C993 — Universal coarsening criterion

Question

When can a componentwise conversion be replaced by one factor per merged block?

Inputs

{
  "coarsening": "sum fine components into declared blocks",
  "conversion": "diagonal factors k_i"
}

Sources

[
  "C991",
  "C992",
  "File63 retained partitions"
]

Opened utc

2026-09-28T03:54:01.790588+00:00

Results

{
  "necessity": "Test each unit vector e_i: the fine factor k_i must equal its containing coarse-block factor.",
  "sufficiency": "If factors are constant within each block, factoring that common value out of the block sum proves equality.",
  "criterion": "C·diag(k_i)=diag(k_B)·C iff k_i=k_B within every block"
}

Finding

Coarsening commutes with conversion for every vector exactly when the factors are constant inside each merged block. Equality on one supplied total is weaker.

Reassessment

Exhibit the information hidden by one equal-total partial/whole comparison.

Checks

{
  "uniform_example": true,
  "mixed_not_universal": true
}

Closed utc

2026-09-28T03:54:01.791014+00:00

Record provenance

Exact record hash: e154b01f56e7b52fcf5a1193c7078496c77ac689a15b3616b00dbaced8f3cb8f

Source journal: Research_Cycles/C0932_C1131/journal.json