Witness indicators are orthogonal idempotents
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Let \(K\) be a field and \(W=\{0,1\}^m\). The full witness algebra is \(B=K^W\) with pointwise addition and multiplication. For each witness \(v\), the indicator \(\delta_v\) is one at \(v\) and zero elsewhere. Then
The four indicator functions form a basis of \(K^4\). General \(m\) has \(2^m\) such basis elements. A verifier is the sum of the indicators of its accepted witnesses, and pointwise multiplication models intersection of accepting sets.
This algebra retains exact witness compatibility: disjoint singleton predicates multiply to zero. A summary that preserves that product operation while retaining every singleton must preserve the same orthogonal structure. The following dimension theorem says exactly how much fixed linear space that requires.
Exact \(m=2\) multiplication table; general product-algebra identities hold over any field. No characteristic-two assumption is needed for the dimension proof.