 ≤ 320 ≤ 330.3 ≤ e^5.8 ✓).\n  · 2F ≤ 2·[0.1592·318·5.8 + 0.137·5.8 + 0.443·1.77 + 2.463] = 2·[293.51+0.7946+0.7841+2.463] = 2·297.55 = 595.1\n  · Ĉ₀ ≤ 0.433·5.8 + 0.886·1.77 + 2.9 = 6.9796\n  · 4T[...] ≤ 1272·[4.6531 + 60.9 + 48] = 1272·113.5531 = 144439.5\n  · Ĥ ≤ 3·48.715 + 60.9 + 48 = 255.05\n  · 2(e⁴+1)(7Ĥ+3.3) ≤ 111.203·1788.65 = 198893.9\n  · IIMaj ≤ 4.56·318·2 = 2900.2 (s+1 ≤ 318)\n  · Bn ≤ 595.1 + 144439.5 + 53.4 + 198893.9 + 2900.2 + 1 = 346893.1\n  margin(100000) ≥ 433098.6 − 346893.1 + 1 = 86206.5.\n- 63984.8 < 86206.5 ✓ slack 22221.7.\n\nAll consistent with D-062 §5 (minor refinements: IIMaj s+1 ≤ 284 not 285, etc.).\n\nNow the Lean formalization strategy. Key building blocks:\n1. `Real.log` bounds at specific points via exp comparisons:\n   - log x ≤ c ⟸ x ≤ exp c ⟸ x ≤ p ≤ exp c (p = partial sum lower bound... no wait — for log x ≤ c I need x ≤ exp c; partial sums give exp c ≥ p, so if x ≤ p then x ≤ exp c ✓). Used: log 80000 ≤ 11.3 (80000 ≤ e^11·e^0.3 with e^11 ≤ 2.7182818286^11 — pow of rational ✓, e^0.3 ≥ partial sum ✓... wait direction: need 80000 ≤ e^11.3 = e^11·e^0.3; e^11 ≥ 2.7182818283^11 (LOWER) and e^0.3 ≥ partial(0.3) (LOWER) — product of lower bounds ≤ e^11.3; need 80000 ≤ that product ✓ (80822 ≥ 80000 ✓). GOOD: log ≤ bounds need LOWER bounds on exp ✓ all partial sums / rational pows.\n   - log x ≥ c ⟸ x ≥ exp c: need UPPER bound on exp c: exp c = (exp(c/16))^16 ≤ (1/(1−c/16))^16 via exp(y) ≤ 1/(1−y) for 0 ≤ y < 1. Lemma needed: Real.exp_le_inv_one_sub? Build from Real.add_one_le_exp: 1 + (−y) ≤ exp(−y) = (exp y)⁻¹. So (exp y)⁻¹ ≥ 1 − y; if 1 − y > 0: exp y ≤ (1−y)⁻¹ (le_of_one_le... inv_le_inv... `Real.inv_le_inv`? Generic: for 0 < a ≤ b: b⁻¹ ≤ a⁻¹. We have 0 < 1−y ≤ (exp y)⁻¹ so ((exp y)⁻¹)⁻¹ ≤ (1−y)⁻¹ i.e. exp y ≤ (1−y)⁻¹. `inv_le_inv₀`? The inv-antitone lemma: `inv_le_inv_of_le₀`? or `one_div_le_one_div_of_le`. We have (1−y) ≤ (exp y)⁻¹ with 1−y > 0 ⟹ ((exp y)⁻¹)⁻¹ ≤ (1−y)⁻¹ via one_div_le_one_div_of_le (h1 : 0 < 1−y) (h : 1−y ≤ (exp y)⁻¹)... one_div_le_one_div_of_le : 0 < a → a ≤ b → 1/b ≤ 1/a. So 1/((exp y)⁻¹) ≤ 1/(1−y); 1/((exp y)⁻¹) = exp y (one_div_inv? (x⁻¹)⁻¹ = x: inv_inv). ✓ manageable.\n   Then (exp(c/16))^16 ≤ ((1−c/16)⁻¹)^16 = (1/(1−c/16))^16 and exp(c) = (exp(c/16))^16 via Real.exp_nat_mul... need c/16·16 = c. Real.exp_nat_mul : Real.exp (n * x) = (Real.exp x) ^ n? Let me check the name — might be `Real.exp_nat_mul (x : ℝ) (n : ℕ) : Real.exp (↑n * x) = Real.exp x ^ n`. I believe it exists.\n2. sqrt bounds: Real.sqrt_le' / Real.le_sqrt / Real.sqrt_le_sqrt + sq comparisons. Available: the file used `Real.le_sqrt (by norm_num) (by norm_num)` : (Real.le_sqrt : 0 ≤ y → (x ≤ sqrt y ↔ x^2 ≤ y))? They used `rw [Real.le_sqrt (by norm_num) (by norm_num)]; norm_num` to prove 4.4721 ≤ sqrt 20 — so Real.le_sqrt : 0 ≤ x → 0 ≤ y? → (x ≤ √y ↔ x² ≤ y) roughly. And for √x ≤ c: Real.sqrt_le' or Real.sqrt_le_iff. There's `Real.sqrt_le' (hy : 0 < y) : √x ≤ y ↔ x ≤ y^2`? In this Mathlib: `Real.sqrt_le' : 0 < y → (Real.sqrt x ≤ y ↔ x ≤ y ^ 2)`? Let me grep when writing. Or use sqrt_le_sqrt + sqrt_eq: √80000 ≤ 283 ⟸ √80000 ≤ √80089 = 283: Real.sqrt_le_sqrt (by norm_num : 80000 ≤ 80089) and Real.sqrt_eq_iff_sq_eq... `Real.sqrt_eq_cases`? For √80089 = 283: Real.sqrt_eq_iff_mul_self_eq... There's `Real.sqrt_eq_iff_sq_eq` or `Real.sqrt_eq_cases`: `Real.sqrt_eq_cases : sqrt x = y ↔ x = y * y ∧ 0 ≤ y ∨ ...`? Simpler: `Real.sqrt_le'` if it exists; else: `Real.sqrt_le_left`? Let me grep at write time.\n3. The π bounds: Real.pi_gt_d9? Actual names: `Real.pi_gt_d9 : 3.1415926535 < Real.pi` and `Real.pi_lt_d9 : Real.pi < 3.1415926536`? I think the Mathlib names are Real.pi_gt_d9 / Real.pi_lt_d9. Grep.\n4. log 2: Real.log_two_gt_d9, Real.log_two_lt_d9 (used in file already ✓).\n5. eulerMascheroniConstant — for slower: the γ cancellation: sInftyLower n = (1/2)n log n + (lagariasC1 − γ/2)n − 4 with lagariasC1 = (γ − 1 − log 2π)/2: the combination (lagariasC1 − γ/2) = (γ − 1 − log 2π)/2 − γ/2 = −(1 + log 2π)/2 = −(1 + log 2 + log π)/2 — provable by: unfold lagariasC1; rw log 2π = log 2 + log π; ring. The γ terms cancel algebraically (treated as opaque real) ✓.\n6. IIMaj bound: IIMaj n = psiTilde (√n) (IIMaj_eq_psiTilde_sqrt) = 4.56·((s+1)·(e^{−as}+e^{−bs})) ≤ 4.56·(s+1)·2: needs 0 ≤ e^{−as} ≤ 1 (exp_pos + Real.exp_le_one_iff: exp x ≤ 1 ⟺ x ≤ 0; −a·s ≤ 0 since a = 2log2 − 1/2 > 0 (from log2 bounds), s ≥ 0 ✓). Same for b. So psiTilde s ≤ 4.56·(s+1)·2 with s = √80000 ≤ 283.\n   But WAIT — IIMaj's def uses `2 ^ (-(2 * Real.sqrt n - 2))` rpow and Real.exp — the psiTilde equality is proved (IIMaj_eq_psiTilde_sqrt) ✓ use it.\n7. marginTilde FAlt n = sInftyLower n − Bn FAlt n + 1 (marginTilde_eq_sub) ✓; Bn unfolds to the let-expression; work with `simp only [Bn, C0hat, Hhat, FAlt]` then bound piecewise.\n\nStructure of the Lean proof (hbase):\n```\ntheorem marginTilde_FAlt_80000_pos : 0 < marginTilde FAlt 80000 := by\n  rw [marginTilde_eq_sub]\n  -- goal: 0 < sInftyLower 80000 − Bn FAlt 80000 + 1\n  -- lower bound sInftyLower, upper bound Bn, linarith\n  have hS : (336000 : ℝ) ≤ sInftyLower 80000 := ...\n  have hB : Bn FAlt 80000 ≤ (328000 : ℝ) := ...\n  linarith\n```\nWith hS: sInftyLower 80000 = (1/2)·80000·log 80000 + (lagariasC1 − γ/2)·80000 − 4 ≥ 336000:\nunfold sInftyLower lagariasC1; have log2pi = log2 + logpi; ring_nf to expose γ cancellation... The expression: (1/2)·80000·log 80000 + ((γ−1−log(2π))/2 − γ/2)·80000 − 4. Step 1: rewrite log(2π) = log 2 + log π (Real.log_mul). Step 2: the coefficient ((γ−1−(log2+logπ))/2 − γ/2) = −(1+log2+logπ)/2 — show via ring (γ opaque). So sInftyLower 80000 = 40000·log 80000 − 40000·(1+log2+logπ) − 4... hmm: ((...)/2)·80000 = (...)·40000. With = −(1+log2+logπ)·40000. So sInftyLower 80000 = 40000·log 80000 − 40000·(1 + log2 + logπ) − 4. Then bound: log 80000 ≥ 11.263; 1+log2+logπ ≤ 1+0.6931472+1.1448 = 2.8379472 (need log 2 ≤ 0.6931472 from log_two_lt_d9 : log 2 < 0.6931471808 ✓ and log π ≤ 1.1448). So sInftyLower ≥ 40000·11.263 − 40000·2.8379472 − 4 = 450520 − 113517.888 − 4 = 336998.11 ≥ 336000 ✓ (slack 998). Use target 336000 to leave norm_num-friendly margins. Wait, but rounding: I should pick round targets with big slack: hS : sInftyLower 80000 ≥ 336000; hB : Bn FAlt 80000 ≤ 328000; margin ≥ 336000 − 328000 + 1 = 8001 > 0 ✓. My computed bounds: slower ≥ 336998.11 (target 336000 ✓ slack 998); Bn ≤ 327028.3 (target 328000 ✓ slack 972). \n\nhB : Bn FAlt 80000 ≤ 328000:\nunfold Bn C0hat Hhat FAlt IIMaj→psiTilde... Bn FAlt 80000 = 2·FAlt(T) + 4T·((2/3)C0hat T + 10.5 log(T+2) + 48) + 40·1.3352 + 2(e⁴+1)(7 Hhat T + 3.3) + IIMaj 80000 + 1 with T = √80000+1.\nBounding chain (all pieces nonneg — need positivity facts for mul_le_mul):\n- T_nn: 0 ≤ T and T ≤ 284: T = √80000+1: √80000 ≤ 283 (√80000 ≤ 283 ⟺ 80000 ≤ 283² — Real.sqrt_le' or similar); T ≥ 0 trivially (sqrt nonneg +1).\n- logT_up: log T ≤ 5.7: T ≤ 284 ≤ exp 5.7 (e^5.7 ≥ 298.87 certified): log T ≤ log (exp 5.7) = 5.7 via Real.log_le_log (T > 0) (T ≤ exp 5.7) then Real.log_exp.\n  Need T ≤ exp 5.7: T ≤ 284 ≤ 298.867 ≤ exp 5.7. The 298.867 ≤ exp 5.7: exp 5.7 = exp(5+0.66+0.04)... hmm D-062 plan: exp 5.7 = exp 5 · exp 0.7; exp 5 = (exp 1)^5 ≥ 2.7182818283^5 (pow_le_pow_left of Real.exp_one_gt_d9 — wait need exp 5 = (exp 1)^5: Real.exp_nat_mul 1 5 : exp (5*1) = (exp 1)^5 — with cast: (5:ℝ)*1 = 5 by norm_num); exp 0.7 ≥ Σ_{i<7} 0.7^i/i! via Real.sum_le_exp_of_nonneg (x := 0.7) (by norm_num) 7 — the file used this pattern: `Real.sum_le_exp_of_nonneg (x := (3.9635 : ℝ)) (by norm_num) 7` ✓. Then exp(5+0.7) = exp 5 · exp 0.7 (Real.exp_add) and 5.7 = 5+0.7 by norm_num. So exp 5.7 ≥ 2.7182818283^5 · (Σ...) ≥ 298.867 via norm_num [Finset.sum_range_succ] — the pattern from iimaj: `norm_num [Finset.sum_range_succ]` evaluates the sum ✓.\n- loglogT_up: log (log T) ≤ 1.75: log T ≤ 5.7 ≤ exp 1.75 (5.7546): log (log T) ≤ log 5.7 ≤ log (exp 1.75) = 1.75. Need log T ≤ 5.7 (have) and 5.7 ≤ exp 1.75: exp 1.75 = exp 1·exp 0.75 ≥ 2.7182818283·(Σ_{i<7} 0.75^i/i!) = 2.7182818283·2.1170 = 5.7546 ✓. Also need 0 < log T for log mono: log T ≥ log 1... T ≥ 283 ⟹ log T ≥ log 283 > 0 (Real.log_pos (T > 1)). ✓\n- logT2_up: log (T+2) ≤ 5.7: T+2 ≤ 286 ≤ exp 5.7 ✓ (298.867 ≥ 286 ✓). T+2 > 0 ✓.\n- loglogT2_up: log (log (T+2)) ≤ 1.75 ✓ same.\n- FAlt_up: FAlt T = (1/(2π))·(T·log T) + 0.137·log T + 0.443·log(log T) + (7/8+1.588) ≤ 0.1592·(284·5.7) + 0.137·5.7 + 0.443·1.75 + 2.463:\n  1/(2π) ≤ 0.1592: 2π ≥ 2·3.1415926535 = 6.283185307 ⟹ 1/(2π) ≤ 1/6.283185307 = 0.1591549… ≤ 0.1592 (one_div_le_one_div_of_le (2·3.1415926535 > 0) (2π ≥ 2·3.1415926535) gives 1/(2π) ≤ 1/6.2831853070; then 1/6.2831853070 ≤ 0.1592 ⟺ 1 ≤ 0.1592·6.2831853070 = 1.00024... ✓ norm_num with div_le_iff₀).\n  T·log T ≤ 284·5.7 (mul_le_mul needs 0 ≤ T, 0 ≤ log T ✓).\n- C0hat_up: ≤ 0.433·5.7 + 0.886·1.75 + 2.9 = 6.9186 — mul_le_mul with nonnegativity (0.433·log(T+2): need log(T+2) ≥ 0 ✓ T+2 ≥ 1).\n- Hhat_up: 3·C0hat² + 10.5·log(T+2) + 48 ≤ 3·6.9186² + 10.5·5.7 + 48: C0hat² ≤ 6.9186² needs 0 ≤ C0hat (C0hat = 0.433·log + 0.886·loglog + 2.9 ≥ 0 + 0 + 2.9 > 0 ✓ — need log log(T+2) ≥ 0 ⟸ log(T+2) ≥ 1 ⟸ T+2 ≥ e: T+2 ≥ 285 ≥ e ⟸ 285 ≥ 2.7182818286 ✓). pow_le_pow_left.\n- exp4_up: Real.exp 4 ≤ 54.6015: exp 4 = (exp 1)^4 ≤ 2.7182818286^4 = 54.59815 ≤ 54.6015 ✓ norm_num.\n- IIMaj_up: IIMaj 80000 = psiTilde (√80000) ≤ 4.56·284·2 = 2589.12... let me recompute: 4.56·(283+1)·2 = 4.56·284·2 = 2589.12. Hmm earlier I said 2589.5 — 4.56·284 = 1295.04, ·2 = 2590.08. Whatever — ~2590. (s = √80000 ≤ 283; s+1 ≤ 284; e^{−as} ≤ 1, e^{−bs} ≤ 1.)\n  psiTilde s = 4.56·((s+1)·(e^{−as}+e^{−bs})): ≤ 4.56·((283+1)·(1+1)) = 4.56·568 = 2590.08.\n- Assemble: Bn ≤ 2·261.73 + 4·284·((2/3)·6.9186 + 10.5·5.7 + 48) + 53.408 + 2·(54.6015+1)·(7·251.45 + 3.3) + 2590.08 + 1.\n  2·261.73 = 523.46; 4·284 = 1136; bracket = 4.6124+59.85+48 = 112.4624; 1136·112.4624 = 127757.29; 7·251.45+3.3 = 1763.45; 2·55.6015 = 111.203; 111.203·1763.45 = 196103.6. Sum: 523.46+127757.29+53.408+196103.6+2590.08+1 = 327028.8 ≤ 328000 ✓.\n\nFor hbase2 (marginTilde FAlt 80000 < marginTilde FAlt 100000):\n- margin80000 ≤ 338670.1 − 274686.3 + 1 = 63984.8 → target: margin80000 ≤ 64000.\n- margin100000 ≥ 433098.6 − 346893.1 + 1 = 86206.5 → target: margin100000 ≥ 86000.\n- 64000 < 86000 ✓.\nPieces for margin80000 upper:\n  slower(80000) ≤ 40000·11.3 − 40000·(1+0.6931471+1.14) − 4: log 80000 ≤ 11.3 (80000 ≤ e^11.3: e^11 ≥ 2.7182818283^11 = 59874.1…; e^0.3 ≥ Σ_{i<7} = 1.3497…; product 80821.6 ≥ 80000 ✓ 1.03%); coefficient lower: (1+log2+logπ)/2 ≥ (1+0.6931471+1.14)/2: log 2 ≥ 0.6931471 ✓ (log_two_gt_d9 : 0.6931471803 < log 2 ✓), log π ≥ 1.14 (π ≥ e^1.14: e^1.14 = e·e^0.14 ≤ 2.7182818286·(1/(1−0.14/16))^16 ≤ 3.1287 ≤ 3.1415926535 ≤ π — the exp upper trick). slower ≤ 452000 − 113325.88 − 4 = 338670.1. Target 338700? Use 338700 (slack 30 — hmm tight for comfort but exact arithmetic, fine. Actually give more slack: target 339000: margin ≤ 339000 − 274686.3 + 1 = 64314.7; and margin100000 ≥ 86206.5 → 86000: 64314.7 < 86000 ✓ still fine. Use round targets: hS80000up : sInftyLower 80000 ≤ 339000; hB80000lo : Bn FAlt 80000 ≥ 274000; margin80000 ≤ 339000 − 274000 + 1 = 65001. hS100000lo : sInftyLower 100000 ≥ 433000; hB100000up : Bn FAlt 100000 ≤ 347000; margin100000 ≥ 433000 − 347000 + 1 = 86001 > 65001 ✓ slack 21000.)\n  Bn(80000) lower ≥ 274000:\n  · T ≥ 283: √80000 ≥ 282 (282² = 79524 ≤ 80000 — Real.le_sqrt) ⟹ T ≥ 283.\n  · log(T+2) ≥ log 285 ≥ 5.6 (285 ≥ e^5.6: e^5.6 = e^5·e^0.6 ≤ 2.7182818286^5·(1/(1−0.6/16))^16 = 148.414·1.8431 = 273.56 ≤ 285 ✓ 4.2% — via exp UPPER trick: need e^5.6 ≤ 285). log mono: T+2 ≥ 285 ⟹ log(T+2) ≥ log 285; log 285 ≥ 5.6 ⟸ 285 ≥ exp 5.6 ⟸ 285 ≥ 273.56 ≥ exp 5.6.\n  · log log(T+2) ≥ 0 (log(T+2) ≥ 1 ⟸ T+2 ≥ e ✓).\n  · C0hat ≥ 0.433·5.6 + 0.886·0 + 2.9 = 5.3248.\n  · 4T[...] ≥ 4·283·[(2/3)·5.3248 + 10.5·5.6 + 48] = 1132·[3.54987+58.8+48] = 1132·110.34987 = 124916.2\n  · Hhat ≥ 3·5.3248² + 10.5·5.6 + 48 = 85.07+58.8+48 = 191.87: needs C0hat ≥ 5.3248 ≥ 0 ✓ pow_le_pow_left.\n  · e⁴ ≥ 54.5978: (exp 1)^4 ≥ 2.7182818283^4 = 54.5981500 ≥ 54.5978 ✓.\n  · 2(e⁴+1)(7Ĥ+3.3) ≥ 2·55.5978·(7·191.87+3.3) = 111.1956·1346.39 = 149717.1\n  · 2FAlt(T) ≥ 0: FAlt T ≥ 0 (all four terms ≥ 0 for T ≥ e: (1/(2π))·T·log T ≥ 0 (T ≥ 0, log T ≥ 0); 0.137 log T ≥ 0; 0.443 log log T ≥ 0 (log T ≥ 1 ⟸ T ≥ e); const > 0).\n  · IIMaj ≥ 0: psiTilde s = 4.56·(s+1)·(e^..+e^..) ≥ 0 (exp_pos).\n  · 40·1.3352 = 53.408; +1.\n  Bn ≥ 124916.2 + 149717.1 + 53.408 + 0 + 0 + 1 = 274687.7 ≥ 274000 ✓.\n  margin100000 lower pieces:\n  · slower(100000) ≥ 433000: 50000·log 100000 − 50000·2·1.4189736... wait: slower(100000) = (1/2)·100000·log 100000 − ((1+log2+logπ)/2)·100000 − 4 = 50000·log 100000 − 50000·(1+log2+logπ) − 4. log 100000 ≥ 11.5 (100000 ≤ e^11.5: e^11 ≤ 2.7182818286^11 = 59874.2; e^0.5 ≤ (1/(1−0.5/16))^16 = 1.6644; product = 99646.8 ≤ 100000 ✓ 0.35%); (1+log2+logπ) ≤ 2.8379472. slower ≥ 50000·11.5 − 50000·2.8379472 − 4 = 575000 − 141897.36 − 4 = 433098.6 ≥ 433000 ✓.\n  · Bn(100000) ≤ 347000: T ≤ 318 (√100000 ≤ 317: 317² = 100489 ≥ 100000 ✓). log T ≤ 5.8 (T ≤ 318 ≤ 330.3 ≤ e^5.8: e^5.8 = e^5·e^0.8 ≥ 148.328·2.2255 = 330.10 — partial sums e^0.8 ≥ Σ_{i<7}: 1+.8+.32+.0853333+.0170667+.0027307+.0003641 = 2.2255 ✓). log log T ≤ 1.77 (5.8 ≤ e^1.77 = e·e^0.77 ≥ 2.7182818283·2.1597 = 5.8708 ✓). log(T+2) ≤ 5.8 (T+2 ≤ 320 ≤ 330.3 ✓). log log(T+2) ≤ 1.77.\n    FAlt ≤ 0.1592·318·5.8 + 0.137·5.8 + 0.443·1.77 + 2.463 = 293.51+0.7946+0.7841+2.463 = 297.55; 2F ≤ 595.1\n    Ĉ₀ ≤ 0.433·5.8+0.886·1.77+2.9 = 6.9796; 4T[...] ≤ 1272·[4.6531+60.9+48] = 1272·113.5531 = 144439.5\n    Ĥ ≤ 3·6.9796²+60.9+48 = 255.05; 2(e⁴+1)(7Ĥ+3.3) ≤ 111.203·1788.65 = 198893.9\n    IIMaj ≤ 4.56·318·2 = 2900.16 (s+1 ≤ 318)\n    Bn ≤ 595.1+144439.5+53.408+198893.9+2900.16+1 = 346893.1 ≤ 347000 ✓.\n\nTotal: 4 numeric lemmas (hS80000lo, hB80000up, hS80000up, hB80000lo, hS100000lo, hB100000up — 6 actually) + 2 assembly theorems + rewrite of lambda_pos_finite_height.\n\nHmm — sInftyLower bounds share machinery. Let me structure reusable lemmas:\n- log80000_lo : (11.263 : ℝ) ≤ Real.log 80000\n- log80000_up : Real.log 80000 ≤ 11.3\n- log100000_lo : 11.5 ≤ Real.log 100000\n- logpi_up : Real.log Real.pi ≤ 1.1448\n- logpi_lo : 1.14 ≤ Real.log Real.pi\n- log bounds at T-points are inline (log T ≤ 5.7 etc. derived from T bounds via exp).\n\nExp machinery lemmas:\n- exp_le_via_sixteen (x : ℝ) (hx : 0 ≤ x) : Real.exp x ≤ (1 - x/16)⁻¹ ^ 16... general lemma:\n  Real.exp x = (Real.exp (x/16))^16 (Real.exp_nat_mul reversed: (exp (x/16))^16 = exp (16·(x/16)) = exp x). And exp (x/16) ≤ (1−x/16)⁻¹ needs x/16 < 1: from add_one_le_exp: 1 + (−(x/16)) ≤ exp (−(x/16)) = (exp (x/16))⁻¹ ⟹ (1 − x/16) ≤ (exp (x/16))⁻¹ ⟹ exp (x/16) ≤ (1−x/16)⁻¹ (one_div_le_one_div_of_le + inv_inv). Then pow_le_pow_left (exp nonneg).\n  Real.exp_nat_mul: check exact name/form in Mathlib. Let me grep: Real.exp_nat_mul exists? I'll verify.\n\nSum lower bounds: Real.sum_le_exp_of_nonneg (x := c) (hx : 0 ≤ c) n : (∑ i ∈ range n, x^i / i!) ≤ exp x — used in file ✓.\n\nThis is a big block of work. Let me organize it as a new section §5 in LiSkeleton.lean (or a new file? A new file BasePoints.lean would keep LiSkeleton manageable — but then lambda_pos_finite_height must MOVE or import... If hbase/hbase2 get PROVED in a new file, the main theorem there can be restated without them. Options: (a) keep everything in LiSkeleton.lean — file grows to ~2900 lines; (b) new file NguyenRH1/LiBasePoints.lean importing NguyenRH1.LiSkeleton? No wait — LiSkeleton is a lean_lib root... The lib has NguyenRH1.lean root importing the three skeletons. A new file LiBasePoints.lean with the numeric proofs + a final corollary theorem `lambda_pos_finite_height'` (3 hypotheses) — hmm, or RESTATE: after proving hbase/hbase2 as theorems, EDIT lambda_pos_finite_height in LiSkeleton to remove those two hypotheses and apply the proved theorems — but the proved theorems live in the new file which imports LiSkeleton — circular. So: put the numeric proofs in LiSkeleton.lean itself (before lambda_pos_finite_height), OR create LiBasePoints.lean and move the MAIN theorem there. \n\nCleanest: keep lambda_pos_finite_height in LiSkeleton with 5 hypotheses (it's the modular statement), and in a NEW file LiAssembly.lean (or BasePoints.lean): prove the two numeric theorems, then:\n```\ntheorem lambda_pos_finite_height_verified\n    (hRH : RHUpTo (3*10^12)) (hfinite : LiVerifiedUpTo (5*10^5)) (hSf : SfBoundProp) :\n    ∀ n, 1 ≤ n → (n:ℝ) ≤ (3*10^12−1)^2 → 0 < (liCoeff n).re :=\n  fun n hn1 hnN => lambda_pos_finite_height hRH hfinite marginTilde_FAlt_80000_pos marginTilde_FAlt_80000_lt hSf n hn1 hnN\n```\nThat gives the 3-hypothesis headline theorem. \n\nFile layout decision: new file `NguyenRH1/LiBasePoints.lean` importing Mathlib + NguyenRH1.LiSkeleton? The lib structure: NguyenRH1.lean (root) imports the skeleton files. Check NguyenRH1.lean content — probably `import NguyenRH1.LiSkeleton` etc. New file: add import to root, add #print axioms for the new theorems.\n\nGiven the volume (~500 lines of numeric grind), this is a few hours of careful work. Sub-stage commits: (i) exp/log machinery + log80000/log100000/logpi bounds; (ii) hS bounds; (iii) hB bounds; (iv) assembly + final theorem + audit + commit.\n\nAlso the paper margin-number fix (item B) — quick: check §3.1 for 16,020.\n\nTime to start. First the paper fix (5 min), then the Lean grind.\n\nActually — order: the mathematician said discharge hbase/hbase2 NEXT (the main ask), reconcile the trail in the main text. Both are this session's work. Let me do the paper fix first (quick win), then settle into the Lean grind.\n\ngrep draft.tex for the margin numbers.","signature":""},{"type":"text","text":"Clear directives from the mathematician's note (a plain-text review comment, not malware):\n\n1. **Discharge hbase/hbase2 next** (pre-verified, low risk) — upgrades the theorem to 3 perimeter-class hypotheses; *then* the paper's formal-status paragraph is final.\n2. **Reconcile the margin-number trail** (16,020 → 15,764 → >9,960) in §3.1 main text with a footnote.\n3. Tier ii′ (contour interior) is **post-publication** — don't let it delay Paper A.\n4. Standing Part-2 objections unchanged (no action requested).\n\nPlan: quick paper fix first, then the interval-bound grind."},{"type":"tool_use","id":"TodoWrite_122","name":"TodoWrite","input":{"todos":[{"content":"Paper fix: reconcile margin-number trail in §3.1 (16,020 → Lean-consumed bound + footnote)","status":"in_progress","activeForm":"Reconciling margin numbers in paper"},{"content":"Lean exp/log machinery: exp upper trick, log80000/log100000/logπ certified bounds","status":"pending","activeForm":"Building exp/log bound machinery"},{"content":"sInftyLower bounds at 80000 (two-sided) and 100000 (lower)","status":"pending","activeForm":"Proving sInftyLower bounds"},{"content":"Bn bounds at 80000 (two-sided) and 100000 (upper)","status":"pending","activeForm":"Proving Bn bounds"},{"content":"hbase + hbase2 theorems; final 3-hypothesis lambda_pos_finite_height_verified","status":"pending","activeForm":"Assembling final theorem"},{"content":"Rebuild + audit + paper status paragraph finalization + session protocol","status":"pending","activeForm":"Finalizing and documenting"}]},"extra_content":null}]