Spaces:
Running
Running
| <html lang="en"> | |
| <head> | |
| <meta charset="utf-8"> | |
| <meta name="viewport" content="width=device-width, initial-scale=1"> | |
| <title>Densest-k-Subgraph theorem audit</title> | |
| <style> | |
| :root { --ink:#172026; --muted:#52606a; --paper:#f5f1e8; --card:#fff; --navy:#173f5f; --green:#1b7651; --gold:#b06a12; } | |
| * { box-sizing:border-box; } | |
| body { margin:0; color:var(--ink); background:var(--paper); font-family:ui-sans-serif,system-ui,sans-serif; } | |
| main { max-width:1200px; margin:auto; padding:42px; } | |
| header { border-left:8px solid var(--navy); padding-left:22px; } | |
| h1 { margin:0 0 8px; font-size:clamp(2rem,5vw,4rem); line-height:1.02; } | |
| h2 { color:var(--navy); margin-top:0; } | |
| p,li { line-height:1.48; } | |
| .sub { color:var(--muted); font-size:1.1rem; } | |
| .grid { display:grid; grid-template-columns:repeat(3,1fr); gap:18px; margin:30px 0; } | |
| .card { background:var(--card); border:1px solid #d8d1c4; border-radius:14px; padding:23px; box-shadow:0 8px 22px #17202610; } | |
| .metric { font-size:2rem; font-weight:850; color:var(--green); } | |
| .formula { color:var(--green); font-weight:760; overflow-wrap:anywhere; } | |
| .correction { border-top:5px solid var(--gold); grid-column:1/-1; } | |
| code { background:#edf1f3; border-radius:4px; padding:2px 5px; } | |
| footer { color:var(--muted); border-top:1px solid #cbc3b5; padding-top:16px; } | |
| @media(max-width:800px){ main{padding:24px}.grid{grid-template-columns:1fr}.correction{grid-column:auto} } | |
| </style> | |
| </head> | |
| <body> | |
| <main> | |
| <header><h1>Densest-k-Subgraph: universal proof audit</h1><p class="sub">VQt4w3lElX · tight relaxation · strict landscape · finite exact convergence</p></header> | |
| <section class="grid"> | |
| <article class="card"><h2>C1 · Tightness</h2><div class="metric">Universal</div><p class="formula">Δg = δΔv + (λ-aᵢⱼ)δ² ≥ 0</p><p>Pairwise rounding strictly decreases the fractional-coordinate count and terminates at a no-worse integral point.</p></article> | |
| <article class="card"><h2>C2 · Dichotomy</h2><div class="metric">Universal</div><p class="formula">dᵀ(A+λI)d = 2(λ-aᵢⱼ) > 0</p><p>KKT gives integral local maxima or an explicit zero-gradient, positive-curvature saddle direction.</p></article> | |
| <article class="card"><h2>C3 · Algorithm 1</h2><div class="metric">35 updates</div><p class="formula">4,039 nodes · 88,234 edges · gap 0</p><p>The equation-level proof closes, and the original SNAP Facebook protocol takes the predicted final γ=1 jump to exact integrality.</p></article> | |
| <article class="card correction"><h2>Negative control</h2><p>The constant-2kL denominator remains fractional at all 4,039 coordinates after 50,000 updates with gap 0.131515. The legacy eigenvector/snap optimizer remains disclosed as non-Algorithm-1 corroboration.</p></article> | |
| </section> | |
| <footer>35/35 tests pass · all proof DAGs valid · original paper-scale graph · CPU only</footer> | |
| </main> | |
| </body> | |
| </html> | |