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

Proposition: Allowed reference rebasing preserves membership

Certificate patterns that do not transfer › Class-invariant information and reference rebasing · lines 1771–1796 · proof included or linked

II@2#prop:rebase

Back to manuscript

Keep the external domain, response prescription, and complete contact space fixed. If

m^{'} = m + ℓ \sum ρ_{ℓ} n_{ℓ}

with constant allowed coefficients, then

[m^{'}] = [m] in R^{X} / C .

Global membership is equivalent for $m, m^{'}$ . On every finite family, fitting vectors transform by $α \mapsto α + ρ$ , separators have the same target pairing, and augmented ranks are unchanged.

Proof. The difference lies in $C$ . At finite evaluation, $m^{'}_{S} = m_{S} + N_{S} ρ$ . This proves the fitting statement. A left annihilator of $N_{S}$ annihilates the difference, and an elementary augmented-column operation preserves rank.

Displayed formulas in this passage

unlabeled · line 1776
$m^{'} = m + ℓ \sum ρ_{ℓ} n_{ℓ}$
unlabeled · line 1780
$[m^{'}] = [m] in R^{X} / C .$

Nearby in this section

section: Class-invariant information and reference rebasing