Manuscript Hawk02-D@1 · Hawk02-D.tex · proposition · e6e4b72f

Proposition: Rank obstruction for identity-target Gram tests

Debugging the proposed imports and physical interpretation › A Gram certificate has a rank obstruction · lines 1716–1728 · proof included or linked

Hawk02-D@1#prop:gram

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Suppose a proposed certificate is $κ = ∥ G - I_{d r} ∥$ , where $G$ is the Gram matrix of $d r$ vectors in the accessible Hilbert space $R$ . If $d r > dim R$ , then $κ \geq 1$ .

Proof. The Gram matrix has rank at most $dim R < d r$ . Choose a unit vector $v$ in its kernel. Then $(G - I) v = - v$ , so $∥ G - I ∥ \geq 1$ .

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section: A Gram certificate has a rank obstruction