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graph-theory
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| <html lang="es"> | |
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| <title>GraphKind — Universos del color refinement</title> | |
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| </style> | |
| </head> | |
| <body> | |
| <header> | |
| <div class="wrap"> | |
| <div class="kicker">GraphKind · preprint privado · 2026</div> | |
| <h1>Sobre los universos del<br><em>color refinement</em></h1> | |
| <p class="lede"> | |
| Cuándo sobrevive un invariante estructural a un cambio de | |
| observación: la clasificación de las operaciones que preservan el | |
| perfil del color refinement bajo involución — y su prueba. | |
| </p> | |
| <div class="cta"> | |
| <a class="btn primary" href="paper/PAPER.md">Leer el paper</a> | |
| <a class="btn" href="paper/PRUEBA-CARACTERIZACION.md">La prueba</a> | |
| <a class="btn" href="reproduce.sh">Reproducir</a> | |
| <a class="btn" href="index.en.html">EN</a> | |
| </div> | |
| <div class="meta"> | |
| <span><b>2 131 019</b> grafos (T4)</span> | |
| <span><b>34/34 · 102/102</b> la ley</span> | |
| <span><b>96</b> exactas en n=4</span> | |
| <span><b>CI</b> verde</span> | |
| </div> | |
| </div> | |
| </header> | |
| <section> | |
| <div class="wrap"> | |
| <h2>El problema</h2> | |
| <p> | |
| El <strong>color refinement</strong> (1-WL) resume un grafo en una | |
| partición de sus vértices. ¿Qué operaciones dejan esa partición | |
| intacta? | |
| </p> | |
| <figure> | |
| <svg viewBox="0 0 720 240" width="100%" role="img" | |
| aria-label="G y su complemento comparten la misma partición"> | |
| <g stroke="var(--line)" stroke-width="2"> | |
| <!-- G: P4 + aislado --> | |
| <line x1="60" y1="90" x2="120" y2="90"/> | |
| <line x1="120" y1="90" x2="180" y2="90"/> | |
| <line x1="180" y1="90" x2="240" y2="90"/> | |
| </g> | |
| <g> | |
| <circle cx="60" cy="90" r="16" fill="var(--gold)"/> | |
| <circle cx="120" cy="90" r="16" fill="var(--blue)"/> | |
| <circle cx="180" cy="90" r="16" fill="var(--blue)"/> | |
| <circle cx="240" cy="90" r="16" fill="var(--gold)"/> | |
| <circle cx="150" cy="180" r="16" fill="var(--accent2)"/> | |
| </g> | |
| <text x="150" y="222" text-anchor="middle" font-size="15" | |
| fill="var(--muted)">G</text> | |
| <!-- flecha --> | |
| <text x="300" y="98" font-size="26" fill="var(--muted)">⟶</text> | |
| <text x="300" y="126" font-size="12.5" fill="var(--muted)" | |
| text-anchor="middle">complemento</text> | |
| <!-- Gbar: complemento --> | |
| <g stroke="var(--line)" stroke-width="2"> | |
| <line x1="400" y1="90" x2="460" y2="90"/> | |
| <line x1="400" y1="90" x2="520" y2="90"/> | |
| <line x1="400" y1="90" x2="580" y2="90"/> | |
| <line x1="460" y1="90" x2="580" y2="90"/> | |
| <line x1="520" y1="90" x2="580" y2="90"/> | |
| <line x1="400" y1="90" x2="490" y2="180"/> | |
| <line x1="460" y1="90" x2="490" y2="180"/> | |
| <line x1="520" y1="90" x2="490" y2="180"/> | |
| <line x1="580" y1="90" x2="490" y2="180"/> | |
| </g> | |
| <g> | |
| <circle cx="400" cy="90" r="16" fill="var(--gold)"/> | |
| <circle cx="460" cy="90" r="16" fill="var(--blue)"/> | |
| <circle cx="520" cy="90" r="16" fill="var(--blue)"/> | |
| <circle cx="580" cy="90" r="16" fill="var(--gold)"/> | |
| <circle cx="490" cy="180" r="16" fill="var(--accent2)"/> | |
| </g> | |
| <text x="490" y="222" text-anchor="middle" font-size="15" | |
| fill="var(--muted)">Ḡ</text> | |
| <text x="660" y="86" font-size="15" fill="var(--accent)" | |
| font-family="ui-monospace,monospace">I(G)=I(Ḡ)</text> | |
| </svg> | |
| <figcaption> | |
| T4: el color refinement de un grafo y el de su complemento | |
| inducen <em>la misma partición</em> (mismos colores = mismas | |
| clases). Descubierto por el motor, verificado en todos los | |
| grafos de n≤7 y con prueba escrita. | |
| </figcaption> | |
| </figure> | |
| <p> | |
| La pregunta que sigue: si el observador comprime los conteos de | |
| vecinos con una función <code>f</code> y la operación es una | |
| involución <code>ι</code>, ¿en qué universos <code>(f, ι)</code> | |
| sigue viviendo T4? | |
| </p> | |
| </div> | |
| </section> | |
| <section> | |
| <div class="wrap"> | |
| <h2>El resultado</h2> | |
| <div class="formula"> | |
| <div class="big">T4(ι, f) VIVE ⟺ ι ∈ Sₙ·K y | |
| (ι cambia el par ⟹ f(1) ≠ f(2))</div> | |
| <div class="muted">K = {id, comp_dir, transp, comp_transp} ≅ Z₂×Z₂</div> | |
| </div> | |
| <figure> | |
| <svg viewBox="0 0 720 210" width="100%" role="img" | |
| aria-label="El grupo de Klein: las dos mitades"> | |
| <g font-size="14" font-family="ui-monospace,monospace"> | |
| <rect x="20" y="30" width="150" height="64" rx="12" | |
| fill="var(--soft)" stroke="var(--line)"/> | |
| <text x="95" y="58" text-anchor="middle" fill="var(--ink)">id</text> | |
| <text x="95" y="78" text-anchor="middle" fill="var(--muted)" | |
| font-size="11.5">swap no · comp no</text> | |
| <rect x="190" y="30" width="150" height="64" rx="12" | |
| fill="var(--soft)" stroke="var(--line)"/> | |
| <text x="265" y="58" text-anchor="middle" fill="var(--ink)">transp</text> | |
| <text x="265" y="78" text-anchor="middle" fill="var(--muted)" | |
| font-size="11.5">swap sí · comp no</text> | |
| <rect x="360" y="30" width="150" height="64" rx="12" | |
| fill="var(--soft)" stroke="var(--line)"/> | |
| <text x="435" y="58" text-anchor="middle" fill="var(--ink)">comp_dir</text> | |
| <text x="435" y="78" text-anchor="middle" fill="var(--muted)" | |
| font-size="11.5">swap no · comp sí</text> | |
| <rect x="530" y="30" width="170" height="64" rx="12" | |
| fill="var(--soft)" stroke="var(--line)"/> | |
| <text x="615" y="58" text-anchor="middle" fill="var(--ink)">comp_transp</text> | |
| <text x="615" y="78" text-anchor="middle" fill="var(--muted)" | |
| font-size="11.5">swap sí · comp sí</text> | |
| </g> | |
| <text x="180" y="140" text-anchor="middle" font-size="13" | |
| fill="var(--accent)">viven con toda f</text> | |
| <line x1="20" y1="118" x2="340" y2="118" stroke="var(--accent)" | |
| stroke-width="2"/> | |
| <text x="530" y="140" text-anchor="middle" font-size="13" | |
| fill="var(--blue)">viven ⟺ f(1) ≠ f(2)</text> | |
| <line x1="360" y1="118" x2="700" y2="118" stroke="var(--blue)" | |
| stroke-width="2"/> | |
| <text x="360" y="180" font-size="13.5" fill="var(--muted)"> | |
| La uniformidad une las dos mitades: fuera de K, todo muere | |
| (aunque preserve el par). | |
| </text> | |
| </svg> | |
| <figcaption> | |
| El grupo de Klein: dos mitades, dos condiciones. La operación | |
| debe ser global (relabel × K); si cambia el par conectado, la | |
| compresión debe distinguir 1 de 2. | |
| </figcaption> | |
| </figure> | |
| <h3>El caso particular: la ley del conteo</h3> | |
| <div class="formula">T4 VIVE ⟺ f(1) ≠ f(2)</div> | |
| <p> | |
| En el complemento, un vértice con <strong>1</strong> vecino de una | |
| clase pasa a tener <code>|c|−1</code> (que puede ser | |
| <strong>2</strong>): si <code>f</code> colapsa 1 y 2, la dualidad | |
| muere. Lo que <code>f</code> haga con 3, 4, 5… es irrelevante. | |
| </p> | |
| <table> | |
| <tr><th>resultado</th><th>verificación</th></tr> | |
| <tr><td>La ley <code>f(1)≠f(2)</code></td> | |
| <td class="num">34/34 · 102/102</td></tr> | |
| <tr><td>Caracterización, n=4 exhaustivo</td> | |
| <td class="num">96 exactas (24·4), cero fuera</td></tr> | |
| <tr><td>T4 bajo la jerarquía k-FWL (≡ (k+1)-WL)</td> | |
| <td class="num">51/51 en k=1,2,3</td></tr> | |
| <tr><td>Cascada del observador (n≤7, 781 875 pares)</td> | |
| <td class="num">26 → 0 (k=1 → k=2)</td></tr> | |
| <tr><td>Datos reales</td> | |
| <td class="num">moléculas 100/100 · CST 94/94</td></tr> | |
| </table> | |
| <figure> | |
| <svg viewBox="0 0 720 260" width="100%" role="img" | |
| aria-label="Cascada del observador"> | |
| <line x1="80" y1="210" x2="690" y2="210" stroke="var(--line)"/> | |
| <line x1="80" y1="20" x2="80" y2="210" stroke="var(--line)"/> | |
| <text x="60" y="40" text-anchor="end" font-size="12" | |
| fill="var(--muted)">26</text> | |
| <text x="60" y="214" text-anchor="end" font-size="12" | |
| fill="var(--muted)">0</text> | |
| <text x="120" y="232" font-size="13" fill="var(--muted)">k = 1</text> | |
| <text x="380" y="232" font-size="13" fill="var(--muted)">k = 2</text> | |
| <text x="630" y="232" font-size="13" fill="var(--muted)">k = 3</text> | |
| <polyline points="120,40 380,205 640,205" | |
| fill="none" stroke="var(--accent)" stroke-width="3"/> | |
| <circle cx="120" cy="40" r="7" fill="var(--accent2)"/> | |
| <circle cx="380" cy="205" r="7" fill="var(--accent)"/> | |
| <circle cx="640" cy="205" r="7" fill="var(--accent)"/> | |
| <text x="140" y="34" font-size="14" fill="var(--ink)"> | |
| 26 pares 1-WL-equivalentes</text> | |
| <text x="400" y="198" font-size="14" fill="var(--ink)"> | |
| 0: 2-WL los resuelve todos</text> | |
| <text x="420" y="60" font-size="13" fill="var(--muted)"> | |
| C6 vs 2·C3: k*=2</text> | |
| <text x="420" y="80" font-size="13" fill="var(--muted)"> | |
| Rook vs Shrikhande: k*=3</text> | |
| </svg> | |
| <figcaption> | |
| La cascada: la aleatoriedad aparente (pares no separados) cae a | |
| cero al subir el observador. En n≤7, k*=2; los pares canónicos | |
| necesitan k=2 y k=3. | |
| </figcaption> | |
| </figure> | |
| </div> | |
| </section> | |
| <section> | |
| <div class="wrap"> | |
| <h2>El cambio de tipo</h2> | |
| <p> | |
| Cambiando el <em>tipo</em> del invariante, T4 cambia de categoría | |
| matemática — y cada fila es un tipo distinto de teorema: | |
| </p> | |
| <div class="grid"> | |
| <div class="card"> | |
| <div class="t">Estándar</div> | |
| <div class="s">invariante: partición · igualdad</div> | |
| <div class="v ok">T4 vive (probado)</div> | |
| </div> | |
| <div class="card"> | |
| <div class="t">Z₃ determinista</div> | |
| <div class="s">invariante: órbita · períodos</div> | |
| <div class="v no">T4 muere (máx. 52.4%)</div> | |
| </div> | |
| <div class="card"> | |
| <div class="t">Aleatorio (proceso)</div> | |
| <div class="s">invariante: distribución · WL asíncrono</div> | |
| <div class="v no">T4 muere (18.6%)</div> | |
| </div> | |
| <div class="card"> | |
| <div class="t">Aleatorio (objeto)</div> | |
| <div class="s">G(n,p) vs G(n,1−p)</div> | |
| <div class="v ok">T4 vive (casi seguro)</div> | |
| </div> | |
| <div class="card"> | |
| <div class="t">Jerarquía k-FWL</div> | |
| <div class="s">observadores más fuertes</div> | |
| <div class="v ok">T4_k 51/51</div> | |
| </div> | |
| <div class="card"> | |
| <div class="t">Involuciones aleatorias</div> | |
| <div class="s">invariante: medida del ensemble</div> | |
| <div class="v">umbral = medida</div> | |
| </div> | |
| </div> | |
| </div> | |
| </section> | |
| <section> | |
| <div class="wrap"> | |
| <h2>Cómo se encontró</h2> | |
| <p> | |
| El motor <strong>propuso y verificó T4</strong> (banco de | |
| conjeturas ciego, 68/68), la verificación masiva lo sostuvo y una | |
| prueba escrita cerró el caso estándar. Después el laboratorio | |
| parametrizó el universo y dejó que <strong>los datos eligieran la | |
| ley</strong>: dos hipótesis humanas fueron refutadas antes de | |
| llegar a la caracterización. | |
| </p> | |
| <p> | |
| Una <strong>búsqueda autónoma de construcciones</strong> encontró | |
| el mejor testigo válido (<code>ciclo</code>, 99.63%) — mejor que | |
| la construcción humana. Y los errores quedaron registrados: | |
| </p> | |
| <div class="timeline"> | |
| <div class="ev fix"><b>1. Multiset de tamaños</b> | |
| invariante débil (C6 y 2·C3 lo comparten) → colores.</div> | |
| <div class="ev fix"><b>2. Comparar colores entre grafos</b> | |
| nunca coinciden → comparar particiones.</div> | |
| <div class="ev fix"><b>3. Índices en la coloración inicial</b> | |
| trivializaba todo → grados + adyacencias.</div> | |
| <div class="ev fix"><b>4. MultiseTs separados</b> | |
| k-WL débil → pares correlacionados (= k-FWL).</div> | |
| <div class="ev fix"><b>5. El Shrikhande mal construido</b> | |
| era el Rook dos veces → cazado por <code>is_isomorphic</code>.</div> | |
| </div> | |
| <p> | |
| Cinco bugs propios cazados por controles, ninguno anunciado antes | |
| de verificar. <strong>El método es parte del resultado.</strong> | |
| </p> | |
| </div> | |
| </section> | |
| <section> | |
| <div class="wrap"> | |
| <h2>Reproducibilidad</h2> | |
| <p> | |
| Todo el paper se reproduce con un comando. Cada verificación lleva | |
| un <strong>assert</strong> con el número exacto: si algo no | |
| coincide, falla. | |
| </p> | |
| <pre>git clone https://github.com/cripto-bot/graphkind-universos | |
| cd graphkind-universos | |
| python3 -m venv .venv && .venv/bin/pip install -r requirements.txt | |
| bash reproduce.sh # V1–V9 (~10-15 min) | |
| bash reproduce.sh --rapido # V1, V3, V5, V8 (~3 min)</pre> | |
| <p> | |
| <span class="pill">estado</span> | |
| <strong>(⇐) probada</strong> · <strong>(⇒) verificada</strong> | |
| (n≤4 exhaustivo, n=5 dirigido, datos reales) · | |
| argumento ∀n y revisión externa: <em>abiertos, declarados</em>. | |
| </p> | |
| </div> | |
| </section> | |
| <footer> | |
| <div class="wrap"> | |
| GraphKind · preprint — © 2026 Juri (cripto-bot) · CC-BY-4.0. | |
| No distribuir sin autorización. Este documento no reclama prioridad | |
| sobre problemas abiertos: reporta una caracterización verificada en | |
| un dominio acotado, con sus límites declarados. | |
| </div> | |
| </footer> | |
| </body> | |
| </html> | |