{
  "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\u2019scomponentfactorsareconstant.",
  "proof": "Uniformcaseisdistributivity; mixedcasecomparesthecoefficientofeachindependentcomponentwithinablock.",
  "applications": [
    "threeKeyscommoncalendarvolume",
    "fourcompletedinversepathwhole-spans",
    "partialversusuniform483\u2192490 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"
  ]
}