Manuscript XXII-B@1 · XXII-B.tex · proposition · 9d5c569c

Proposition: Conditional assembly, not operational derivation

Cone interface and hypothesis-ledger syntax › The conditional classification route · lines 714–740 · proof included or linked

XXII-B@1#prop:classification

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Suppose that, for a particular datum $D$ , the proposed extraction returns the representation of Definition def:representation, with $dim V_{D} = m \geq 3$ . If (eq:homogeneity) and (eq:strict-convexity) hold, then there exists $T_{D} \in GL (V_{D}, R^{m})$ such that

T_{D} (K_{D}) = L_{m} .

In particular,

p ⪯_{D} q ⟺ T_{D} (ι_{D} (q) - ι_{D} (p)) \in L_{m} .

Proof. The classification of strictly convex homogeneous cones gives the linear isomorphism in dimension at least three ItoLourenco (Theorem 7). Apply (eq:faithfulness) and the invertibility of $T_{D}$ . This is the conditional route recorded in En08-A CheungEn08A (thm:weaker) and XXII-A CheungXXIIA (thm:classification).

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unlabeled · line 721
$T_{D} (K_{D}) = L_{m} .$
unlabeled · line 725
$p ⪯_{D} q ⟺ T_{D} (ι_{D} (q) - ι_{D} (p)) \in L_{m} .$

Proposition: Conditional assembly, not operational derivation

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