from research import *
from algebra import Matrix
s=begin(1259,'Derive the necessary and sufficient coefficient-five calibration','When do the row gaps equal5(C+2T) and5C?',{},['C1254','C1256','Strategy §4.1'])
H=Matrix([[10,5],[0,5]]);coeff=list(H.inv()*Matrix([F(25,2),F(13,2)]));u=100;T,C=[v*u for v in coeff]
a=artifact('model/regular_kernel_five_calibration.json',json.dumps(clean({'equations':['25u/2=10T+5C','13u/2=5C'],'solution':{'T_over_u':coeff[0],'C_over_u':coeff[1]},'source_values':{'u':u,'T':T,'C':C},'proof':'Invert thetwoequations inT,C; determinant50 is nonzero, so conditionsare necessaryandsufficient.','not_constrained':'SojournS doesnotappear','source_cost':'The rowmasks do not entailthese ratios; theyare additionalobserved cross-sourcecalibration relations.'}),indent=2)+'\n')
finish(s,{'calibration':a,'ratios':coeff},'The fivefoldexpression holds exactly whenT=3u/5 andC=13u/10. These match60 and130 at u100, but the ratios are additional source relations; the row supports alone do not generate them.','Separate compatibility ofa commoncoefficient fromthe choice thatcoefficient equalsfive.',{'ratios':coeff==[F(3,5),F(13,10)],'sourcevalues':T==60 and C==130,'reconstruction':F(25,2)*u==5*(C+2*T) and F(13,2)*u==5*C})