from research import *
s=begin(1307,'Register Key compatibility with complete path sums','When does applyingKeys before oraftergrouping give thesamefield?',{},['C1306','C984–C1007 inherited','inherited/model/completed_path_interface_ledger.json'])
r={'id':'Key_sum_compatibility','class':'identity_with_domain_condition','uniform_statement':'k sum(w_i)=sum(k w_i)','anchored_statement':'Applytospansaboutthedeclaredpivot;do notmultiplyrawBClabels.','mixed_statement':'ForblocksumB anddiagonalfactorsK,thereexistsblockfactorKbar withBK=Kbar B forallinputs iff eachblock’scomponentfactorsareconstant.','proof':'Uniformcaseisdistributivity; mixedcasecomparesthecoefficientofeachindependentcomponentwithinablock.','applications':['threeKeyscommoncalendarvolume','fourcompletedinversepathwhole-spans','partialversusuniform483→490 comparison'],'limits':['Rationalwhole-spancompletiondoesnotmaketheproperprefixesintegral','DifferentcomponentKeysneedsource-appointedcutroles','Equalendpoint490doesnotimplyequalinteriors'],'evidence':['inherited/model/completed_path_prefix_field.json','inherited/model/retained_part_complete_field.json','inherited/model/mixed_calendar_complete_field.json']}
a=artifact('model/relation_Key_sums.json',json.dumps(r,indent=2)+'\n')
finish(s,{'Key_sum_relation':a},'UniformKeys commute withsumming aselectedpath. DifferentcomponentKeys commute withblockgroupingonlyunder aconstant-factorcondition withinthatblock, so retainedcutsremain explanatorydata.','Register anchoredcomposition andintegerintermediate-domain conditions.',{'bound_evidence':all((ROOT/p).exists() for p in r['evidence']),'three_limits':len(r['limits'])==3})