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

Finite certificate alternative

Finite certificate principles › Membership or separation · lines 2459–2475

II@2#fig:finite-alternative

General certificate principle — not the present three-row result.

(a)Finite alternative

contact image

\operatorname{im} N

u_{\mathrm{in}}

inside:

N\alpha=u_{\mathrm{in}}

, ranks equal

u_{\mathrm{out}}

outside:

zN=0,\ zu_{\mathrm{out}}=1

, augmented rank increases by 1

Exactly one alternative for a fixed target.

(b)At most r+1 supporting rows

$r$ pivot rows $R$ , one extra row $j$ , original columns $B$ .

M=B_R,\qquad \delta=u_j-b_j M^{-1}u_R\ne 0

z_R=-b_j M^{-1},\qquad z_j=1

Other row entries are zero. Normalize by $\delta$ .

\det\begin{pmatrix}M&u_R\\ b_j&u_j\end{pmatrix}=\det(M)\,\delta

No outside target or separator is asserted for the family studied here.

The finite alternative is exact linear algebra. Its global interpretation additionally depends on the response contract and coverage.

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