Let \(P^2=P\) and \(Q=I-P\). Inserting \(I=P+Q\) gives the exact defect
Read the diagram in words
Follow the route from the retained coordinate into the discarded coordinate and back. Compressing each operation separately erases this intermediate route. In the displayed swap example, both individually compressed steps vanish while their full two-step composition returns the retained state.
Take \(P={\operatorname{diag}}(1,0)\) and
Then \(PLP=PMP=0\), while \(PLMP=P\). The missing term \(PLQMP=P\) is the full return through the discarded coordinate.
The running example exhibits the same mechanism in \(PL^2P\):
Starting from \((1,0)\), separate scalar compression predicts \(y_2=1/4\). The exact value is \(1/2\).
The mathematical point
The obstruction is an omitted route through the operator composition. Its size depends jointly on outgoing and returning transfer blocks.
Scope
The algebra requires idempotence. An orthogonal split additionally gives the familiar Hilbert-space energy interpretation.
Formal route
LinearClosure.lean; norm bounds in Normed.lean.