The Scalar Dagger Lift of the Backward Equalizer Is Rank One and Cannot Recover the Prime-Exclusion Ward Form
The linear zero-state equalizer has the canonical positive lift
But a dagger square of one scalar covector has rank at most one. At the
smallest prime-two module on labels (1,2,3), the native exclusion Ward form
is diag(1,0,1) and has rank two. No scalar weighting or seam coefficient can
remove this mismatch.
Therefore the backward witness must retain labelwise or pro-valued covectors
before forming its dagger or exterior square. Aggregating first and squaring
later cannot recover the arithmetic Ward tower. More generally, at least
rank(P_{p-free}) independent backward observers are required on an
unrestricted cutoff module, so no fixed finite observer family suffices as
the cutoff grows.
Research packet:
research/grothendieck/the-scalar-dagger-lift-of-the-backward-equalizer-is-rank-one-and-cannot-recover-the-prime-exclusion-ward-form.md
Exact checker:
research/grothendieck/checkers/check_scalar_dagger_ward_rank_obstruction.py
The checker passes 5/5 gates.