Exact decomposition and learned transfer

C2 asks whether a Boolean function can be separated into a row factor exclusive-or a column factor, and whether learned models recognize this structure at a larger size. The distinction matters because an exact mathematical test can succeed even when a learned model fails to generalize. For this decomposition family, combine the row and column factors by exclusive-or. With both factors zero, one in zero-first order, the operation is different from the and outer product used in other matrix factorizations. Keep the reconstruction operator visible. The four combinations give zero, one, one, zero. The resulting matrix is exclusive-or. Its mixed-input cells are one, and its equal-input cells are zero. Replacing exclusive-or with and would produce a different matrix, so these factor families must not be conflated. To understand the corner correction, use the matrix one zero, zero one in zero-first order. Its first column gives g equal one zero. Exclusive-or each first-row entry with the corner one, giving h equal zero one. The non-anchor lower-right cell reconstructs as zero exclusive-or one, or one. The four-cell parity is one exclusive-or zero exclusive-or zero exclusive-or one, which is zero. Anchors alone are not enough; test every remaining cell. The remaining cells still need an exact test; fitting the anchors alone is not sufficient. C2 uses a balanced split of the variables into row and column groups. A positive matrix has each cell equal to its row factor exclusive-or its column factor. The exact detector verifies this relation for every cell. The learned models make proposals whose held-out accuracy is evaluated separately. A two-variable exclusive-or gives a small positive example of this additive family. With A increasing from zero to one down the rows, and B increasing from zero to one across the columns, the rows are zero, one and one, zero. Both factors are zero, one. Combining each row factor with each column factor by exclusive-or reconstructs all four cells exactly. AND is a negative example for this family. At the bottom-right cell, the anchored parity check gives one exclusive-or zero exclusive-or zero exclusive-or zero, which is one instead of zero. For larger matrices, that anchored parity must be zero at every cell. The detector constructs the factors and checks exact reconstruction; these small grids illustrate the rule, not a new measured result. In the retained C2 study, the exact detector succeeded, but the learned models failed the predeclared comparison and larger-size transfer criteria. Training used four, six, and eight variables. Confirmation used ten variables, so performance at training sizes could not establish success on the larger matrices. The learned models were not promoted. Exact checking protected accepted outputs, but it did not establish a learned accuracy or speed advantage. Apply the anchored test to and. Both factors become all zero, so the reconstruction is all zero. The lower-right target cell is one and fails. This rejects exclusive-or separation for this partition; it does not imply that and has no useful factorization in another family. For a larger split, use row factors zero, one, one, zero and column factors zero, zero, one, one. A zero row factor copies h; a one row factor complements it. This constructs the four displayed rows and provides sixteen exact entries to check. Flip one non-anchor cell while keeping the first row and column unchanged. The anchored factors remain the same, but full reconstruction now disagrees at that cell. An exact test can detect this mismatch. Learning to propose good factors on unseen widths is a separate empirical question from deterministic full-cell checking. Work a second complete case with row_AB 1, column_CD 2. Keep this assignment fixed throughout the calculation. Each symbol retains the meaning already introduced; a new value does not create a new logical rule. Use zero-based row index one, corresponding to A B zero one, and column index two, corresponding to C D one zero. The row factor is one and the column factor is one. Their exclusive-or is zero, which is the entry in the second displayed row and third displayed column. This case establishes a specific contrast. The factor operation is XOR. Indices follow the declared partition and order. Keep the input, intermediate values, and conclusion together when comparing it with the previous case. The calculated answer belongs to this complete assignment under the stated rule. C2's exact additive decomposition rule and its learned recognition results support separate conclusions. A model needs its own held-out evidence at the intended size; it cannot inherit the exact detector's success.