from research import *
s=begin(1002,'Domain covariance under translation','Does translating both pivots and the whole field preserve the domain?',{'translation':215},['C998','C1001'])
t=s['inputs']['translation'];r,m=68,529;x=r+2*m;a,b=14006,4836
y=a+E*(x-a);z=b+E*(y-b);yy=(a+t)+E*((x+t)-(a+t));zz=(b+t)+E*(yy-(b+t))
finish(s,{'translated_residue':(r+t)%m,'unchanged_period':m,'first_output_shift':yy-y,'second_output_shift':zz-z,'general_rule':'Translate points and both pivots together; residue shifts by t.'},'Whole-frame translation preserves the domain obstruction and period while shifting the residue. This is the same structural covariance seen in the SP cap capacities.','Represent a nonintegral source field exactly instead of changing its values.',{'stage_shifts':yy-y==zz-z==t,'same_pivot_difference':(b+t)-(a+t)==b-a})