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C2 asks whether a Boolean function can be separated
into a row factor exclusive-or a column factor,

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and whether learned models recognize this structure at a larger size.

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The distinction matters because an exact mathematical test
can succeed even when a learned model fails to generalize.

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For this decomposition family, combine the row and column factors by exclusive-or.

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With both factors zero, one in zero-first order, the operation is different from the and outer product used in other matrix factorizations.

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Keep the reconstruction operator visible.

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The four combinations give zero, one, one, zero.

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The resulting matrix is exclusive-or.

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Its mixed-input cells are one, and its equal-input cells are zero.

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Replacing exclusive-or with and would produce a different matrix, so these factor families must not be conflated.

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To understand the corner correction, use the matrix one zero, zero one in zero-first order.

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Its first column gives g equal one zero.

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Exclusive-or each first-row entry with the corner one, giving h equal zero one.

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The non-anchor lower-right cell reconstructs as zero exclusive-or one, or one.

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The four-cell parity is one exclusive-or zero exclusive-or zero exclusive-or one, which is zero.

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Anchors alone are not enough; test every remaining cell.

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The remaining cells still need an exact test; fitting the anchors alone is not sufficient.

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C2 uses a balanced split of the variables into row and column groups.

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A positive matrix has each cell equal to its
row factor exclusive-or its column factor.

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The exact detector verifies this relation for every cell.

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The learned models make proposals whose
held-out accuracy is evaluated separately.

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A two-variable exclusive-or gives a small
positive example of this additive family.

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With A increasing from zero to one down the rows,

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and B increasing from zero to one across the
columns, the rows are zero, one and one, zero.

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Both factors are zero, one.

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Combining each row factor with each column factor
by exclusive-or reconstructs all four cells exactly.

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AND is a negative example for this family.

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At the bottom-right cell, the anchored parity check gives
one exclusive-or zero exclusive-or zero exclusive-or zero,

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which is one instead of zero.

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For larger matrices, that anchored parity must be zero at every cell.

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The detector constructs the factors and checks exact reconstruction;

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these small grids illustrate the rule, not a new measured result.

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In the retained C2 study, the exact detector succeeded,

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but the learned models failed the predeclared
comparison and larger-size transfer criteria.

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Training used four, six, and eight variables.

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Confirmation used ten variables, so performance at training
sizes could not establish success on the larger matrices.

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The learned models were not promoted.

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Exact checking protected accepted outputs, but it did
not establish a learned accuracy or speed advantage.

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Apply the anchored test to and.

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Both factors become all zero, so the reconstruction is all zero.

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The lower-right target cell is one and fails.

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This rejects exclusive-or separation for this partition; it does not imply that and has no

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useful factorization in another family.

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For a larger split, use row factors zero, one, one, zero and column

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factors zero, zero, one, one.

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A zero row factor copies h; a one row factor complements it.

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This constructs the four displayed rows and provides sixteen exact entries to check.

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Flip one non-anchor cell while keeping the first row and column unchanged.

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The anchored factors remain the same, but full reconstruction now disagrees at that cell.

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An exact test can detect this mismatch.

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Learning to propose good factors on unseen widths is a separate empirical question from

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deterministic full-cell checking.

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Work a second complete case with row_AB 1, column_CD 2.

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Keep this assignment fixed throughout the calculation.

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Each symbol retains the meaning already introduced; a new value does not

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create a new logical rule.

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Use zero-based row index one, corresponding to A B zero one, and column index two,

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corresponding to C D one zero.

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The row factor is one and the column factor is one.

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Their exclusive-or is zero, which is the entry in the second displayed

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row and third displayed column.

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This case establishes a specific contrast.

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The factor operation is XOR.

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Indices follow the declared partition and order.

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Keep the input, intermediate values, and conclusion together when comparing

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it with the previous case.

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The calculated answer belongs to this complete assignment under the stated rule.

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The calculated answer belongs to this complete assignment under the stated rule.

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C2's exact additive decomposition rule and its learned
recognition results support separate conclusions.

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A model needs its own held-out evidence at the intended
size; it cannot inherit the exact detector's success.