from research import *
s=begin(1250,'Determine the arithmetic freedom in the430/490 bridge','Does the integer bridge430m−490n=130 have only the source solution?',{},['C1238','File51a §7A.2'])
base=[6,5];step=[49,43];examples=[[base[0]+step[0]*k,base[1]+step[1]*k] for k in [0,1]]
a=artifact('model/Moses_cycle_bridge_integer_family.json',json.dumps({'equation':'43m−49n=13','source_solution':base,'all_integer_solutions':'m=6+49k,n=5+43k for integerk','proof':'Subtract theknownsolution:43(m−6)=49(n−5);coprimality of43and49 forces the stated steps.','positive_solutions':'k>=0; the source solution is the smallest positive pair in this fixed coefficientequation.','examples':examples,'limitation':'The equationdoesnot identifyMoses orauthorizeotherchronologicalendpoints; fullsourcepath2580 supplies the application.'},indent=2)+'\n')
finish(s,{'bridge_family':a},'The bridge has infinitely many positive integer solutions, with6/5 the smallest in this fixed430/490 equation. That arithmetic property is separate from the source path’s identification ofMoses.','Assess the minimal source packet that regenerates the completeMosesfamily rather than just itscyclelabels.',{'source_equation':430*6-490*5==130,'step_homogeneous':430*49-490*43==0,'examples':all(430*m-490*n==130 for m,n in examples)})