Verification Suite for the paper "Projective Steenrod--Milnor Hulls: Frobenius Periodicity and Quantum Density Obstructions"

### Contents of this Deposit `manuscript_exact_verification.py` is the primary Python 3 verification script accompanying the revised manuscript. It implements exact arithmetic over \(\mathbb{F}_4=\mathbb{F}_2[\omega]/(\omega^2+\omega+1)\), with elements encoded as \(0,1,\omega,\omega^2\), and uses the normalized projective ordering of \(\mathbb{P}^2(\mathbb{F}_4)\) given by \((1,a,b)\), \((0,1,b)\), and \((0,0,1)\), where \(a,b\in\mathbb{F}_4\). The script constructs the formal Milnor-primitive matrices \(Q_n:R_d\to R_{d+\delta_n}\) before projective evaluation, computes evaluated Steenrod-kernel codes, Hermitian duals, Hermitian hulls, canonical hull-CSS parameters, explicit finite-field check matrices, low-weight span certificates, MacWilliams dual weight-enumerator prefixes, and support-rank distance certificates. It also includes the additional checks required by the revised manuscript: the corrected degree-six Margolis homology values \(\dim H_6(P_3;Q_0)=0\) and \(\dim H_6(P_3;Q_1)=1\), the formal/evaluated distinction for the degree-five \(Q_1\)-kernel, the stabilizer-sector degeneracy certificate for the \([[21,2,6/6]]_4\) code, and the joint Singleton-defect checks under the stated AQMDS convention. `manuscript_exact_verification_output.txt` is the pre-computed reference log generated by executing `python3 manuscript_exact_verification.py --full`. It records the normalized projective length, structural tower dimensions, Hermitian dual dimensions, Hermitian hull dimensions, the nonsaturated \(Q_0\)- and \(Q_1\)-tower entries, low-weight spanning thresholds, degree-six weight-enumerator prefixes for \(K_{1,6}\), \(K_{1,6}^{\perp_H}\), \(\operatorname{Hull}_H(K_{1,6})\), and \(\operatorname{Hull}_H(K_{1,6})^{\perp_H}\), the rank and sparsity data for the explicit \(H_X,H_Z\) check matrices, the corrected Margolis homology certificates, the degree-five \(Q_1\) evaluation-vanishing certificate, the stabilizer degeneracy certificate, and the Singleton-defect certificates for the canonical hull-CSS codes appearing in the revised manuscript. `README.md` gives the system dependencies, the finite-field conventions, the projective point ordering, and quick-start execution instructions for reproducing both the default and full verification logs. `steenrod_milnor_hulls_code.zip` is a convenience archive bundling the Python verifier, the reference output, and the README for local replication. ### Execution Modes The default command `python3 manuscript_exact_verification.py` performs the lightweight exact checks used for routine reproducibility. It verifies the structural tower dimensions, Hermitian dual dimensions, Hermitian hull dimensions, stabilization assertions, the explicit \([[21,2,6/6]]_4\) check matrices, the corrected degree-six Margolis homology statements, the formal/evaluated degree-five \(Q_1\) distinction, and the stabilizer degeneracy certificate. The full command `python3 manuscript_exact_verification.py --full` additionally prints and checks the nonsaturated tower data, low-weight spanning certificates, degree-six MacWilliams weight-enumerator prefixes, explicit check-matrix rank and sparsity data, corrected Margolis homology certificates, formal/evaluated \(Q_1\) degree-five certificate, stabilizer degeneracy certificate, and canonical-hull Singleton defects. This is the mode used to generate `manuscript_exact_verification_output.txt`. ### Version Note for the manuscript This deposit supersedes the earlier verification description. In particular, the manuscript uses the Margolis differential with incoming degree \(r-\delta_n\). Therefore, in degree six, the \(Q_0\)-incoming map is \(Q_0:R_5\to R_6\), not \(Q_0:R_3\to R_6\), and the script certifies \(\dim H_6(P_3;Q_0)=13-13=0\). For \(Q_1\), the script certifies \(\dim H_6(P_3;Q_1)=11-10=1\). The script also certifies that the degree-six code is degenerate in the stabilizer sense because \(C_Z=K_{1,6}^{\perp_H}\) contains nonzero words of weight \(5<d_Z=6\). ### System Requirements Python 3.10 or later is required, together with NumPy. The only external Python dependency is NumPy, installable by `pip install numpy`. The code uses exact finite-field tables for addition, multiplication, inversion, conjugation, row reduction, nullspaces, and Hermitian products over \(\mathbb{F}_4\). No floating-point approximations, randomized searches, machine-learning procedures, or empirical datasets are used. ### Reproducibility Scope All arithmetic takes place strictly over \(\mathbb{F}_4\). The script constructs the formal Steenrod-Milnor action matrices first and only then applies projective evaluation, so no identity such as \(x^4=x\) is imposed inside the formal polynomial ring. The output is deterministic for the stated monomial ordering and projective point ordering. The support indices of the displayed \([[21,2,6/6]]_4\) check matrices are tied to this ordering.