from research import *
s=begin(1001,'Whole-field placement and difference conditions','What makes every point of a source field belong to one admissible domain?',{},['C998','C1000'])
proof={'criterion':'All x_i lie in r+mZ iff x_0≡r mod m and every x_i−x_0 is divisible by m.','difference_condition':'Controls field shape up to translation.','placement_condition':'One retained point fixes the residue class.'}
old={'C480':{'points':[2148,3206],'residue':32,'period':529},'fixed12026':{'points':[4114,3056],'residue':388,'period':529}}
for v in old.values():v['width_ok']=(v['points'][1]-v['points'][0])%v['period']==0;v['placement_ok']=all((x-v['residue'])%v['period']==0 for x in v['points'])
finish(s,{'criterion':proof,'inherited_witnesses_reclassified':old},'A whole-field test separates the difference lattice from one placement residue. The inherited529-width examples share the width condition but differ in placement.','Check how this domain transforms under a whole-frame translation.',{'both_widths':all(v['width_ok'] for v in old.values()),'placement_distinction':old['C480']['placement_ok'] and not old['fixed12026']['placement_ok']})