from research import *
s=begin(1064,'Common-pivot gap obstruction','Can paired endpoints990 or1980 apart both have integral Key images about one integer pivot?',{'gaps':[990,1980],'denominators':[23,69,299]},['File52c paired branches','C850–C858','C998'])
rows=[{'gap':d,'residues':{str(q):d%q for q in s['inputs']['denominators']}} for d in s['inputs']['gaps']]
finish(s,{'gap_tests':rows,'proof':'Two integral images require both x−a and y−a divisible by the reduced denominator, hence require x−y divisible by it.','scope':'necessary-condition theorem; no pivot or target search'},'Neither long-core gap allows both paired endpoints to expand integrally about any one common integer Key pivot. The impossibility follows from their difference alone.','State the corresponding symbolic conditions for the remaining branch forms.',{'all_obstructed':all(d%q!=0 for d in [990,1980] for q in [23,69,299]),'mod23':[d%23 for d in [990,1980]]==[1,2]})