"""Exact first fractional-block comparisons at the second scale. Authored and self-checked by GPT-6 Astra (OpenAI), Ultra. CC0. The accompanying proof establishes the reductions for all ranks and fields. No floating-point test is used to certify a mathematical inequality. """ from fractions import Fraction as Q from math import factorial from pathlib import Path import argparse import json def case_data(): xi = Q(101, 100) # name,c1,c3,c4,a,q,zeta2,A0 minimum,d minimum,Siegel ratio, # normal factor,counterhypothesis multiplier,prefactor upper,target return [ ('I.1', Q(7,10), Q(6,5), 14, 7, 2, Q(5101,5000), Q(3,2), 2, Q(2,25), Q(3,2)+xi/2, Q(11,10), 627, 1438), ('I.2', Q(7,10), Q(6,5), 14, 7, 2, Q(5101,5000), Q(3,2), 1, Q(2,25), Q(3,2)+xi/2, Q(51,50), 631, 648), ('II', Q(7,10), Q(6,5), 14, 7, 2, Q(5101,5000), Q(5,2), 2, Q(1,134), Q(5,4)+xi/4, Q(21,20), 627, 690), ('III.1', Q(7,10), Q(8,5), 14, 7, 2, Q(2083,2000), Q(3,4), 2, Q(10,71), xi, Q(109,100), 452, 495), ('III.2', Q(7,10), Q(8,5), 14, 7, 2, Q(2083,2000), Q(3,4), 1, Q(10,71), xi, Q(6,5), 454, 557), ('IV', Q(7,10), Q(3), 14, 7, 2, Q(1321,1250), Q(1,2), 2, Q(4,23), Q(7,6)+xi/6, Q(13,10), 1763, 2418), ('V', Q(41,50), Q(3,5), 6, 13, 3, Q(507,500), Q(2), 2, Q(5,211), 1+xi/2, Q(7,6), 343, 406), ] def exp_bounds(c): lower = sum(c**k / factorial(k) for k in range(11)) upper = lower + c**11 / (factorial(11) * (1-c/12)) return lower, upper def calculate(): lam, xi = Q(59,58), Q(101,100) rows = [] for name,c1,c3,c4,a,q,zeta,amin,dmin,w,nfactor,chi,pref,target in case_data(): c2 = Q(a,q*q)/zeta c = Q(27,50) if q == 2 else Q(4,5) ell = Q(7,50) if q == 2 else Q(11,50) rmin = 3 if q == 2 else 4 assert chi*pref < target r0 = rmin radius_upper = (1-c/Q(r0+1))**(-r0-1) + 1/(c3*q*dmin*(r0+1)**2) assert radius_upper < q comparisons = [] for r in list(range(rmin,8)) + [8]: tail = r == 8 y = Q(1,r+1) eta, smin = 1-c*y, c3*q*dmin*(r+1)**2 tmin = Q(12,29)*(r+1)*12**r br = 1500*12**(r-3) eps = Q(8*r+15,9*(r+1)*br) rem = Q(1,79)+Q(1,71928*br) hminus = eta**(r+1) hplus = 1/exp_bounds(c)[0] if tail else hminus cr = Q(1) if tail else 1-Q(483,824)*y common = ((1/hminus+1/smin)/c2 + w/(2*c2)*(1+1/(2*smin)) + y*y/(c2*c3*q) + (1+w/2)*c1*eps+(1+w)*c1*rem) b = lam*(1+ell+w+ell*Q(q,q-1)) k = (Q(107,103)+Q(3,8)*y)*hplus+w*cr arith = { 'minimum': common+xi*(b*amin/c4+k/c3)/(1+amin), 'infinity': common+xi*b/c4, } if tail: gamma = (2*c*q/exp_bounds(c)[1]*sum(Q(1,q**j) for j in range(4)) - Q(q*9*17,q**8)/smin) else: gamma = (2*c*q/Q(q**r)*sum((q*eta)**j for j in range(r+1)) - Q(q*(r+1),q**r)/smin *(1-hminus+2*sum(eta**j-hminus for j in range(1,r+1)))) normal = lam*nfactor/(c4*q**r) input_lower = c1*q*chi-xi*(1-hminus+2/tmin)/(q**r*c3*(1+amin)) for endpoint, arithmetic in arith.items(): gap_a, gap_i = gamma-normal-arithmetic, input_lower-arithmetic assert gap_a > Q(1,2), (name,r,endpoint,'mass',float(gap_a)) assert gap_i > Q(1,1000), (name,r,endpoint,'input',float(gap_i)) comparisons.append({ 'ranks': 'all >= 8' if tail else r, 'A0_endpoint': endpoint, 'arithmetic_upper': str(arithmetic), 'normal_loss_upper': str(normal), 'mass_lower': str(gamma), 'input_lower': str(input_lower), 'mass_gap': str(gap_a), 'input_gap': str(gap_i), }) bounds = {'I.1':(Q(99,100),Q(3,25)),'I.2':(Q(99,100),Q(3,2000)),'II':(Q(109,100),Q(3,20)),'III.1':(Q(93,100),Q(1,20)),'III.2':(Q(93,100),Q(19,100)),'IV':(Q(51,50),Q(11,25)),'V':(Q(27,25),Q(69,100))} assert all(Q(v['mass_gap'])>bounds[name][0] and Q(v['input_gap'])>bounds[name][1] for v in comparisons) rows.append({'case':name,'siegel_ratio_upper':str(w), 'counterhypothesis_multiplier':str(chi), 'scaled_prefactor_upper':str(chi*pref), 'stated_prefactor':target,'radius_upper':str(radius_upper), 'comparisons':comparisons, 'minimum_mass_gap':str(min(Q(v['mass_gap']) for v in comparisons)), 'minimum_input_gap':str(min(Q(v['input_gap']) for v in comparisons))}) # Exact elementary lower bounds on p^(f A0) in the seven rows. assert 3**3 > 5**2 assert 5**5 > 50**2 assert 5**3 > 3**4 assert Q(7) > Q(5,2)**2 rho_bounds = [Q(5),Q(5),Q(50),Q(3),Q(3),Q(5,2),Q(16)] for row, lower in zip(rows,rho_bounds): assert 1/(Q(27,10)*lower-1) == Q(row['siegel_ratio_upper']) return {'state':'all_exact_comparisons_passed', 'scope':'First fractional block at second scale with strengthened kernel choice and casewise chi*U counterhypothesis; actual degree at most q^r*d retained. No contracted-stage conclusion.', 'finite_ranks':34,'uniform_tail_rows':7,'endpoint_comparisons':82, 'strict_mass_gap':'1/2','strict_input_gap':'1/1000','cases':rows} def render(result, output): import matplotlib matplotlib.use('Agg') import matplotlib.pyplot as plt fig, axes = plt.subplots(1,2,figsize=(12,4.7),layout='constrained') labels = [r['case'] for r in result['cases']] x = list(range(7)) axes[0].bar(x,[float(Q(r['siegel_ratio_upper'])) for r in result['cases']],color='#267d69') axes[0].axhline(10/17,color='#a44436',linestyle='--',label='Earlier sufficient bound 10/17') axes[0].set(xticks=x,xticklabels=labels,ylabel='Upper bound for row / surplus ratio', title='Extra selected columns lower coefficient height') axes[0].legend(fontsize=8) axes[1].bar(x,[float(Q(r['minimum_input_gap'])) for r in result['cases']],color='#175e86') axes[1].set(yscale='log',xticks=x,xticklabels=labels, ylabel='Strict lower bound for input minus arithmetic cost', title='Full root-field cost: all ranks in each case') axes[1].axhline(.001,color='#a44436',linestyle='--',label='Proved uniform margin 1/1000') axes[1].legend(fontsize=8) for ax in axes: ax.grid(axis='y',alpha=.2) fig.suptitle('The first second-scale fractional zero block',fontsize=14) fig.savefig(output/'yu-second-fractional-profile.png',dpi=160) plt.close(fig) if __name__ == '__main__': parser = argparse.ArgumentParser() parser.add_argument('--output',type=Path,default=Path(__file__).resolve().parent) parser.add_argument('--figure',action='store_true') args = parser.parse_args() result = calculate() args.output.mkdir(parents=True,exist_ok=True) (args.output/'second-fractional-profile-check.json').write_text(json.dumps(result,indent=2)+'\n',encoding='utf-8') if args.figure: render(result,args.output) print(json.dumps({k:result[k] for k in ('state','finite_ranks','uniform_tail_rows','endpoint_comparisons','strict_mass_gap','strict_input_gap')}))