Quadratic projection can destroy safety
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At \(m=3\), variables are ordered by source labels \(v=1,2,3,4,5,6\). Assignment index \(13\) is read least-significant-bit first:
The active labels have syndrome \(1+3+4=6\), not \(a=7\).
The three linear forms evaluate to \(L=(0,1,1)\), so \(P_3=L_1L_2L_3=0\). The quadratic grade has pairs \((1,6)\), \((2,5)\), \((3,4)\), so
Thus \(Q_3\notin I_{P_3}\): homogeneous extraction does not preserve the positive-support ideal. The whole polynomial is supported on its own accepting set, but its isolated homogeneous components need not be.
The new script checks all 64 inputs here, finds 24 false positives, and confirms that 13 is the first in the stated integer ordering. The frozen \(m=4\) control similarly has 7,616 false positives among its \(2^{14}\) assignments. These are separate finite statements with exact ranges.
Fresh complete \(m=3\) truth-table check; baseline \(m=4\) check was reproduced by the full source verifier. Safety failure is already established by the single valid assignment above.