[{"type":"thinking","thinking":"Now 12.2: the base points. First, let me get the EXACT values from E-028/E-029 data to know what to prove. margiñ(80000) and margiñ′(80000) with the FAlt majorant (not the Trudgian F!). E-029 used the Trudgian F + S_b presumably. With FAlt the margin is slightly smaller (by ≈ 0.4516·2·T ≈ 256 at n=80000). Let me compute the exact marginTilde FAlt 80000 value numerically first with a quick Python check, then design the Lean proof with appropriate crude bounds.\n\nmarginTilde FAlt 80000 = sInftyLower(80000) − Bn FAlt 80000 + 1.\n\nLet me compute precisely in Python (float64 fine for sizing).","signature":""},{"type":"text","text":"Now the base points. First, exact float values of `marginTilde FAlt 80000` and its derivative to size the proof margins."},{"type":"tool_use","id":"Bash_77","name":"Bash","input":{"command":"cd D:/Python/Quantum && .venv/Scripts/python.exe -c \"\nimport math\nfrom mpmath import mp, mpf, log, exp, sqrt, pi, euler\nmp.mp.dps = 50\nn = mpf(80000)\nT = sqrt(n) + 1\na1 = (log(4*pi) + euler)/2\nC1 = (euler - 1 - log(2*pi))/2\ndef C0hat(t): return mpf('0.433')*log(t+2) + mpf('0.886')*log(log(t+2)) + mpf('2.9')\ndef Hhat(t): return 3*C0hat(t)**2 + mpf('10.5')*log(t+2) + 48\ndef IIMaj(n): return 2*(sqrt(n)+1)*(exp(sqrt(n)/2)+1)*mpf('0.57')*2**(-(2*sqrt(n)-2))\ndef FAlt(t): return (1/(2*pi))*(t*log(t)) + mpf('0.137')*log(t) + mpf('0.443')*log(log(t)) + (mpf(7)/8 + mpf('1.588'))\ndef FTru(t): return (t/(2*pi))*log(t/(2*pi*exp(1))) + mpf(7)/mpf(8) + mpf('0.137')*log(t) + mpf('0.443')*log(log(t)) + mpf('1.588')\ndef Bn(F, n):\n    t = sqrt(n)+1\n    return 2*F(t) + 4*t*((mpf(2)/3)*C0hat(t) + mpf('10.5')*log(t+2) + 48) + 40*mpf('1.3352') + 2*(exp(4)+1)*(7*Hhat(t)+mpf('3.3')) + IIMaj(n) + 1\ndef slower(n): return mpf('0.5')*n*log(n) + (C1 - euler/2)*n - 4\ndef margin(F, n): return slower(n) - Bn(F, n) + 1\nprint('T           =', T)\nprint('slower      =', slower(n))\nprint('Bn FAlt     =', Bn(FAlt, n))\nprint('margin FAlt =', margin(FAlt, n))\nprint('margin FTru =', margin(FTru, n))\nprint('FAlt-FTru   =', FAlt(T)-FTru(T))\n# derivative of margin wrt n at n=80000 (central difference, h=0.5)\nh = mpf('0.5')\nprint('marginprime FAlt ~', (margin(FAlt, n+h)-margin(FAlt, n-h))/(2*h))\nprint('marginprime FTru ~', (margin(FTru, n+h)-margin(FTru, n-h))/(2*h))\n\"\n","description":"Compute exact base-point values for FAlt margin"},"extra_content":null}]