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Manuscript IX-C@1 · IX-C.tex · section · 03ce8e6b

Completed results

Retained obstructions and the unresolved release › Completed results · lines 1388–1430

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{}1.12

Object	Status
$B_{f} ⋆_{1} W_{g}$	Constructed on the named first-stage domain
$W_{g} ⋆_{1} B_{f^{'}}$	Constructed on the named reverse domain
Three one-edge polynomial outputs	Existence and admission proved from inherited closure
Three two-edge labelled expressions	Identified; sharp existence/admission unresolved here
Scalar map on $L_{2} \times B$	Exact descent test proved; test not discharged
Scalar map on $B \times R_{2}$	Exact descent test proved; test not discharged
$(B_{f} ⋆_{1} W_{g}) ⋆ B_{f^{'}}$	Unresolved; second-stage map not constructed
$B_{f} ⋆ (W_{g} ⋆_{1} B_{f^{'}})$	Unresolved; second-stage map not constructed
Associativity through order three	Exact criterion proved; construction hypotheses unresolved
Second-stage involution	Unresolved for nonconstant maps
Quantum presentation independence	Conditional on actual descent; not released as completed
VII-C obstruction	Retained under its original hypotheses
IX-D and interacting products	Not opened
G2	Untouched; original conditional boundary retained
V's three quantum objects	Separation only; not used in proofs

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theorem: Theorem: IX-C reduction package and release status