Faithful linear gluing algebras
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CLAIM C06 This chapter states an exponential dimension result for a specified summary model. The qualifications are part of the theorem's meaning: a lower bound on an algebra's dimension is not automatically a lower bound on the length of a circuit describing one of its elements.
18.1 The witness algebra
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Let \(K\) be a field and let \(W={\{0,1\}}^m\). The full witness algebra is
with pointwise addition and multiplication. For each witness \(v\), let \(\delta_v\) be its indicator function. These form a basis of \(\mathcal B\) and satisfy
In characteristic two, this is equivalently the Boolean quotient algebra
Theorem 18.1: Dimension of a faithful multiplicative summary
Let \(\Phi:\mathcal B\to A\) be a \(K\)-linear multiplicative map into a \(K\)-algebra. Suppose
Then
No unital hypothesis on \(\Phi\) is required.
Proof
Write \(e_v=\Phi(\delta_v)\). Multiplicativity makes these nonzero orthogonal idempotents. If
multiply the equality by \(e_u\). Orthogonality annihilates every term except \(c_ue_u\), which must be zero. Since \(K\) is a field and \(e_u\ne0\), \(c_u=0\). This holds for every \(u\), so the \(2^m\) vectors \(e_v\) are linearly independent.
Corollary 18.2: Scalar unital summaries are evaluations
Every unital \(K\)-linear multiplicative map \(\Phi:K^W\to K\) is evaluation at exactly one witness.
Proof
An idempotent scalar in a field is either zero or one. Orthogonality allows at most one \(\Phi(\delta_v)\) to be one, and preservation of the unit requires exactly one. If it occurs at \(v_0\), linearity gives
For \(m\ge1\), such a scalar summary loses at least one singleton witness other than \(v_0\). At \(m=0\), the unique empty witness is preserved by the identity map. The dimension theorem is an elementary product-algebra fact, not a claim of new algebra or a universal computational lower bound.