from fractions import Fraction as Q
def run(load,check):
 reg=load('C286_DATA.json')['target_register'];cases=load('C396_DATA.json')['hits'];rows=[];good={500:[],502:[]}
 def find(r,role,rail,shift):
  gear=r['gear']-(shift-500)//2
  return [s for s in reg if s['role']==role and s['gear']==gear and s.get('rail')==rail and s['center']==r['center']-shift and all(s.get(k)==r.get(k) for k in ['tradition','Cainan','Terah60','full430'])]
 for shift in [500,502]:
  for i,h in enumerate(cases):
   slots=[]
   for name,r in h['records'].items():
    role='Shem death' if name=='J_Flood' else 'Shem birth';rail=h['records']['higher_Noah'].get('rail') if name=='J_Flood' else r.get('rail');hits=find(r,role,rail,shift)
    slots.append({'role_in_loop':name,'source':r,'required_role':role,'required_Gear':r['gear']-(shift-500)//2,'required_coordinate':r['center']-shift,'matching_records':hits})
   for k,r in enumerate(h['P_return_source_records']):
    hits=find(r,'Shem death','primary' if r['tradition']=='SP' else None,shift);slots.append({'role_in_loop':'P_pivot_'+str(k),'source':r,'required_role':'Shem death','required_Gear':r['gear']-(shift-500)//2,'required_coordinate':r['center']-shift,'matching_records':hits})
   passed=all(len(s['matching_records'])==1 for s in slots)
   check('binding boundary '+str((shift,i)),passed==(shift==500 or h['lower_Gear']==2))
   if not passed:check('only unavailable Gear0 slots '+str((shift,i)),all(s['required_Gear']==0 for s in slots if not s['matching_records']))
   loop=[h[k]-shift for k in ['E_Noah_pivot','E_input','E_output','J_return_pivot']]
   if passed:
    good[shift].append(loop);V,x,y,A=loop;check('admitted loop arithmetic '+str((shift,i)),Q(25,23)*(x-V)==y-V and Q(300,299)*(y-A)==x-A)
   rows.append({'shift':shift,'mode':'500/100' if shift==500 else '502/98','case':i,'original_lower_Gear':h['lower_Gear'],'slots':slots,'all_required_records_sourced':passed,'coordinate_loop':loop if passed else None})
 a={tuple(r) for r in good[500]};c={tuple(r) for r in good[502]}
 check('502 outputs are a subset of500 source configurations',len(a)==6 and len(c)==3 and c<a)
 complete502=[r for r in rows if r['shift']==502 and r['all_required_records_sourced'] and any(s['role_in_loop'].startswith('P_pivot') for s in r['slots'])]
 check('one complete502 coordinate graph',len(complete502)==1 and complete502[0]['coordinate_loop']==[2556,2533,2531,1933])
 return {'step':'C407','binding_trials':rows,'counts':{'labelled_binding_trials':len(rows),'admitted500_labelled_loops':len(good[500]),'admitted502_labelled_loops':len(good[502]),'admitted502_coordinate_loops':len(c),'complete502_E_P_J_graphs':len(complete502),'new_coordinate_loops_from502':0,'unavailable502_Gear0_cases':sum(r['shift']==502 and not r['all_required_records_sourced'] for r in rows)},'interpretation':'Both bindings remain valid.500 preserves allG1–G3 source loops;502 lowers every required target Gear byone, so only the originalG2–G3 segment survives as targetG1–G2. Missing Gear0 is a domain boundary, not rejection of502/98. No−1 companion was imported into MT or LXX, and Flood-close identity remains independent.'}