[{"type":"thinking","thinking":"The main text §3.1 Step 4 says: margin(8×10⁴) = 16 020 > 0 with positive derivative; hence margin > 0 on [8×10⁴, N₁]. And line 471 (numerics section) says \"with margin 16 020 and positive derivative\".\n\nThe Lean theorem consumes the FAlt margin: margiñ(8×10⁴) = 15 764 (exact FAlt value, dps-50 computed: 15763.79). The mathematician's instruction: \"Make the main-text number match whichever bound the Lean theorem actually consumes, with the trail in a footnote.\"\n\nThe Lean theorem consumes: marginTilde FAlt 80000 > 0 and marginTilde FAlt 80000 < marginTilde FAlt 100000 — the FAlt margin values: 15 764 (at 8×10⁴) and ~93 000 (at 10⁵). \n\nEdit plan for Step 4:\n- Change \"margin(8×10⁴) = 16 020 > 0 with positive derivative; hence margin(n) > 0 for all n ∈ [8×10⁴, N₁]\" to use the FAlt value 15 764 and the two-point secant argument (matching the Lean assembly), with a footnote carrying the trail: 16 020 (E-028, Trudgian F + S_b majorant) → 15 764 (FAlt majorant, the form machine-verified) → > 9 960 (dps-40-certified rational interval lower bound consumed by the Lean discharge... well, hbase not yet discharged — but the footnote can say the certified bound is ≥ 9 960 — hmm, careful: don't claim the Lean discharge before it's done. The footnote trail: \"16 020 with the Trudgian F+S_b majorant (E-028); 15 764 with the mildly enlarged majorant used in the Lean development; the certified interval lower bound 9 960 is what the formal discharge consumes.\")\n\nBut wait — the main text should stay accurate about the pen-and-paper theorem too. The pen-and-paper R-004 used margin(8×10⁴) = 16 020 with positive derivative (E-028/E-029). The Lean version uses FAlt + two-point. The main text describes the PROOF SKETCH of R-004. The mathematician wants the main-text number to match the Lean-consumed bound with the trail in a footnote.\n\nRewrite Step 4:\n```\n\\emph{Step 4 (margin).}\n$\\mathrm{margin}(n) := \\tfrac12 n\\log n + (C_1 - \\gamma/2)n - 4 - B(n) + 1$\nis convex for $n \\ge 20$ (proved: chain-rule reduction to eleven\none-variable sign conditions, each discharged elementarily — D-057),\nand, with the mildly enlarged majorant of the formal development,\n$\\mathrm{margin}(8\\times10^4) = 15\\,764 > 0$ and\n$\\mathrm{margin}(10^5) > \\mathrm{margin}(8\\times10^4)$; by convexity\n(secant slopes increase) hence\n$\\mathrm{margin}(n) > 0$ for all $n \\in [10^5, N_1]$\n(computed to $n = 4\\times10^5$ on a unit grid and to $N_1$ on a\nlogarithmic grid; $\\mathrm{margin}(N_1) = 2.458\\times 10^{26}$).\\footnote{The\nmargin value at $8\\times10^4$ has a documented trail: $16\\,020$ with\nthe Trudgian $F + S_b$ majorant (E-028); $15\\,764$ with the mildly\nenlarged majorant consumed by the Lean development (a negative-weight\nterm dropped so that every convexity block has provably negative\n$\\psi$, at a cost of $\\approx 0.9\\sqrt n$); and $> 9\\,960$ as the\ncertified rational interval lower bound used in the machine discharge.}\n```\n\nHmm wait — \"the certified rational interval lower bound used in the machine discharge\" — the discharge hasn't landed yet. The mathematician said the discharge is \"low risk, close\" and to hold the formal-status paragraph until then. Plan: discharge FIRST, then finalize paper text. But I'm doing the paper fix first... Let me reorder: do the Lean discharge NOW, then write the paper reconciliation with the final state (three hypotheses). That's cleaner — single paper edit with the final story.\n\nActually the mathematician's exact words: \"Consider holding the paper's formal-status paragraph until then; it's close.\" — the formal-status paragraph = the appendix paragraph I wrote. So: after discharge, update both §3.1 Step 4 + the appendix paragraph (five→three hypotheses). Do the Lean work first, then paper. Reorder todos.\n\nAlso Step 5's overlap statement \"The ranges overlap N₀ = 8×10⁴ by a factor 6.25\" — with the regime split at 10⁵, the overlap factor becomes 5×10⁵/10⁵ = 5. Hmm — the Lean assembly uses 10⁵ as the split. The pen-and-paper theorem used 8×10⁴. For consistency the main text should say the split is at 10⁵ with overlap factor 5 vs Misguich–Voros. Let me adjust Step 5 accordingly: \"The ranges overlap the split point 10⁵ by a factor 5.\" Wait — actually the main text can keep N₀ = 8×10⁴ as the crossover (margin crossover happens at 75 451, adopted N₀ = 80 000 for the pen-and-paper) — but the formal version uses 10⁵. For paper consistency: make Step 4/5 use the 10⁵ split (the Lean-consumed form) and keep 8×10⁴ in the footnote trail. The theorem statement (λ_n > 0 for n ≤ N₁) is unaffected.\n\nLet me now get to the Lean grind. First verify the Mathlib lemma names I need:\n1. Real.add_one_le_exp\n2. Real.exp_nat_mul (name/form)\n3. Real.sum_le_exp_of_nonneg (known ✓ used in file)\n4. Real.log_le_log signature\n5. Real.sqrt_le' / Real.le_sqrt forms\n6. Real.pi_gt_d9 / Real.pi_lt_d9 exact names\n7. Real.exp_one_gt_d9 / Real.exp_one_lt_d9 exact names (used in file: Real.exp_one_lt_d9 ✓)\n8. one_div_le_one_div_of_le ✓ (used in file)","signature":""},{"type":"text","text":"Reordering: the mathematician said to hold the paper's formal-status paragraph until the discharge lands — so Lean first, paper reconciliation second (single final edit). Verifying the Mathlib lemma names I need for the grind."},{"type":"tool_use","id":"Bash_125","name":"Bash","input":{"command":"cd D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"theorem Real.add_one_le_exp\\|theorem add_one_le_exp\" Mathlib/Analysis/ | head -3; grep -rn \"theorem Real.exp_nat_mul\\|exp_nat_mul\" Mathlib/Analysis/SpecialFunctions/Exp*.lean Mathlib/Analysis/SpecialFunctions/Exp/*.lean 2>/dev/null | head -3; grep -rn \"pi_gt_d9\\|pi_lt_d9\" Mathlib/Analysis/SpecialFunctions/Trigonometric/*.lean 2>/dev/null | head -4","description":"Verify Mathlib exp/pi lemma names"},"extra_content":null}]