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Vtree Alignment and Higher-Order Four-Corner
Defects in Exact Boolean Factorization
Research draft
29 September 2026
Abstract
We isolate two structural limits in exact AND-factor extraction
from vtree-structured Boolean representations. First, maximizing the
number of factors exposed by exact vtree cuts is a complete objec-
tive: its global optimum equals the semantic prime-factor count, while
hard carrier-subtree constraints are compatible with semantic prime
alignment exactly when the carrier and semantic block families are
laminar. Second, a nonzero four-corner defect with varying supports
U and W proves that some semantic prime block meets both supports,
but this cannot in general be reduced to a pairwise interaction. The
family supported only on 0 n and 1 n is semantically prime although
every fixed-context two-variable slice has rank at most one.
1 Setup
Let Π f denote the canonical finest partition of the variable set for which a
nonfalse Boolean function factors as a conjunction of functions on disjoint
blocks. For a vtree T , let ET (f ) be the finest partition exposed recursively
by the exact compressed-SDD cut criterion.
Theorem 1 (Exposure optimum). For every nonfalse Boolean function f ,
max
T
|ET (f )| = |Πf |.
Proof. Every exposed factorization is valid, hence is a coarsening of Π f .
Conversely, build a binary vtree with one subtree for each block of Π f and
combine those subtrees arbitrarily. Every internal split is a union of prime
blocks against its complement and therefore factors; recursion reaches all
prime blocks.
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Theorem 2 (Carrier/semantic compatibility) . Let Π and Σ be partitions
of the same ground set. A binary rooted leaf-labelled tree can contain every
block of both partitions as a subtree leaf set if and only if the combined block
family Π ∪ Σ is laminar.
Proof. Subtree leaf sets form a laminar family, giving necessity. Conversely,
represent the inclusion poset of a finite laminar family as a rooted tree and
refine nodes of degree greater than two by inserting binary nodes.
2 Defect support
Consider a cut A | B and a nonzero four-corner determinant witnessed by
row assignments a0, a1 and column assignments b0, b1. Let U ⊆ A and W ⊆
B be the coordinates on which the two row and two column assignments
differ.
Lemma 1 (Support crossing). Some block of Πf intersects both U and W .
Proof. If not, after fixing all nonvarying coordinates every prime factor de-
pends on row variables only, column variables only, or neither. Its 2 × 2
corner table is rank one. Entrywise multiplication of rank-one outer prod-
ucts remains rank one, contradicting the nonzero determinant.
Corollary 1. If U = {x} and W = {y}, then x and y are in the same
semantic prime block.
Theorem 3 (No universal pairwise localization) . For n ≥ 3, let fn(x) =
1 exactly for x = 0 n and x = 1 n. Then Πfn consists of one block, but
every two-variable restriction obtained by fixing all remaining coordinates
has matrix rank at most one. For every nontrivial cut A | B, a four-corner
defect exists only when the witness changes all coordinates of A and all
coordinates of B.
Proof. A nontrivial product decomposition would make the two-point sat-
isfying relation a nontrivial Cartesian product, which would contain mixed
combinations; hence the function is prime. A fixed-context two-variable
slice contains at most one of 0 n, 1n, so it has at most one nonzero entry.
Across A | B, the only two nonzero matrix entries are the all-zero and all-
one row/column pairs; a nonzero 2 × 2 minor must select both, forcing all
cut-side coordinates to change.
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3 Interpretation
The results support hyperedge-valued defect evidence rather than automatic
pairwise must-link edges. They also show that carrier blocks should not be
hard-wired into vtrees when they cross emerging semantic factors.
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