"""Exact certificates for the second-scale integer extension, TR-BAKER-10. Authored and self-checked by GPT-6 Astra (OpenAI), Ultra. CC0. The lesson proves the uniform reductions. This script verifies their rational coefficients, endpoints and exponential-series bounds. """ from fractions import Fraction as Q from math import comb, factorial from pathlib import Path import argparse import json def calculate(): import sympy as sp y = sp.symbols('y') lam, xi = Q(59, 58), Q(101, 100) eps, rem = Q(13, 18000), Q(1, 79) + Q(1, 71928 * 1500) # name,c1,c3,c4,a,q,zeta2,A0min,dmin,normal factor, # strict finite-step lower bound, strict tail-at-five lower bound cases = [ ('I.1', Q(7,10), Q(6,5), 14, 7, 2, Q(5101,5000), Q(3,2), 2, Q(3,2)+xi/2, Q(7,10), Q(12,5)), ('I.2', Q(7,10), Q(6,5), 14, 7, 2, Q(5101,5000), Q(3,2), 1, Q(3,2)+xi/2, Q(69,100), Q(119,50)), ('II', Q(7,10), Q(6,5), 14, 7, 2, Q(5101,5000), Q(5,2), 2, Q(5,4)+xi/4, Q(93,100), Q(67,25)), ('III.1', Q(7,10), Q(8,5), 14, 7, 2, Q(2083,2000), Q(3,4), 2, xi, Q(73,100), Q(67,50)), ('III.2', Q(7,10), Q(8,5), 14, 7, 2, Q(2083,2000), Q(3,4), 1, xi, Q(18,25), Q(33,25)), ('IV', Q(7,10), Q(3), 14, 7, 2, Q(1321,1250), Q(1,2), 2, Q(7,6)+xi/6, Q(93,100), Q(41,50)), ('V', Q(41,50), Q(3,5), 6, 13, 3, Q(507,500), Q(2), 2, 1+xi/2, Q(9,2), Q(57,10)), ] results, total_coefficients = [], 0 for name,c1,c3,c4,a,q,zeta,amin,dmin,nfactor,finite_bound,tail_bound in cases: c2 = Q(a,q*q)/zeta c5 = Q(27,50) if q == 2 else Q(4,5) ell = Q(7,50) if q == 2 else Q(11,50) gamma, normal = c3*q*dmin, lam*nfactor/(c1*c4) polynomials = [] for j in range(1,5): eta = 1-c5*y analytic = 2*c5*q**(j+1)/c1*eta**j-c5*q/(c1*gamma)*eta**j*y*y common = (Q(q**(j+1))/(c1*c2) + Q(10,17)/(2*c1*c2)*(1+y*y/(2*gamma)) + y*y/2+Q(22,17)*eps+Q(27,17)*rem+normal) b = lam*(1+(j+1)*ell+Q(10,17)) k = (Q(107,103)+Q(3,8)*y)*eta**(j+1)+Q(10,17)*(1-Q(483,824)*y) interval = Q(1,max(4,j+2)) for endpoint in ('minimum','infinity'): ratio = b/c4 if endpoint == 'infinity' else (b*amin/c4+k/c3)/(1+amin) poly = sp.Poly(sp.expand(analytic-common-xi/c1*ratio),y) n = j+2 assert poly.degree() <= n power = [Q(poly.nth(k))*interval**k for k in range(n+1)] bernstein = [sum(power[k]*Q(comb(i,k),comb(n,k)) for k in range(i+1)) for i in range(n+1)] reconstructed = sum(bernstein[i]*comb(n,i)*(y/interval)**i *(1-y/interval)**(n-i) for i in range(n+1)) assert sp.expand(reconstructed-poly.as_expr()) == 0 assert min(bernstein) > finite_bound > Q(1,2) total_coefficients += len(bernstein) polynomials.append({'step':j,'A0_endpoint':endpoint,'interval_upper':str(interval), 'degree':n,'bernstein_coefficients':list(map(str,bernstein)), 'minimum_coefficient':str(min(bernstein))}) eta_lower = Q(29,50) if q == 2 else Q(4,9) alpha = (2*c5*eta_lower-1/c2)/c1 constant = (c5*q/(16*c1*gamma) + Q(10,17)/(2*c1*c2)*(1+Q(1,32)/gamma) + Q(1,32)+Q(22,17)*eps+Q(27,17)*rem+normal + xi/c1*(lam*(1+Q(10,17))/c4 +(Q(107,103)+Q(10,17))/(c3*(1+amin)))) slope = xi*lam*ell/(c1*c4) value_at_five = alpha*q**6-constant-6*slope increment_at_five = alpha*(q-1)*q**6-slope assert alpha > 0 and increment_at_five > 0 assert value_at_five > tail_bound > Q(1,2) results.append({'case':name,'finite_lower_bound':str(finite_bound), 'finite_polynomials':polynomials,'tail_alpha':str(alpha), 'tail_constant':str(constant),'tail_slope':str(slope), 'tail_value_at_five':str(value_at_five), 'tail_increment_at_five':str(increment_at_five), 'tail_lower_bound':str(tail_bound)}) exponential_bounds = [] for c, upper in [(Q(27,50),Q(50,29)),(Q(4,5),Q(9,4))]: series_upper = sum(c**k/factorial(k) for k in range(7)) series_upper += c**7/factorial(7)/(1-c/8) assert series_upper < upper exponential_bounds.append({'exponent':str(c),'series_upper':str(series_upper), 'strict_upper':str(upper),'gap':str(upper-series_upper)}) assert total_coefficients == 308 assert Q(725,82944) < Q(1,100) assert Q(101,100)*Q(115,48)*Q(27,50)/(Q(7,10)*2) == Q(20907,22400) < Q(15,16) assert Q(101,100)*Q(115,48)*Q(4,5)/(Q(7,10)*3) == Q(2323,2520) < Q(15,16) return {'state':'exact_uniform_reductions_passed', 'scope':'All depth-zero integer stages through q^r S, order eta^r T, every r>=3 and every stated field case. No fractional or contracted-stage assertion.', 'cases':results,'finite_polynomials':56,'bernstein_coefficients':308, 'exponential_bounds':exponential_bounds,'later_step_gap_lower_bound':'1/2', 'local_input_upper_bound':'15/16'} def render(result, output): import matplotlib matplotlib.use('Agg') import matplotlib.pyplot as plt fig,axes=plt.subplots(1,2,figsize=(12,4.8),layout='constrained') ax=axes[0] stages=list(range(7));eta=Q(1)-Q(27,50)/7 ax.plot(stages,[2**j for j in stages],marker='o',color='#175e86',label=r'Radius $2^j S$') ax.set_yscale('log',base=2) ax.set(xlabel='Integer stage j',ylabel='Radius / S',xticks=stages, title='Example: r = 6, q = 2, c5 = 27/50') ax.set_yticks([1,2,4,8,16,32,64],labels=['1','2','4','8','16','32','64']) right=ax.twinx() right.plot(stages,[float(eta**j) for j in stages],marker='s',color='#bd6b22',label=r'Order $\eta^j T$') right.set(ylabel='Prepared total order / T',ylim=(.55,1.025)) ax.yaxis.label.set_color('#175e86');ax.tick_params(axis='y',labelcolor='#175e86') right.yaxis.label.set_color('#bd6b22');right.tick_params(axis='y',labelcolor='#bd6b22') ax.grid(alpha=.2) ax=axes[1] labels=[row['case'] for row in result['cases']] finite=[min(float(Q(p['minimum_coefficient'])) for p in row['finite_polynomials']) for row in result['cases']] tail=[float(Q(row['tail_value_at_five'])) for row in result['cases']] xs=list(range(7)) ax.bar([x-.19 for x in xs],finite,width=.38,label='Steps 1–4: coefficient minimum',color='#267d69') ax.bar([x+.19 for x in xs],tail,width=.38,label='Steps ≥5: tail lower bound',color='#175e86') ax.axhline(.5,color='#a44436',linestyle='--',label='Common strict gap 1/2') ax.set(xticks=xs,xticklabels=labels,ylabel='Gap lower bound / [Zd/(et)]', title='Uniform bounds for every later integer step') ax.legend(fontsize=8);ax.grid(axis='y',alpha=.2) fig.suptitle('Exact zeros on every enlarged integer interval',fontsize=14) fig.savefig(output/'yu-second-integer-extension.png',dpi=160) plt.close(fig) def main(): parser=argparse.ArgumentParser();parser.add_argument('--output-dir',type=Path,default=Path(__file__).parent) output=parser.parse_args().output_dir;output.mkdir(parents=True,exist_ok=True) result=calculate() (output/'second-integer-extension-certificate.json').write_text(json.dumps(result,indent=2)+'\n') render(result,output) print(json.dumps({k:result[k] for k in ['state','finite_polynomials','bernstein_coefficients']})) if __name__=='__main__':main()