"""Exact comparisons for the later second-scale fractional and contraction steps. GPT-6 Astra (OpenAI), Ultra. Independent mathematics and figure; CC0. The accompanying proof supplies uniform reductions and actual-family hypotheses. Rational arithmetic only; no sampled depth argument. """ from fractions import Fraction as Q from pathlib import Path import json from yu_second_fractional_profile import case_data, exp_bounds 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) logq_upper=Q(7,10) if q==2 else Q(11,10) rmin=3 if q==2 else 4 comparisons=[] for r in range(rmin,9): tail=r==8 y=Q(1,r+1);eta=1-c*y smin=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 qbase=q*hminus assert qbase>1 logQ_lower=2*(qbase-1)/(qbase+1) torus=(1/hminus+1/smin)/c2 other=(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**2+w*(1 if tail else 1-Q(483,824)*y) depth_b=3*lam*logq_upper/(c4*logQ_lower) if tail: gamma=(2*c*q/exp_bounds(c)[1]*sum(Q(1,q**j) for j in range(4)) -Q(2*q*9*(q+7),q**8)/smin) else: gamma=(2*c*q/Q(q**r)*(sum((q*eta)**j for j in range(r+1))+q-2) -Q(q*(r+1),q**r)/smin *(2*(q+1)*(eta**2-hminus)+2*sum(eta**j-hminus for j in range(3,r+1)))) normal=lam*nfactor/(c4*q**r) input_lower=c1*q*chi-xi*(1-hminus+c*(1+2*hplus)*y+1/tmin)/(q**r*c3*(1+amin)) for depth_endpoint in ['zero','maximum']: base=torus if depth_endpoint=='zero' else torus/32768 linear=b/c4+(depth_b if depth_endpoint=='maximum' else 0) arith={'minimum':base+other+xi*(linear*amin+k/c3)/(1+amin), 'infinity' if tail else 'rank':base+other+xi*linear if tail else base+other+xi*(linear*r+k/c3)/(1+r)} for endpoint,bound in arith.items(): ga,gi=gamma-normal-bound,input_lower-bound gc=2*c*q*(q-1)-lam*nfactor/c4-bound ic=c1*chi*q**(r+1)-xi*(3*c*y+1/tmin)/(c3*(1+amin))-bound assert ga>Q(7,10) and gi>Q(2,25),(name,r,depth_endpoint,endpoint) assert gc>Q(7,10) and ic>9,(name,r,depth_endpoint,endpoint) comparisons.append({'ranks':'all >= 8' if tail else r, 'depth_endpoint':depth_endpoint,'A0_endpoint':endpoint, 'arithmetic_upper':str(bound),'mass_lower':str(gamma), 'normal_loss_upper':str(normal),'input_lower':str(input_lower), 'mass_gap':str(ga),'input_gap':str(gi), 'contraction_mass_gap':str(gc),'contraction_input_gap':str(ic)}) stated={'I.1':(Q(107,100),Q(1,5)), 'I.2':(Q(107,100),Q(2,25)), 'II':(Q(11,10),Q(21,100)), 'III.1':(Q(103,100),Q(13,100)), 'III.2':(Q(51,50),Q(7,25)), 'IV':(Q(26,25),Q(49,100)), 'V':(Q(18,25),Q(39,100))}[name] assert all(Q(v['mass_gap'])>stated[0] and Q(v['input_gap'])>stated[1] for v in comparisons) rows.append({'case':name,'comparisons':comparisons, 'stated_mass_gap':str(stated[0]),'stated_input_gap':str(stated[1]), '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)), 'minimum_contraction_mass_gap':str(min(Q(v['contraction_mass_gap']) for v in comparisons)), 'minimum_contraction_input_gap':str(min(Q(v['contraction_input_gap']) for v in comparisons))}) assert Q(173,200)**4>Q(1,3) and Q(21,25)**5>Q(1,3) assert Q(27,50)*Q(1,3)*Q(12,29)*12**3>1 return {'state':'all_exact_later_fractional_and_contraction_comparisons_passed','scope':'Theorems207–208: I>=1, I log Q<=3B, c T_(I+1)/(r+1)>=1. Full root degree and rational denominators retained; finite ranks use the proved A0=1 lhs=(q-1)*(1-hf)+(eta-hf)+sum((q**j-q**(j-1))*(eta**j-hf) for j in range(2,r+1)) rhs=c/(r+1)*(sum((q*eta)**j for j in range(r+1))+q-2) assert lhs==rhs return {'scope':'Exact counting illustration, not a number-field parameter instance.', 'S_I':str(si),'T_I':str(ti),'output_order':output,'radii':radii, 'multiplicities':mu,'counts':counts,'deleted_increment':mu[0]-mu[1], 'mass':mass,'layered_mass':layered} def render(result,example,path): import matplotlib matplotlib.use('Agg') import matplotlib.pyplot as plt fig,axes=plt.subplots(1,2,figsize=(12,4.9),layout='constrained',gridspec_kw={'width_ratios':[1.2,1]}) ax=axes[0] bars=ax.bar(range(4),example['multiplicities'],color=['#175e86','#267d69','#72519a','#b26836']) for bar,n in zip(bars,example['counts']): ax.text(bar.get_x()+bar.get_width()/2,bar.get_height()+6,f'{n} nodes\norder weight {int(bar.get_height())}',ha='center',fontsize=8) ax.set(xticks=range(4),xticklabels=['Inner odd\n|s| <= 38','Inner even\n|s| <= 38','Annulus\n38 < |s| <= 75','Annulus\n75 < |s| <= 150'],ylim=(0,355),ylabel='Interpolation multiplicity',title='Exact mixed profile: total degree 37,530') ax.tick_params(axis='x',labelsize=8) ax=axes[1] labels=[v['case'] for v in result['cases']] bounds=[Q(v['stated_input_gap']) for v in result['cases']] bars=ax.bar(range(7),[float(v) for v in bounds],color='#175e86') for bar,v in zip(bars,bounds):ax.text(bar.get_x()+bar.get_width()/2,bar.get_height()+.012,str(v),ha='center',fontsize=8) ax.axhline(.08,ls='--',color='#a44436',label='Uniform strict margin 2/25') ax.set(xticks=range(7),xticklabels=labels,ylim=(0,.56),ylabel='Proved lower bound in normalized valuation units',title='Fractional input margins\nthrough the admissible depths') ax.tick_params(axis='x',labelsize=8);ax.legend(fontsize=8,loc='upper left') for ax in axes:ax.grid(axis='y',alpha=.18);ax.set_axisbelow(True) fig.savefig(path,dpi=160);plt.close(fig) if __name__=='__main__': result=calculate() example=counting_example();result['counting_example']=example Path(__file__).with_name('second-later-fractional-check.json').write_text(json.dumps(result,indent=2)+'\n') import sys if '--figure' in sys.argv:render(result,example,Path(__file__).with_name('yu-second-later-fractional.png')) print(json.dumps({'state':result['state'],'comparisons':result['endpoint_comparisons'], 'gaps':[{k:r[k] for k in ['case','minimum_mass_gap','minimum_input_gap']} for r in result['cases']]}))