The finite-copy
bound atlas.
The current full 4–20-copy triple-column registry is presented together, with separate final 21–24-copy refinements and a clearly labeled symbolic 25–100 finite-copy continuation.
NIGHTFALL Werner bounds: 4–20 copies
The complete latest approved 17-row comparison. Parameter α=−t; larger t is a stronger sufficient undistillability bound. Qutrit NPT begins at t>1/3.
| Copies | Best qutrit | Theorem-only qutrit | Best all-dimension |
|---|---|---|---|
| 4 | 0.49791 | 0.49790 | 0.49375 |
| 5 | 0.47189 | 0.47000 | 0.45000 |
| 6 | 0.44819 | ≥0.36872 | 0.36872 |
| 7 | 0.43140 | 0.42980 | 0.31429 |
| 8 | 0.41889 | 0.41750 | 0.27973 |
| 9 | 0.40920 | 0.40800 | 0.24932 |
| 10 | 0.40148 | 0.40030 | 0.22472 |
| 11 | 0.39500 | 0.39420 | 0.20441 |
| 12 | 0.38980 | 0.38900 | 0.18744 |
| 13 | 0.38540 | 0.38470 | 0.17304 |
| 14 | 0.38160 | 0.38090 | 0.16067 |
| 15 | 0.37849 | 0.37752 | 0.14996 |
| 16 | 0.37564 | 0.37492 | 0.14059 |
| 17 | 0.37313 | 0.37226 | 0.13176 |
| 18 | 0.37089 | 0.37007 | 0.12444 |
| 19 | 0.36812 | 0.36812 | 0.11789 |
| 20 | 0.36636 | 0.36636 | 0.11200 |
Best qutrit 4–18 depends on quantitative source margins inferred from rounded published calculations. The theorem-only column uses source theorem statements without those inferred margins. At k=6, “≥0.36872” is a theorem-only lower guarantee, not a sharp endpoint. All-dimension values are NPT in a given dimension d only when t>1/d. Original JSON and external upstream SDP witness replay remain open.
Higher historical six-to-nine-copy working certificates remain separately archived pending a source-premise and certificate-lineage audit; they are outside this latest approved numerical registry.
Newer refinements: 21–24 copies
Final October 28 source-conditional exact-rational checker results. The 21-copy value is inherited from 22 by padding.
| Copies | Qutrit threshold | Evidence |
|---|---|---|
| 21 | 0.36119 | Published-theorem transfer |
| 22 | 0.36119 | Published-theorem transfer |
| 23 | 0.35151 | Published-theorem transfer |
| 24 | 0.34001 | Published-theorem transfer |
Counts 25–100 are a symbolic B24-dependent analytic continuation, not 76 separately certified numerical LP bounds. The exact B24 seed and original coefficient checker files await a public mirror.
Every individual finite count
An analytic projector argument gives a copy-dependent t(d,k)>1/d at every finite k. With the 24-copy qutrit seed, the reported 25-copy increment is approximately 5.77×10⁻¹⁶. Its small magnitude illustrates the distinction between mathematical existence and competitive numerical bounds. The unrestricted fixed-parameter all-copy problem remains open.
A restricted all-copy class
A separate separable-order compression manuscript reports a 13.3× increase in encoded Alice-side witness coverage at 21 copies. This is a restricted witness-sector statement, not an unrestricted threshold improvement. Full specialist proof review is pending.
Explicit failed experiment
A floating-point 25-copy candidate at t=0.34 failed exact coefficient feasibility and is excluded from the certified ledger.
What reviewers need to check
- Replay the upstream quantitative four-/five-copy qutrit source margins with their correct physical normalization.
- Reconstruct exact source rows, rational weights and positive residuals for copies 19–24, including the separability and copy-budget gates.
- Inspect the corrected trace-difference coefficient in every dimension-uniform eight-to-sixteen-copy certificate.
- Reconstruct the exact 24-copy coercivity seed and audit the analytic finite-k block induction.
- Assess novelty independently, keeping restricted encoded witness sectors separate from unrestricted theorems.