C.2 Finite census tables and witnesses
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C.2.1 Status key
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Every table in this appendix is generated from exact finite output in the accompanying JSON files. “Restricted size” and “ambient size” refer to the enumerated quotient groups, not necessarily the full unitary groups including phases and translations.
C.2.2 Clifford census summary
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| Qubits | Metric | SWAP | Restricted size | Ambient size | \(\operatorname{diam}_R\) | \(\operatorname{diam}_A\) | Defects |
|---|---|---|---|---|---|---|---|
| 1 | total | no | 2 | 6 | 1 | 3 | 0 |
| 1 | entangling | no | 2 | 6 | 0 | 0 | 0 |
| 2 | total | no | 48 | 720 | 5 | 8 | 0 |
| 2 | entangling | no | 48 | 720 | 3 | 3 | 4 |
| 3 | total | no | 10,752 | 1,451,520 | 8 | 12 | 24 |
| 3 | total | yes | 10,752 | 1,451,520 | 7 | 11 | 0 |
| 3 | entangling | no | 10,752 | 1,451,520 | 6 | 6 | 1,624 |
C.2.3 Clifford defect distributions
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| Model | Distance pair | Count |
|---|---|---|
| Two qubits, entangling | \(3\to2\) | 4 |
| Three qubits, total | \(7\to6\) | 6 |
| Three qubits, total | \(8\to7\) | 18 |
| Three qubits, entangling | \(3\to2\) | 24 |
| Three qubits, entangling | \(4\to3\) | 432 |
| Three qubits, entangling | \(5\to4\) | 1,056 |
| Three qubits, entangling | \(6\to4\) | 48 |
| Three qubits, entangling | \(6\to5\) | 64 |
C.2.4 Affine-versus-Toffoli census
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| Bits | Metric | Restricted size | Ambient size | Restricted diameter | Defects |
|---|---|---|---|---|---|
| 3 | total | 1,344 | 40,320 | 7 | 0 |
| 3 | entangling | 1,344 | 40,320 | 6 | 0 |
C.2.5 Linear atom modular checks
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| Modulus | Restricted word length | Ambient word length | Both match target |
|---|---|---|---|
| 2 | 7 | 6 | yes |
| 3 | 7 | 6 | yes |
| 5 | 7 | 6 | yes |
| 8 | 7 | 6 | yes |
These checks confirm the symbolic words. Exact optimality over general rings is proved separately in the core chapter.
C.2.6 Core binary six-dimensional census
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For \(U_6({\mathbb F}_2)\) there are \(2^{15}=32{,}768\) targets. The complete comparison is:
| Restricted distance | Ambient distance | Number of targets |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 15 |
| 2 | 2 | 125 |
| 3 | 3 | 720 |
| 4 | 4 | 2,914 |
| 5 | 5 | 7,851 |
| 6 | 6 | 12,656 |
| 7 | 6 | 20 |
| 7 | 7 | 7,926 |
| 8 | 8 | 540 |
The twenty strict defects are exactly the analytic three-cross, one-phase motifs.
C.2.7 Endpoint Green coefficients for the canonical atom
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Restricted word counts for degrees zero through ten are
Ambient word counts for degrees zero through eight are
Thus first arrival occurs at seven and six, respectively.
C.2.8 Machine-readable locations
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The release contains:
results/atlas_results.json,results/clifford_n3_total.json,results/clifford_n3_total_swap.json,results/clifford_n3_entangling.json,the complete unitriangular supplement under
core_extract/supplement/.
The JSON schema records model, metric, group sizes, diameters, defect distributions, and selected witnesses.