λEnter Calculus CompanionRegistered manuscripts, kept distinct

Manuscript II@2 · II.tex · figure · 50d64def

Finite and global certificates

Complete matrix encoding and the finite certificate › Rank factorization with an original minor · lines 1657–1675

II@2#fig:certificate-scope

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Functions on all of X\mathcal Xm∈C⟺∃α ∀β∈X: m(β)=∑ℓαℓnℓ(β)m\in\mathcal C\Longleftrightarrow \exists\alpha\ \forall\beta\in\mathcal X:\ m(\beta)=\sum_\ell\alpha_\ell n_\ell(\beta)

One constant vector α\alpha.

evaluation
Evaluations on a finite family SS
Finite fit NSα=mSN_S\alpha=m_S
Complete finite separator zNS=0,zmS≠0zN_S=0,\quad zm_S\ne 0
separator → global nonmembership, by complete contact coverage
finite fit ↛ global membership without a global identity or injective evaluation
SEP-yes
SEP-no
SEP-finite — this result
SEP-open

The scope of the finite certificate. Dashed transport is not an established implication without its additional hypothesis.

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