from fractions import Fraction as Q
from math import lcm
def run(load,check):
 K=529;prev=load('PREVIOUS_DATA.json');reg=load('C286_DATA.json')['target_register'];coords=sorted({r['center'] for r in reg});trials=[];maps=[];maxdiff=0
 for c in prev['vertices']:
  for name,p,j in [('P',1,0),('J',0,1)]:
   form=next(f for f in prev['normal_form_classes'] if f['start']==f['end']==c and (f['E_count'],f['P_count'],f['J_count'])==(1,p,j));slope,inter=map(Q,form['affine_coefficients']);hits=[];md=max(abs(x-c) for x in coords);maxdiff=max(maxdiff,md)
   check('circuit denominator exceeds source range '+str((c,name)),slope.denominator in [1587,6877] and md<slope.denominator and inter==(1-slope)*c)
   for x in coords:
    y=slope*x+inter;member=y in coords
    trials.append({'source_vertex':c,'circuit':name,'input':x,'output':str(y),'source_member':member})
    if member:hits.append(x)
   check('only the basepoint remains in source menu '+str((c,name)),hits==[c])
   spans={str(n)+'K':str(slope*n*K) for n in [2,17]}
   want={'2K':'3500/3','17K':'29750/3'} if name=='P' else {'2K':'15000/13','17K':'127500/13'}
   check('formal529 span images '+str((c,name)),spans==want and all(Q(v).denominator>1 for v in spans.values()))
   maps.append({'source_vertex':c,'circuit':name,'multiplier':str(slope),'reduced_denominator':slope.denominator,'maximum_source_displacement':md,'source_preserving_inputs':hits,'formal_span_images':spans,'K_lattice_index_divisor':slope.denominator})
 check('complete finite source evaluation',len(coords)==373 and len(trials)==2238 and sum(t['source_member'] for t in trials)==6)
 check('common529 lattice index',lcm(1587,6877)==39*K==20631 and K*lcm(1587,6877)==10913799)
 check('all longer positive circuits also exceed menu',K*K>maxdiff)
 return {'step':'C401','source_coordinate_range':[min(coords),max(coords)],'generator_maps':maps,'source_trials':trials,'counts':{'unique_circuit_generators':6,'regular_coordinate_trials':2238,'source_preserving_inputs_across_maps':6,'nonfixed_source_images':0,'preserved2K_or17K_spans':0},'positive_circuit_reduced_denominator':'For p P-circuits and j J-circuits, D=3^max(p-j,0)*13^j*529^(p+j). The numerator contains no factor23. For p+j>=2, D>=529^2, already beyond all source displacements; the one-circuit denominators1587/6877 also exceed them.','K_lattice_condition':'For the P generator, c+Kn maps to another c+K integer point iff1587 dividesn; for J iff6877 dividesn. Both conditions require20631=39K dividesn. This is a domain sublattice, not preservation of the entireK lattice.','minimum_nonzero_common_K_lattice_displacement':10913799,'interpretation':'Every nonempty positive circuit preserves exactly its selected basepoint in the frozen regular coordinate menu. The local source-path grammar is valid, but its affine extensions are not actions on the whole source field or the original529 lattice. Formal large lattice displacements are arithmetic diagnostics, not dates or source endpoints.'}