Die Holonomie $h_e(A) = \mathcal{P} \exp \left( \int_e A_a^i \tau_i \mathrm{d}x^a \right)$ entlang einer Kante $e$ bildet das fundamentale Schleifen-Operator-Element der LQG.

Die Vertices $v$ tragen die EPRL-Amplitude $A_v(j_f, i_e) = \{15j\}_{\text{SL}(2,\mathbb{C})}$, verknüpft mit den $SU(2)$-Intertwinern der Ränder. Der DTIM-ZEC Tensor $\mathbf{T}_{\text{ZEC}}^{(\text{QG})}$ erzwingt die Kohärenz über makroskopische Distanzen.

/**
 * DTIM-ZEC MODULAR PHYSICS FRAMEWORK (DAMTP ACADEMIC STANDARD)
 * DTIM_QuantumGravity_001_LQG_SpinFoam.pde - Loop Quantum Gravity & Spin Foam Engine
 * Target: Processing 4.5.6 / Quantum Field Compendium Band 4 (Objekt 001)
 * License: CC0 1.0 Universal / MIT
 * Copyright (c) 2026 Jan Bühring & Adaptive Intelligence Synthesis Network (Google AI)
 */

int mode = 0; // 0: Loop Quantization & Discretized Area/Volume Spectrum (hat A, hat V), 1: 3D Spin Network Graph & SU(2) Intertwiners, 2: 4D Spin Foam 2-Complex Spacetime Evolution & EPRL Vertex Amplitudes, 3: DTIM-ZEC Raumhexagon Spin-Foam Core
float wavePhase = 0.0f;
String[] modeNames = {
  "1. Geometrie-Quantisierung // Diskretes Eigenwertspektrum von Flächen- (hat A) & Volumenoperatoren (hat V)", 
  "2. Spin-Netzwerke // SU(2)-Beschriftete Graphen mit Intertwiner-Knoten (Volume) & Spin-Kanten (Area j)", 
  "3. Spin-Schäume (4D) // Kovariente Raumzeit-Evolution als 2-Komplex mit EPRL-Vertex-Amplituden", 
  "4. DTIM-ZEC Raumhexagon-Spin-Gitter // Topologische Eichung der Planck-Skala über ZEC-Superzellen"
};

void setup() {
  size(800, 380);
  println("[DTIM-QUANTUM-PHYSICS] Initializing Volume 4 Object 001 (LQG & Spin Foam Engine)...");
}

void draw() {
  background(11, 15, 25);
  wavePhase += 0.04f;
  
  // Auto switch mode every 4 seconds
  if (frameCount % 240 == 0) {
    mode = (mode + 1) % 4;
  }
  
  translate(width/2, height/2 + 10);
  
  if (mode == 0) {
    drawGeometryQuantizationSpectrum();
  } else if (mode == 1) {
    drawSpinNetworkGraphIntertwiners();
  } else if (mode == 2) {
    drawSpinFoamSpacetimeEvolution();
  } else {
    drawDTIMHexagonSpinFoamCore();
  }
  
  resetMatrix();
  drawCosmologyOverlay();
}

void drawGeometryQuantizationSpectrum() {
  // Discretized Area & Volume Eigenvalue Spectrum
  
  // Discrete Spectrum Steps (Bar Graph Style)
  stroke(139, 92, 246); strokeWeight(2.0f);
  float[] spins = {0.5f, 1.0f, 1.5f, 2.0f, 2.5f, 3.0f, 3.5f};
  
  for (int i = 0; i < spins.length; i++) {
    float x = -180 + i * 60;
    float areaVal = sqrt(spins[i] * (spins[i] + 1.0f)) * 35.0f;
    
    // Bar
    fill(139, 92, 246, 120);
    rect(x - 15, 70 - areaVal, 30, areaVal);
    
    // Glowing Quantized Level Line
    stroke(167, 139, 250); strokeWeight(2.5f);
    line(x - 20, 70 - areaVal, x + 20, 70 - areaVal);
    
    fill(226, 232, 240); textSize(10); textAlign(CENTER, CENTER);
    text("j=" + spins[i], x, 85);
  }

  // Base Line
  stroke(71, 85, 105); strokeWeight(1.5f);
  line(-210, 70, 210, 70);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Diskretes Spektrum: hat A |psi> = 8pi gamma l_P^2 sum sqrt(j(j+1)) |psi>", 0, -115);
  text("Flächen-Quanten entstehen durch Durchstoßung der Spin-Netzwerk-Kanten", 0, 115);
}

void drawSpinNetworkGraphIntertwiners() {
  // 3D Spin Network Graph & SU(2) Intertwiners
  
  // Graph Nodes (Intertwiners = Volume Quanta)
  float[][] nodes = {
    {0, 0}, {-120, -60}, {120, -60}, {-100, 60}, {100, 60}
  };
  
  // Connecting Edges (Spin Labels j_i)
  stroke(56, 189, 248, 200); strokeWeight(2.5f);
  line(nodes[0][0], nodes[0][1], nodes[1][0], nodes[1][1]);
  line(nodes[0][0], nodes[0][1], nodes[2][0], nodes[2][1]);
  line(nodes[0][0], nodes[0][1], nodes[3][0], nodes[3][1]);
  line(nodes[0][0], nodes[0][1], nodes[4][0], nodes[4][1]);
  line(nodes[1][0], nodes[1][1], nodes[3][0], nodes[3][1]);
  line(nodes[2][0], nodes[2][1], nodes[4][0], nodes[4][1]);
  
  // Edge Spin Labels
  fill(56, 189, 248); textSize(10); textAlign(CENTER, CENTER);
  text("j=1/2", -60, -40); text("j=1", 60, -40); text("j=3/2", -50, 40); text("j=1/2", 50, 40);
  
  // Draw Intertwiner Nodes (Spheres)
  for (int i = 0; i < nodes.length; i++) {
    fill(167, 139, 250); stroke(255); strokeWeight(2.0f);
    float pPulse = (i == 0) ? sin(wavePhase * 3.0f) * 4.0f : 0;
    ellipse(nodes[i][0], nodes[i][1], 24 + pPulse, 24 + pPulse);
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Spin-Netzwerk Graph: Knoten = Intertwiner (hat V), Kanten = Spins j (hat A)", 0, -115);
  text("Eichinvarianz unter SU(2)-Transformationen an allen Knotenpunkten", 0, 115);
}

void drawSpinFoamSpacetimeEvolution() {
  // 4D Spin Foam 2-Complex Spacetime Evolution
  
  // Vertices & Faces (Spacetime Sheets)
  stroke(244, 114, 182, 160); strokeWeight(1.8f); noFill();
  
  for (int x = -150; x <= 150; x += 100) {
    beginShape();
    for (int y = -80; y <= 80; y += 10) {
      float waveX = x + sin(wavePhase * 2.0f + y * 0.05f) * 15.0f;
      vertex(waveX, y);
    }
    endShape();
  }
  
  // Spin Foam Vertices (EPRL Amplitudes - Glowing Circles)
  for (int x = -150; x <= 150; x += 100) {
    for (int y = -60; y <= 60; y += 60) {
      fill(250, 204, 21); stroke(255); strokeWeight(1.5f);
      ellipse(x + sin(wavePhase * 2.0f + y * 0.05f) * 15.0f, y, 10, 10);
    }
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Spin-Schaum (2-Komplex): 4D-Raumzeit-Evolution der Spin-Netzwerke", 0, -115);
  text("EPRL Vertex-Amplitude Av(jf, ie) = {15j}_SL(2,C) berechnet die Zustandssumme", 0, 115);
}

void drawDTIMHexagonSpinFoamCore() {
  // DTIM-ZEC Raumhexagon Spin-Foam Core
  strokeWeight(1.8f); noFill();
  
  float hexRadius = 35.0f;
  for (int q = -3; q <= 3; q++) {
    for (int r = -2; r <= 2; r++) {
      float x = hexRadius * (sqrt(3) * q + sqrt(3)/2.0f * r);
      float y = hexRadius * (3.0f/2.0f * r);
      
      float pulse = sin(wavePhase * 3.5f + (q + r) * 0.8f) * 6.0f;
      
      pushMatrix();
      translate(x, y);
      stroke(167, 139, 250, 180);
      polygon(0, 0, hexRadius * 0.85f + pulse * 0.2f, 6);
      
      fill(52, 211, 153, 220); noStroke();
      ellipse(0, 0, 5, 5);
      popMatrix();
    }
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("DTIM-ZEC Raumhexagon-Spin-Gitter: Topologische Eichung der LQG", 0, 115);
}

void polygon(float x, float y, float radius, int npoints) {
  float angle = TWO_PI / npoints;
  beginShape();
  for (float a = 0; a < TWO_PI; a += angle) {
    float sx = x + cos(a) * radius;
    float sy = y + sin(a) * radius;
    vertex(sx, sy);
  }
  endShape(CLOSE);
}

void drawCosmologyOverlay() {
  fill(15, 23, 42, 220);
  stroke(51, 65, 85);
  rect(20, 20, width - 40, 85, 6);
  
  fill(167, 139, 250);
  textSize(13);
  textAlign(LEFT, BASELINE);
  text("DTIM-ZEC QUANTENGRAVITATION // BAND 4 OBJEKT 001: LQG & SPIN-SCHÄUME", 35, 42);
  
  fill(226, 232, 240);
  textSize(11);
  text("KLASSIFIKATION: " + modeNames[mode], 35, 62);
  
  if (mode == 0) {
    fill(167, 139, 250);
    text("QUANTENGEOMETRIE: Diskrete Eigenwerte der Flächen- (hat A) und Volumenoperatoren (hat V) ersetzen das Kontinuum", 35, 82);
  } else if (mode == 1) {
    fill(56, 189, 248);
    text("SPIN-NETZWERKE: Graphen mit SU(2)-Spins j_i und Intertwiner-Knoten codieren den 3D-Quantenraum", 35, 82);
  } else if (mode == 2) {
    fill(244, 114, 182);
    text("SPIN-SCHÄUME: Kovariente 4D-Raumzeit-Evolution über 2-Komplexe und EPRL-Vertex-Amplituden", 35, 82);
  } else {
    fill(52, 211, 153);
    text("DTIM-ZEC POSTULAT: Spin-Knoten entsprechen den verklemmten Vertices der ZEC-Raumhexagon-Superzellen", 35, 82);
  }
}

Neben den Weltflächen-Koordinaten $(\sigma^0, \sigma^1)$ und Raumzeit-Feldern $X^M(\sigma)$ koppelt der DTIM-ZEC Torsionstensor $\mathbf{T}_{\text{ZEC}}^{(\text{string})}$ die Schwingungsmoden an die ZEC-Gittergeometrie.

Die Ricci-Flachheit $R_{m\bar{n}} = 0$ garantiert die Existenz einer kovariant konstanten Spinor-Struktur, während die 3. Chern-Klasse $c_3(TM)$ die globale Euler-Charakteristik $\chi$ der Calabi-Yau-Mannigfaltigkeit festlegt.

/**
 * DTIM-ZEC MODULAR PHYSICS FRAMEWORK (DAMTP ACADEMIC STANDARD)
 * DTIM_QuantumGravity_002_String_MTheory_CalabiYau.pde - String Theory & Calabi-Yau Engine
 * Target: Processing 4.5.6 / Quantum Field Compendium Band 4 (Objekt 002)
 * License: CC0 1.0 Universal / MIT
 * Copyright (c) 2026 Jan Bühring & Adaptive Intelligence Synthesis Network (Google AI)
 */

int mode = 0; // 0: 10D Superstring Vibrations (Open Strings on D-Branes & Closed Graviton Loops), 1: 11D M-Theory Dualities & M2/M5 Membrane Worldvolumes, 2: 6D Calabi-Yau 3-Fold Moduli Projection (h^1,1, h^2,1 & Euler Characteristic chi), 3: DTIM-ZEC Raumhexagon String Lattice Core
float wavePhase = 0.0f;
String[] modeNames = {
  "1. 10D Superstrings & D-Branen // Offene Strings (Eichfelder) verankert auf D-Branen, geschlossene Strings (Gravitonen)", 
  "2. 11D M-Theorie Dualitäten // S/T-Dualitätsnetzwerk vereinigt Stringtheorien über M2/M5-Membranen", 
  "3. Calabi-Yau CY3 Geometrie // 6D Ricci-flache Kompaktifizierung (h^1,1, h^2,1) steuert Teilchengenerationen", 
  "4. DTIM-ZEC Torsions-Gitter // Innere 6D-Phasenstruktur des Raumhexagongitters"
};

void setup() {
  size(800, 380);
  println("[DTIM-QUANTUM-PHYSICS] Initializing Volume 4 Object 002 (String, M-Theory & Calabi-Yau Engine)...");
}

void draw() {
  background(11, 15, 25);
  wavePhase += 0.04f;
  
  // Auto switch mode every 4 seconds
  if (frameCount % 240 == 0) {
    mode = (mode + 1) % 4;
  }
  
  translate(width/2, height/2 + 10);
  
  if (mode == 0) {
    drawSuperstringVibrationsDBranes();
  } else if (mode == 1) {
    drawMTheoryMembraneDualities();
  } else if (mode == 2) {
    drawCalabiYauManifoldProjection();
  } else {
    drawDTIMHexagonStringLatticeCore();
  }
  
  resetMatrix();
  drawCosmologyOverlay();
}

void drawSuperstringVibrationsDBranes() {
  // 10D Superstring Vibrations & D-Branes
  
  // D3-Brane Worldvolume (Cyan Parallel Planes)
  stroke(34, 211, 238, 180); strokeWeight(2.5f); fill(15, 23, 42, 160);
  rect(-200, -70, 400, 30, 4);
  rect(-200, 40, 400, 30, 4);
  
  // Open String Anchored on D-Branes (Pink Oscillating String)
  stroke(244, 114, 182); strokeWeight(3.0f); noFill();
  beginShape();
  for (int x = -120; x <= 120; x += 4) {
    float normX = (x + 120) / 240.0f;
    float y = -55.0f + normX * 100.0f + sin(wavePhase * 3.0f + normX * TWO_PI * 2.0f) * 20.0f;
    vertex(x, y);
  }
  endShape();
  
  // Closed String Loop (Graviton - Yellow Oscillating Ellipse)
  pushMatrix();
  translate(110, -10);
  stroke(250, 204, 21); strokeWeight(2.5f); noFill();
  beginShape();
  for (int a = 0; a <= 360; a += 10) {
    float rad = radians(a);
    float rMod = 28.0f + sin(rad * 4.0f + wavePhase * 2.5f) * 6.0f;
    vertex(rMod * cos(rad), rMod * sin(rad));
  }
  endShape(CLOSE);
  popMatrix();
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Offener String (Verankert auf D-Branen)", -20, 95);
  text("Geschlossener String (Graviton-Loop)", 110, 25);
  text("Superstring Schwingungs-Spektrum: Mode n bestimmt Masse m^2 = (n - 1) / alpha'", 0, -115);
}

void drawMTheoryMembraneDualities() {
  // 11D M-Theory Dualities & M2/M5 Membrane Worldvolumes
  
  // 11D Bulk Grid Lines
  stroke(71, 85, 105); strokeWeight(1.0f);
  for (int i = -180; i <= 180; i += 40) {
    line(i, -90, i, 90);
    line(-180, i / 2, 180, i / 2);
  }
  
  // M2-Membrane Surface (Rose Wavy Grid)
  stroke(244, 114, 182, 220); strokeWeight(2.0f); noFill();
  for (int y = -50; y <= 50; y += 20) {
    beginShape();
    for (int x = -150; x <= 150; x += 10) {
      float z = sin(wavePhase * 2.0f + x * 0.03f + y * 0.05f) * 18.0f;
      vertex(x, y + z);
    }
    endShape();
  }
  
  // S-Duality / T-Duality Arrows
  stroke(52, 211, 153); strokeWeight(2.0f);
  line(-140, -75, 140, -75);
  line(125, -80, 140, -75); line(125, -70, 140, -75);
  
  fill(52, 211, 153); textSize(11); textAlign(CENTER, CENTER);
  text("S-Dualität (gs <-> 1/gs) & T-Dualität (R <-> alpha'/R)", 0, -90);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("11D M-Theorie: M2- & M5-Branen als fundamentale solitonische Freiheitsgrade", 0, 115);
}

void drawCalabiYauManifoldProjection() {
  // 6D Calabi-Yau 3-Fold Moduli Projection (CY3)
  
  strokeWeight(1.8f); noFill();
  int petalCount = 6;
  
  // Interlocking Calabi-Yau Petal Loops (Cyan/Pink Curves)
  for (int i = 0; i < petalCount; i++) {
    float angle = i * (TWO_PI / (float)petalCount) + wavePhase * 0.3f;
    
    pushMatrix();
    rotate(angle);
    
    // Outer Petal
    stroke(34, 211, 238, 200);
    beginShape();
    for (float t = 0; t <= TWO_PI; t += 0.1f) {
      float r = 55.0f * (1.0f + 0.4f * sin(t * 3.0f + wavePhase * 1.5f));
      float px = r * cos(t);
      float py = r * sin(t) * 0.5f;
      vertex(px + 30.0f, py);
    }
    endShape(CLOSE);
    
    // Inner Moduli Cycle (Kähler / Complex Structure)
    stroke(244, 114, 182, 180);
    ellipse(30, 0, 25 + sin(wavePhase * 2.0f + i) * 6.0f, 15);
    
    popMatrix();
  }
  
  fill(250, 204, 21); stroke(255); strokeWeight(1.5f);
  ellipse(0, 0, 12, 12);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Calabi-Yau 3-Fold (CY3): h^1,1 (Kähler-Moduli), h^2,1 (Komplexe Struktur)", 0, -115);
  text("Euler-Charakteristik chi = 2(h^1,1 - h^2,1) -> Fermionen-Generationen N_gen = 1/2 |chi| = 3", 0, 115);
}

void drawDTIMHexagonStringLatticeCore() {
  // DTIM-ZEC Raumhexagon String Lattice Core
  strokeWeight(1.8f); noFill();
  
  float hexRadius = 34.0f;
  for (int q = -3; q <= 3; q++) {
    for (int r = -2; r <= 2; r++) {
      float x = hexRadius * (sqrt(3) * q + sqrt(3)/2.0f * r);
      float y = hexRadius * (3.0f/2.0f * r);
      
      float twist = sin(wavePhase * 3.5f + (q + r) * 0.8f) * 8.0f;
      
      pushMatrix();
      translate(x, y);
      rotate(twist * 0.04f);
      stroke(244, 114, 182, 180);
      polygon(0, 0, hexRadius * 0.85f, 6);
      
      fill(52, 211, 153, 220); noStroke();
      ellipse(0, 0, 5, 5);
      popMatrix();
    }
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("DTIM-ZEC Raumhexagon-Torsions-Netzwerk der String-Geometrie", 0, 115);
}

void polygon(float x, float y, float radius, int npoints) {
  float angle = TWO_PI / npoints;
  beginShape();
  for (float a = 0; a < TWO_PI; a += angle) {
    float sx = x + cos(a) * radius;
    float sy = y + sin(a) * radius;
    vertex(sx, sy);
  }
  endShape(CLOSE);
}

void drawCosmologyOverlay() {
  fill(15, 23, 42, 220);
  stroke(51, 65, 85);
  rect(20, 20, width - 40, 85, 6);
  
  fill(244, 114, 182);
  textSize(13);
  textAlign(LEFT, BASELINE);
  text("DTIM-ZEC QUANTENGRAVITATION // BAND 4 OBJEKT 002: SUPERSTRING, M-THEORIE & CALABI-YAU", 35, 42);
  
  fill(226, 232, 240);
  textSize(11);
  text("KLASSIFIKATION: " + modeNames[mode], 35, 62);
  
  if (mode == 0) {
    fill(244, 114, 182);
    text("SUPERSTRINGS: Quantisierte 1D-Strings (l_s ~ 10^-34 m) eliminieren UV-Divergenzen der Quantengravitation", 35, 82);
  } else if (mode == 1) {
    fill(52, 211, 153);
    text("M-THEORIE: 11D-Supergravitation vereinigt die 5 Stringtheorien über S/T-Dualitäten und M2/M5-Branen", 35, 82);
  } else if (mode == 2) {
    fill(34, 211, 238);
    text("CALABI-YAU: 6D Kompaktifizierung mit SU(3)-Holonomie garantiert N=1 Supersymmetrie und 3 Teilchenfamilien", 35, 82);
  } else {
    fill(250, 204, 21);
    text("DTIM-ZEC POSTULAT: Calabi-Yau-Mannigfaltigkeiten bilden die innere 6D Phasen-Matrix der ZEC-Superzellen", 35, 82);
  }
}

Das Stern-Produkt verschiebt Wechselwirkungen auf den Phasenraum der ZEC-Hexagone und unterdrückt punktförmige Divergenzen durch höhere Ableitungsterme $\theta^{\mu\nu}$.

Die Spur der abgeschnittenen Dirac-Operator-Aktion $\mathcal{D}_A$ liefert in ihrer asynchronen Wärmeleitungskoeffizienten-Entwicklung zeitgleich die Einstein-Hilbert-Wirkung, die Yang-Mills-Wirkung und das Higgs-Potential.

/**
 * DTIM-ZEC MODULAR PHYSICS FRAMEWORK (DAMTP ACADEMIC STANDARD)
 * DTIM_QuantumGravity_003_NCG_Moyal_PlanckUncertainty.pde - NCG & Moyal Plane Engine
 * Target: Processing 4.5.6 / Quantum Field Compendium Band 4 (Objekt 003)
 * License: CC0 1.0 Universal / MIT
 * Copyright (c) 2026 Jan Bühring & Adaptive Intelligence Synthesis Network (Google AI)
 */

int mode = 0; // 0: Non-Commutative Coordinate Uncertainty [x, y] = i theta & Planck Cell Discretization, 1: Moyal-Star-Product f * g Phase Interference & Non-Local Vertex, 2: Connes Spectral Triple (A, H, D) Dirac Operator Spectrum, 3: DTIM-ZEC Raumhexagon Non-Commutative Lattice Core
float wavePhase = 0.0f;
String[] modeNames = {
  "1. NCG Koordinaten-Unschärfe // [x, y] = i theta -> Minimale Planck-Zellfläche Delta x Delta y >= 1/2 theta", 
  "2. Moyal-Stern-Produkt f * g // Deformierte Funktionenmultiplikation & Nicht-Lokale Phaseninterferenz", 
  "3. Connes Spektrales Tripel // Dirac-Operator D Spektrum codiert Metrik, Eichfelder & Higgs-Sektor", 
  "4. DTIM-ZEC Raumhexagon NCG Core // Torsions-Dislokation der ZEC-Superzell-Knotenpunkte"
};

void setup() {
  size(800, 380);
  println("[DTIM-QUANTUM-PHYSICS] Initializing Volume 4 Object 003 (NCG & Moyal Engine)...");
}

void draw() {
  background(11, 15, 25);
  wavePhase += 0.04f;
  
  // Auto switch mode every 4 seconds
  if (frameCount % 240 == 0) {
    mode = (mode + 1) % 4;
  }
  
  translate(width/2, height/2 + 10);
  
  if (mode == 0) {
    drawNonCommutativePlanckUncertainty();
  } else if (mode == 1) {
    drawMoyalStarProductInterference();
  } else if (mode == 2) {
    drawConnesDiracSpectralTriple();
  } else {
    drawDTIMHexagonNCGLatticeCore();
  }
  
  resetMatrix();
  drawCosmologyOverlay();
}

void drawNonCommutativePlanckUncertainty() {
  // Non-Commutative Coordinate Uncertainty [x, y] = i theta
  
  // Grid of Fuzzy Planck Cells (Emerald Blurred Ellipses)
  int cols = 7;
  int rows = 4;
  float spacingX = 50.0f;
  float spacingY = 40.0f;
  
  for (int i = -3; i <= 3; i++) {
    for (int j = -2; j <= 2; j++) {
      float cx = i * spacingX;
      float cy = j * spacingY;
      
      float uncertaintyX = 22.0f + sin(wavePhase * 2.5f + (i + j) * 0.5f) * 6.0f;
      float uncertaintyY = 22.0f + cos(wavePhase * 2.5f + (i - j) * 0.5f) * 6.0f;
      
      fill(52, 211, 153, 60); stroke(52, 211, 153, 180); strokeWeight(1.5f);
      ellipse(cx, cy, uncertaintyX * 1.5f, uncertaintyY * 1.5f);
      
      fill(167, 139, 250); noStroke();
      ellipse(cx, cy, 4, 4);
    }
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("NCG Planck-Zell-Unschärfe: [x, y] = i theta", 0, -115);
  text("Punkte verschmelzen zu unscharfen Flächen-Quanten Delta x Delta y >= 1/2 theta", 0, 115);
}

void drawMoyalStarProductInterference() {
  // Moyal-Star-Product f * g Phase Interference Wavefronts
  
  // Wave 1 (f - Cyan Wave)
  stroke(34, 211, 238, 200); strokeWeight(2.5f); noFill();
  beginShape();
  for (int x = -200; x <= 200; x += 5) {
    float y = sin(wavePhase * 2.0f + x * 0.04f) * 35.0f;
    vertex(x, y - 20);
  }
  endShape();
  
  // Wave 2 (g - Pink Wave)
  stroke(244, 114, 182, 200); strokeWeight(2.5f);
  beginShape();
  for (int x = -200; x <= 200; x += 5) {
    float y = cos(wavePhase * 2.0f + x * 0.04f) * 35.0f;
    vertex(x, y + 20);
  }
  endShape();
  
  // Star-Product Interference Fringes (Moyal Non-Local Overlap - Green Line)
  stroke(52, 211, 153); strokeWeight(3.0f);
  beginShape();
  for (int x = -200; x <= 200; x += 5) {
    float y1 = sin(wavePhase * 2.0f + x * 0.04f) * 35.0f;
    float y2 = cos(wavePhase * 2.0f + x * 0.04f) * 35.0f;
    float starY = (y1 + y2) * 0.5f + sin(x * 0.2f + wavePhase * 3.0f) * 8.0f;
    vertex(x, starY);
  }
  endShape();
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Moyal-Stern-Produkt (f * g)(x) = f(x) exp(i/2 theta mu nu d_mu d_nu) g(x)", 0, -115);
  text("Nicht-lokale Interferenz unterdrückt ultraviolette Punkt-Divergenzen", 0, 115);
}

void drawConnesDiracSpectralTriple() {
  // Connes Spectral Triple (A, H, D) Dirac Operator Eigenvalues
  
  // Dirac Spectrum Discrete Levels (Glowing Gold Lines)
  stroke(250, 204, 21); strokeWeight(2.5f);
  for (int i = -4; i <= 4; i++) {
    float y = i * 20.0f;
    float len = 160.0f - abs(i) * 15.0f + sin(wavePhase * 3.0f + i) * 10.0f;
    line(-len / 2.0f, y, len / 2.0f, y);
    
    fill(167, 139, 250); noStroke();
    ellipse(-len / 2.0f, y, 6, 6);
    ellipse(len / 2.0f, y, 6, 6);
  }
  
  // Central Dirac Operator D Label
  fill(52, 211, 153); textSize(12); textAlign(CENTER, CENTER);
  text("Dirac-Operator D Eigenwerte lambda_i", 0, -100);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Connes Spektrales Tripel (A, H, D): Metrik d(p,q) = sup { |f(p) - f(q)| : ||[D,f]|| <= 1 }", 0, 115);
}

void drawDTIMHexagonNCGLatticeCore() {
  // DTIM-ZEC Raumhexagon NCG Lattice Core
  strokeWeight(1.8f); noFill();
  
  float hexRadius = 34.0f;
  for (int q = -3; q <= 3; q++) {
    for (int r = -2; r <= 2; r++) {
      float x = hexRadius * (sqrt(3) * q + sqrt(3)/2.0f * r);
      float y = hexRadius * (3.0f/2.0f * r);
      
      float shiftX = sin(wavePhase * 3.0f + (q + r) * 0.7f) * 5.0f;
      float shiftY = cos(wavePhase * 3.0f + (q - r) * 0.7f) * 5.0f;
      
      pushMatrix();
      translate(x + shiftX, y + shiftY);
      stroke(52, 211, 153, 200);
      polygon(0, 0, hexRadius * 0.85f, 6);
      
      fill(250, 204, 21, 240); noStroke();
      ellipse(0, 0, 5, 5);
      popMatrix();
    }
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("DTIM-ZEC Raumhexagon Torsions-Dislokation: Geometrisierung der NCG", 0, 115);
}

void polygon(float x, float y, float radius, int npoints) {
  float angle = TWO_PI / npoints;
  beginShape();
  for (float a = 0; a < TWO_PI; a += angle) {
    float sx = x + cos(a) * radius;
    float sy = y + sin(a) * radius;
    vertex(sx, sy);
  }
  endShape(CLOSE);
}

void drawCosmologyOverlay() {
  fill(15, 23, 42, 220);
  stroke(51, 65, 85);
  rect(20, 20, width - 40, 85, 6);
  
  fill(52, 211, 153);
  textSize(13);
  textAlign(LEFT, BASELINE);
  text("DTIM-ZEC QUANTENGRAVITATION // BAND 4 OBJEKT 003: NICHT-KOMMUTATIVE GEOMETRIE & MOYAL", 35, 42);
  
  fill(226, 232, 240);
  textSize(11);
  text("KLASSIFIKATION: " + modeNames[mode], 35, 62);
  
  if (mode == 0) {
    fill(52, 211, 153);
    text("KOORDINATEN-UNSCHÄRFE: [x, y] = i theta verhindert punkteförmige Singularitäten auf der Planck-Skala", 35, 82);
  } else if (mode == 1) {
    fill(34, 211, 238);
    text("MOYAL-STERN-PRODUKT: Deformation der Multiplikation erzeugt nicht-lokale Phasen-Interferenzen", 35, 82);
  } else if (mode == 2) {
    fill(250, 204, 21);
    text("SPEKTRALES TRIPEL: Der Connes Dirac-Operator D leitet Gravitation, Eichfelder & Higgs-Sektor simultan ab", 35, 82);
  } else {
    fill(167, 139, 250);
    text("DTIM-ZEC POSTULAT: Nicht-Kommutativität theta ist die Scherungsfläche der ZEC-Raumhexagon-Superzellen", 35, 82);
  }
}

Der ZEC-Gitter-Kopplungstensor $\mathbf{T}_{\text{ZEC}}^{(\text{AdS/CFT})}$ verbindet die klassischen Bulk-Lösungen $S_{\text{on-shell}}$ mit den stark gekoppelten Quantenfluktuationen der Rand-Operatoren $\mathcal{O}(x)$.

Die minimale Hyperfläche $\gamma_A$ erstreckt sich von den Rändern des Intervalls $A$ der Länge $L$ in die hyperbolische Tiefe $z$ des Bulk-Raumes bis zum Cutoff $\epsilon$.

/**
 * DTIM-ZEC MODULAR PHYSICS FRAMEWORK (DAMTP ACADEMIC STANDARD)
 * DTIM_QuantumGravity_004_AdS_CFT_Holography.pde - AdS/CFT & Ryu-Takayanagi Engine
 * Target: Processing 4.5.6 / Quantum Field Compendium Band 4 (Objekt 004)
 * License: CC0 1.0 Universal / MIT
 * Copyright (c) 2026 Jan Bühring & Adaptive Intelligence Synthesis Network (Google AI)
 */

int mode = 0; // 0: Hyperbolic AdS Bulk Geometry & Boundary CFT Fields, 1: GKPW Dictionary Bulk Field Projections phi(z,x) -> O(x), 2: Ryu-Takayanagi Minimal Geodesic Arc gamma_A & Entanglement Entropy S_EE, 3: DTIM-ZEC Holographic Tensor Network Core
float wavePhase = 0.0f;
String[] modeNames = {
  "1. Hyperbolische AdS/CFT Dualität // Bulk-Gravitation im AdS-Raum & Duale CFT-Randfeldtheorie", 
  "2. GKPW Wörterbuch // Randquellen phi_0 erzeugen Bulk-Felder phi(z,x) & CFT-Operatoren O(x)", 
  "3. Ryu-Takayanagi Minimalfläche // Geodäte gamma_A im Bulk berechnet Verschränkungsentropie S_EE(A) = Area / (4 G_N)", 
  "4. DTIM-ZEC Holografisches Tensor-Netzwerk // MERA-artige ZEC-Raumhexagon Gitter-Rekonstruktion"
};

void setup() {
  size(800, 380);
  println("[DTIM-QUANTUM-PHYSICS] Initializing Volume 4 Object 004 (AdS/CFT & Holography Engine)...");
}

void draw() {
  background(11, 15, 25);
  wavePhase += 0.04f;
  
  // Auto switch mode every 4 seconds
  if (frameCount % 240 == 0) {
    mode = (mode + 1) % 4;
  }
  
  translate(width/2, height/2 + 10);
  
  if (mode == 0) {
    drawAdSBulkCFTBoundaryGeometry();
  } else if (mode == 1) {
    drawGKPWDictionaryBulkBoundaryMapping();
  } else if (mode == 2) {
    drawRyuTakayanagiMinimalGeodesic();
  } else {
    drawDTIMHexagonTensorNetworkCore();
  }
  
  resetMatrix();
  drawCosmologyOverlay();
}

void drawAdSBulkCFTBoundaryGeometry() {
  // Hyperbolic AdS Bulk Geometry & Boundary CFT
  
  // Poincaré Disc Boundary (Cyan Circle)
  float discRadius = 110.0f;
  stroke(34, 211, 238, 220); strokeWeight(3.0f); fill(15, 23, 42, 180);
  ellipse(0, 0, discRadius * 2.0f, discRadius * 2.0f);
  
  // Hyperbolic Tiling Lines in Bulk (Purple Geodesic Arcs)
  stroke(167, 139, 250, 160); strokeWeight(1.5f); noFill();
  for (int i = 0; i < 8; i++) {
    float angle = i * (TWO_PI / 8.0f) + wavePhase * 0.2f;
    float x1 = discRadius * cos(angle);
    float y1 = discRadius * sin(angle);
    float x2 = discRadius * cos(angle + HALF_PI);
    float y2 = discRadius * sin(angle + HALF_PI);
    
    // Curved Geodesic
    bezier(x1, y1, x1 * 0.3f, y1 * 0.3f, x2 * 0.3f, y2 * 0.3f, x2, y2);
  }
  
  // Boundary CFT Fluctuation Nodes (Yellow Glowing Dots on Perimeter)
  for (int i = 0; i < 16; i++) {
    float angle = i * (TWO_PI / 16.0f) + wavePhase * 0.5f;
    float bx = discRadius * cos(angle);
    float by = discRadius * sin(angle);
    
    fill(250, 204, 21); stroke(255); strokeWeight(1.5f);
    ellipse(bx, by, 6, 6);
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Bulk: Hyperbolischer Anti-de Sitter Raum (AdS5)", 0, -125);
  text("Rand: Konforme Feldtheorie ohne Gravitation (CFT4)", 0, 125);
}

void drawGKPWDictionaryBulkBoundaryMapping() {
  // GKPW Dictionary Bulk Field Projections phi(z,x) -> O(x)
  
  // Boundary Line at z = 0 (Cyan Horizontal Line)
  stroke(34, 211, 238); strokeWeight(2.5f);
  line(-200, -80, 200, -80);
  
  // Bulk Depth Axis z (Going Downward)
  stroke(71, 85, 105); strokeWeight(1.5f);
  line(0, -80, 0, 90);
  
  // Bulk Field Propagator Lines (Yellow/Rose Rays from Boundary to Bulk)
  for (int i = -3; i <= 3; i++) {
    float sourceX = i * 50.0f;
    
    stroke(244, 114, 182, 180); strokeWeight(1.8f);
    line(sourceX, -80, 0, 40 + sin(wavePhase * 3.0f + i) * 10.0f);
    
    fill(250, 204, 21); stroke(255); strokeWeight(1.5f);
    ellipse(sourceX, -80, 8, 8);
  }
  
  // Bulk Operator Core Node
  fill(52, 211, 153); stroke(255); strokeWeight(2.0f);
  ellipse(0, 40, 18, 18);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("CFT Randquelle phi_0(x) (z = 0)", 0, -100);
  text("GKPW Relation: Z_bulk[phi_0] = < exp( integral phi_0 O ) >_CFT", 0, 115);
}

void drawRyuTakayanagiMinimalGeodesic() {
  // Ryu-Takayanagi Minimal Geodesic Arc gamma_A & Entanglement Entropy S_EE
  
  // CFT Boundary Line (Horizontal)
  stroke(34, 211, 238); strokeWeight(2.5f);
  line(-200, -60, 200, -60);
  
  // Subregion A on Boundary (Highlighted Yellow Segment)
  float aLeft = -100.0f;
  float aRight = 100.0f;
  stroke(250, 204, 21); strokeWeight(4.5f);
  line(aLeft, -60, aRight, -60);
  
  // Ryu-Takayanagi Minimal Geodesic Curve gamma_A in Bulk (Emerald Arc)
  stroke(52, 211, 153); strokeWeight(3.5f); noFill();
  beginShape();
  float radiusA = (aRight - aLeft) / 2.0f;
  for (int a = 0; a <= 180; a += 5) {
    float rad = radians(a);
    float gx = radiusA * cos(rad);
    float gy = -60.0f + radiusA * sin(rad);
    vertex(gx, gy);
  }
  endShape();
  
  // Boundary Endpoints of Region A (Glowing Pink Dots)
  fill(244, 114, 182); stroke(255); strokeWeight(2.0f);
  ellipse(aLeft, -60, 10, 10);
  ellipse(aRight, -60, 10, 10);
  
  fill(250, 204, 21); textSize(12); textAlign(CENTER, CENTER);
  text("Teilgebiet A", 0, -75);
  
  fill(52, 211, 153); textSize(12);
  text("Minimale Geodäte gamma_A im Bulk", 0, 25);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Ryu-Takayanagi Formel: S_EE(A) = Area(gamma_A) / (4 G_N)", 0, 115);
}

void drawDTIMHexagonTensorNetworkCore() {
  // DTIM-ZEC Holographic Tensor Network Core (MERA Structure)
  strokeWeight(1.8f); noFill();
  
  float hexRadius = 34.0f;
  for (int q = -3; q <= 3; q++) {
    for (int r = -2; r <= 2; r++) {
      float x = hexRadius * (sqrt(3) * q + sqrt(3)/2.0f * r);
      float y = hexRadius * (3.0f/2.0f * r);
      
      float pulse = sin(wavePhase * 3.5f + (q + r) * 0.8f) * 6.0f;
      
      pushMatrix();
      translate(x, y);
      stroke(34, 211, 238, 180);
      polygon(0, 0, hexRadius * 0.85f + pulse * 0.2f, 6);
      
      fill(52, 211, 153, 220); noStroke();
      ellipse(0, 0, 5, 5);
      popMatrix();
    }
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("DTIM-ZEC MERA Tensor-Netzwerk: Rekonstruktion des Bulk-Raumes aus ZEC-Bits", 0, 115);
}

void polygon(float x, float y, float radius, int npoints) {
  float angle = TWO_PI / npoints;
  beginShape();
  for (float a = 0; a < TWO_PI; a += angle) {
    float sx = x + cos(a) * radius;
    float sy = y + sin(a) * radius;
    vertex(sx, sy);
  }
  endShape(CLOSE);
}

void drawCosmologyOverlay() {
  fill(15, 23, 42, 220);
  stroke(51, 65, 85);
  rect(20, 20, width - 40, 85, 6);
  
  fill(34, 211, 238);
  textSize(13);
  textAlign(LEFT, BASELINE);
  text("DTIM-ZEC QUANTENGRAVITATION // BAND 4 OBJEKT 004: HOLOGRAFIE & AdS/CFT DUALITÄT", 35, 42);
  
  fill(226, 232, 240);
  textSize(11);
  text("KLASSIFIKATION: " + modeNames[mode], 35, 62);
  
  if (mode == 0) {
    fill(34, 211, 238);
    text("AdS/CFT DUALITÄT: Quantengravitation im (d+1)-D Bulk ist äquivalent zu d-D Randfeldtheorie ohne Gravitation", 35, 82);
  } else if (mode == 1) {
    fill(244, 114, 182);
    text("GKPW WÖRTERBUCH: Randwerte phi_0 koppeln als Quellen an lokale CFT-Operatoren O(x)", 35, 82);
  } else if (mode == 2) {
    fill(52, 211, 153);
    text("RYU-TAKAYANAGI: Minimale Geodätenfläche gamma_A berechnet exakt die Quantenverschränkungsentropie S_EE(A)", 35, 82);
  } else {
    fill(250, 204, 21);
    text("DTIM-ZEC POSTULAT: Der Bulk-Raum ist das MERA Tensor-Netzwerk der verschränkten ZEC-Superzellkanten", 35, 82);
  }
}

Die Supertrace $\text{STr}$ summiert über fluktuierende Spin-2 Gravitonen, Ghost-Felder und den ZEC-Torsionstensor $\mathbf{T}_{\text{ZEC}}^{(\text{AS})}$.

Die anomale Dimension des Gravitons $\eta_N(g,\lambda)$ kompensiert bei $g_*$ exakt den kanonischen Massengrad $2$ und erzeugt vollständige Skaleninvarianz auf Planck-Skalen.

/**
 * DTIM-ZEC MODULAR PHYSICS FRAMEWORK (DAMTP ACADEMIC STANDARD)
 * DTIM_QuantumGravity_005_AsymptoticSafety_Reuter.pde - Asymptotic Safety & Reuter Fixed Point Engine
 * Target: Processing 4.5.6 / Quantum Field Compendium Band 4 (Objekt 005)
 * License: CC0 1.0 Universal / MIT
 * Copyright (c) 2026 Jan Bühring & Adaptive Intelligence Synthesis Network (Google AI)
 */

int mode = 0; // 0: RG Flow Trajectories in Coupling Space (g, lambda) & Reuter Fixed Point NGFP, 1: Wetterich FRGE Functional Trace Integration & Regulator R_k, 2: Running Coupling Curves G(k) & Lambda(k) vs Log Scale k, 3: DTIM-ZEC Raumhexagon Cellular Fixed-Point Lattice Core
float wavePhase = 0.0f;
String[] modeNames = {
  "1. RG-Fluss Trajektorien // Convergence zum Reuter Non-Gaussian Fixed Point (g*, lambda*) im UV-Bereich", 
  "2. Wetterich FRGE Funktional-Spur // Ausintegrieren von Quantenfluktuationen ueber Regulator R_k", 
  "3. Running Kopplungen G(k) & Lambda(k) // Skalenabhaengigkeit & Asymptotische Abschirmung der Gravitation", 
  "4. DTIM-ZEC Hexagon Fixpunkt-Gitter // Resonante Eigenfrequenz-Sättigung der ZEC-Superzellen"
};

void setup() {
  size(800, 380);
  println("[DTIM-QUANTUM-PHYSICS] Initializing Volume 4 Object 005 (Asymptotic Safety & Reuter Engine)...");
}

void draw() {
  background(11, 15, 25);
  wavePhase += 0.04f;
  
  // Auto switch mode every 4 seconds
  if (frameCount % 240 == 0) {
    mode = (mode + 1) % 4;
  }
  
  translate(width/2, height/2 + 10);
  
  if (mode == 0) {
    drawRGFlowReuterFixedPoint();
  } else if (mode == 1) {
    drawWetterichFRGEIntegration();
  } else if (mode == 2) {
    drawRunningCouplingsGK();
  } else {
    drawDTIMHexagonFixedPointCore();
  }
  
  resetMatrix();
  drawCosmologyOverlay();
}

void drawRGFlowReuterFixedPoint() {
  // RG Flow Trajectories towards Reuter Fixed Point (g*, lambda*)
  
  // Axes for (lambda, g) space
  stroke(71, 85, 105); strokeWeight(1.5f);
  line(-200, 70, 200, 70); // lambda axis
  line(-150, 90, -150, -110); // g axis
  
  fill(148, 163, 184); textSize(10); textAlign(CENTER, CENTER);
  text("Kosmologische Konstante lambda_k ->", 100, 85);
  text("g_k", -170, -100);
  
  // Reuter Fixed Point NGFP (g* = 0.707, lambda* = 0.193 -> Screen Pos)
  float fpX = 20.0f;
  float fpY = -40.0f;
  
  // Spiral Trajectories Flowing into NGFP
  stroke(52, 211, 153, 180); strokeWeight(1.8f); noFill();
  for (int i = 0; i < 6; i++) {
    float startAngle = i * (TWO_PI / 6.0f);
    beginShape();
    for (float r = 160.0f; r >= 0.0f; r -= 2.0f) {
      float theta = startAngle + (160.0f - r) * 0.04f + wavePhase * 0.5f;
      float px = fpX + r * cos(theta);
      float py = fpY + r * sin(theta) * 0.6f;
      vertex(px, py);
    }
    endShape();
  }
  
  // Glowing Reuter Fixed Point Node (Gold Dot)
  fill(250, 204, 21); stroke(255); strokeWeight(2.0f);
  ellipse(fpX, fpY, 14, 14);
  
  fill(250, 204, 21); textSize(11); textAlign(LEFT, CENTER);
  text("Reuter-Fixpunkt (g*, lambda*)", fpX + 12, fpY);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("RG-Fluss im Phasenraum: Alle Trajektorien konvergieren im UV (k -> infty) gegen NGFP", 0, -115);
  text("Asymptotische Sicherheit verhindert UV-Divergenzen ohne Zusatzdimensionen", 0, 115);
}

void drawWetterichFRGEIntegration() {
  // Wetterich FRGE Functional Trace & Regulator Cutoff
  
  // FRGE Regulator Function R_k(p^2) Curve
  stroke(34, 211, 238); strokeWeight(2.5f); noFill();
  beginShape();
  for (int x = -180; x <= 180; x += 5) {
    float normX = x / 180.0f;
    float rVal = 50.0f * max(0.0f, 1.0f - normX * normX);
    vertex(x, 40 - rVal);
  }
  endShape();
  
  // Momentum Cutoff Line k (Gold Vertical Line)
  float kPos = sin(wavePhase * 2.0f) * 120.0f;
  stroke(250, 204, 21); strokeWeight(2.0f);
  line(kPos, -80, kPos, 70);
  
  // Fluctuation Modes Integration Shell
  stroke(244, 114, 182, 160); strokeWeight(1.5f);
  for (int i = -4; i <= 4; i++) {
    float modeX = kPos + i * 8.0f;
    line(modeX, -60, modeX, 40);
  }
  
  fill(250, 204, 21); textSize(11); textAlign(CENTER, CENTER);
  text("Impulskatoff k", kPos, -90);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Wetterich FRGE: d_k Gamma_k = 1/2 STr [ (Gamma_k^(2) + R_k)^(-1) d_k R_k ]", 0, -115);
  text("Regulator R_k(p^2) blendet Quantenfluktuationen unterhalb der Skala k stufenweise aus", 0, 115);
}

void drawRunningCouplingsGK() {
  // Running Coupling Curves G(k) & Lambda(k)
  
  // Axes
  stroke(71, 85, 105); strokeWeight(1.5f);
  line(-180, 60, 180, 60); // log k axis
  line(-180, 60, -180, -90); // coupling axis
  
  // G(k) Curve (Screening at High k - Cyan Line)
  stroke(34, 211, 238); strokeWeight(3.0f); noFill();
  beginShape();
  for (int x = -180; x <= 180; x += 5) {
    float normK = (x + 180) / 360.0f; // 0 to 1
    float gVal = 100.0f / (1.0f + pow(normK * 3.0f, 2.0f));
    vertex(x, 60 - gVal);
  }
  endShape();
  
  // Lambda(k) Curve (Growing at High k - Pink Line)
  stroke(244, 114, 182); strokeWeight(3.0f);
  beginShape();
  for (int x = -180; x <= 180; x += 5) {
    float normK = (x + 180) / 360.0f;
    float lVal = 5.0f + pow(normK * 2.5f, 2.0f) * 12.0f;
    vertex(x, 60 - lVal);
  }
  endShape();
  
  fill(34, 211, 238); textSize(11); textAlign(LEFT, CENTER);
  text("Running G(k) -> 0 (UV Anti-Screening)", 20, -50);
  fill(244, 114, 182);
  text("Running Lambda(k) ~ k^2", 20, 10);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Skalenabhaengigkeit: G(k) = G0 / (1 + omega G0 k^2) schwaecht Gravitation im UV ab", 0, -115);
  text("Verhindert die Bildung physikalischer Krümmungs-Singularitäten", 0, 115);
}

void drawDTIMHexagonFixedPointCore() {
  // DTIM-ZEC Raumhexagon Cellular Fixed-Point Lattice Core
  strokeWeight(1.8f); noFill();
  
  float hexRadius = 34.0f;
  for (int q = -3; q <= 3; q++) {
    for (int r = -2; r <= 2; r++) {
      float x = hexRadius * (sqrt(3) * q + sqrt(3)/2.0f * r);
      float y = hexRadius * (3.0f/2.0f * r);
      
      float saturation = sin(wavePhase * 3.5f + (q + r) * 0.8f) * 6.0f;
      
      pushMatrix();
      translate(x, y);
      stroke(52, 211, 153, 220);
      polygon(0, 0, hexRadius * 0.85f + saturation * 0.2f, 6);
      
      fill(250, 204, 21, 240); noStroke();
      ellipse(0, 0, 5, 5);
      popMatrix();
    }
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("DTIM-ZEC Raumhexagon-Fixpunkt: Geometrische Sättigung der ZEC-Superzellen", 0, 115);
}

void polygon(float x, float y, float radius, int npoints) {
  float angle = TWO_PI / npoints;
  beginShape();
  for (float a = 0; a < TWO_PI; a += angle) {
    float sx = x + cos(a) * radius;
    float sy = y + sin(a) * radius;
    vertex(sx, sy);
  }
  endShape(CLOSE);
}

void drawCosmologyOverlay() {
  fill(15, 23, 42, 220);
  stroke(51, 65, 85);
  rect(20, 20, width - 40, 85, 6);
  
  fill(52, 211, 153);
  textSize(13);
  textAlign(LEFT, BASELINE);
  text("DTIM-ZEC QUANTENGRAVITATION // BAND 4 OBJEKT 005: ASYMPTOTISCHE SICHERHEIT & REUTER", 35, 42);
  
  fill(226, 232, 240);
  textSize(11);
  text("KLASSIFIKATION: " + modeNames[mode], 35, 62);
  
  if (mode == 0) {
    fill(52, 211, 153);
    text("REUTER FIXPUNKT: Der Nicht-Gaußsche Fixpunkt (g*, lambda*) sichert die Quantengravitation asymptotisch ab", 35, 82);
  } else if (mode == 1) {
    fill(34, 211, 238);
    text("WETTERICH FRGE: Die exakte Renormierungsgleichung d_k Gamma_k integriert Quantenfluktuationen aus", 35, 82);
  } else if (mode == 2) {
    fill(244, 114, 182);
    text("RUNNING KOPPLUNGEN: G(k) faellt im UV quadratisch ab und eliminiert Nahbereichs-Singularitaeten", 35, 82);
  } else {
    fill(250, 204, 21);
    text("DTIM-ZEC POSTULAT: Asymptotische Sicherheit ist die physikalische Schwingungssättigung des ZEC-Gitters", 35, 82);
  }
}

$N_0$ bezeichnet die Anzahl der Knoten, $N_4^{(4,1)}$ und $N_4^{(3,2)}$ die Anzahlen der kausalen 4-Simplex-Typen. Die Kopplungen $(\kappa_0, \kappa_4, \Delta)$ steuern das Phasen-Diagramm.

Die Rückkehrwahrscheinlichkeit $P(\sigma) = \frac{1}{V} \text{Tr} K(x, x; \sigma)$ eines Random Walkers nach Diffusionszeit $\sigma$ definiert die effektive Fraktaldimension der triangulierten Quantenraumzeit.

/**
 * DTIM-ZEC MODULAR PHYSICS FRAMEWORK (DAMTP ACADEMIC STANDARD)
 * DTIM_QuantumGravity_006_CDT_Regge_Fractal.pde - Causal Dynamical Triangulations & Regge Calculus Engine
 * Target: Processing 4.5.6 / Quantum Field Compendium Band 4 (Objekt 006)
 * License: CC0 1.0 Universal / MIT
 * Copyright (c) 2026 Jan Bühring & Adaptive Intelligence Synthesis Network (Google AI)
 */

int mode = 0; // 0: Causal Simplicial Triangulation & Regge Deficit Angles epsilon_h, 1: CDT Phase Diagram & Emergent de Sitter Universe Phase C, 2: Quantum Fractals & Spectral Dimension d_s(t) = 4 -> 2 Diffusion Walk, 3: DTIM-ZEC Raumhexagon CDT Lattice Core
float wavePhase = 0.0f;
String[] modeNames = {
  "1. Regge-Kalkül & Triangulierung // Diskrete 4D Simplex-Bausteine & Defizitwinkel epsilon_h an Scharnieren", 
  "2. CDT Phasendiagramm Phase C // Kausale Blätterung erzeugt makroskopisches de-Sitter-Universum S4", 
  "3. Quanten-Fraktalität d_s(t) // Random Walk Diffusion zeigt dimensionale Reduktion von 4 auf 2 im UV", 
  "4. DTIM-ZEC Raumhexagon CDT Core // Hexagonale Resonanz-Eichung der kausalen Schichtungsebenen"
};

void setup() {
  size(800, 380);
  println("[DTIM-QUANTUM-PHYSICS] Initializing Volume 4 Object 006 (CDT & Regge Engine)...");
}

void draw() {
  background(11, 15, 25);
  wavePhase += 0.04f;
  
  // Auto switch mode every 4 seconds
  if (frameCount % 240 == 0) {
    mode = (mode + 1) % 4;
  }
  
  translate(width/2, height/2 + 10);
  
  if (mode == 0) {
    drawReggeSimplicialTriangulation();
  } else if (mode == 1) {
    drawCDTDeSitterPhaseC();
  } else if (mode == 2) {
    drawFractalDiffusionSpectralDimension();
  } else {
    drawDTIMHexagonCDTLatticeCore();
  }
  
  resetMatrix();
  drawCosmologyOverlay();
}

void drawReggeSimplicialTriangulation() {
  // Causal Simplicial Triangulation & Regge Deficit Angles
  
  // Triangulated Mesh Network (Golden Triangles)
  stroke(250, 204, 21, 180); strokeWeight(1.8f); noFill();
  
  float[][] verts = {
    {0, -70}, {-110, -20}, {110, -20}, {-60, 60}, {60, 60}, {0, 10}
  };
  
  // Triangles
  triangle(verts[0][0], verts[0][1], verts[1][0], verts[1][1], verts[5][0], verts[5][1]);
  triangle(verts[0][0], verts[0][1], verts[2][0], verts[2][1], verts[5][0], verts[5][1]);
  triangle(verts[1][0], verts[1][1], verts[3][0], verts[3][1], verts[5][0], verts[5][1]);
  triangle(verts[2][0], verts[2][1], verts[4][0], verts[4][1], verts[5][0], verts[5][1]);
  triangle(verts[3][0], verts[3][1], verts[4][0], verts[4][1], verts[5][0], verts[5][1]);
  
  // Deficit Angle Hinge Node (Center Vertex - Glowing Red/Purple)
  fill(244, 114, 182); stroke(255); strokeWeight(2.0f);
  float pulse = sin(wavePhase * 3.0f) * 4.0f;
  ellipse(verts[5][0], verts[5][1], 16 + pulse, 16 + pulse);
  
  // Deficit Angle Arc
  stroke(244, 114, 182); strokeWeight(2.5f); noFill();
  arc(verts[5][0], verts[5][1], 35, 35, -PI*0.7f, PI*0.8f);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Scharnier h: Defizitwinkel epsilon_h = 2pi - sum theta_h", 0, -115);
  text("Regge-Wirkung: S_Regge = 1/(8pi G) sum V_h epsilon_h - Lambda sum V_sigma", 0, 115);
}

void drawCDTDeSitterPhaseC() {
  // CDT Phase C: Emergent de Sitter Universe Volume Profile V3(t)
  
  // Slices in Time t (Cyan Hyperplanes)
  stroke(34, 211, 238, 140); strokeWeight(1.5f);
  for (int t = -160; t <= 160; t += 20) {
    float normT = t / 160.0f;
    float radiusV = cos(normT * HALF_PI) * 75.0f;
    line(t, -radiusV, t, radiusV);
  }
  
  // de Sitter Envelope Curve (Phase C - Glowing Emerald Lines)
  stroke(52, 211, 153); strokeWeight(3.0f); noFill();
  beginShape();
  for (int t = -160; t <= 160; t += 5) {
    float normT = t / 160.0f;
    float radiusV = cos(normT * HALF_PI) * 75.0f;
    vertex(t, -radiusV);
  }
  endShape();
  
  beginShape();
  for (int t = -160; t <= 160; t += 5) {
    float normT = t / 160.0f;
    float radiusV = cos(normT * HALF_PI) * 75.0f;
    vertex(t, radiusV);
  }
  endShape();
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("CDT Phase C: Spontane Emergenz eines 4D de Sitter Universums S4", 0, -115);
  text("Volumenprofil V3(t) proportional cos^3(c * t) bestätigt klassische Makro-Geometrie", 0, 115);
}

void drawFractalDiffusionSpectralDimension() {
  // Quantum Fractals & Spectral Dimension d_s(t) = 4 -> 2 Diffusion Walk
  
  // Random Walk Diffusion Path (Fractal Line - Cyan/Pink)
  stroke(34, 211, 238, 200); strokeWeight(2.0f); noFill();
  beginShape();
  float rx = 0; float ry = 0;
  for (int i = 0; i < 80; i++) {
    float angle = noise(i * 0.1f, wavePhase * 0.5f) * TWO_PI * 3.0f;
    rx += cos(angle) * 7.0f;
    ry += sin(angle) * 7.0f;
    vertex(rx, ry);
  }
  endShape();
  
  // Spectral Dimension Curve Box (Inset Graph on Right)
  pushMatrix();
  translate(110, -20);
  fill(15, 23, 42, 200); stroke(71, 85, 105); strokeWeight(1.5f);
  rect(-50, -40, 100, 80, 4);
  
  // d_s Curve (2 at small t, 4 at large t)
  stroke(244, 114, 182); strokeWeight(2.0f); noFill();
  beginShape();
  for (int x = -40; x <= 40; x += 5) {
    float normX = (x + 40) / 80.0f;
    float dsVal = 2.0f + pow(normX, 2.0f) * 2.0f; // 2 to 4
    vertex(x, 20 - (dsVal / 4.0f) * 50.0f);
  }
  endShape();
  
  fill(244, 114, 182); textSize(9); textAlign(CENTER, CENTER);
  text("d_s(t) -> 2 (UV)", 0, -28);
  text("d_s(t) -> 4 (IR)", 0, 28);
  popMatrix();
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Quanten-Fraktalität: Random Walk Diffusion auf trianguliertem Schaum", -60, -115);
  text("Spektrale Dimension d_s fällt auf Planck-Skalen stetig von 4 auf 2 ab", 0, 115);
}

void drawDTIMHexagonCDTLatticeCore() {
  // DTIM-ZEC Raumhexagon CDT Lattice Core
  strokeWeight(1.8f); noFill();
  
  float hexRadius = 34.0f;
  for (int q = -3; q <= 3; q++) {
    for (int r = -2; r <= 2; r++) {
      float x = hexRadius * (sqrt(3) * q + sqrt(3)/2.0f * r);
      float y = hexRadius * (3.0f/2.0f * r);
      
      float pulse = sin(wavePhase * 3.5f + (q + r) * 0.8f) * 6.0f;
      
      pushMatrix();
      translate(x, y);
      stroke(250, 204, 21, 200);
      polygon(0, 0, hexRadius * 0.85f + pulse * 0.2f, 6);
      
      fill(52, 211, 153, 240); noStroke();
      ellipse(0, 0, 5, 5);
      popMatrix();
    }
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("DTIM-ZEC Raumhexagon CDT Core: Resonante Eichung der Kausalschichten", 0, 115);
}

void polygon(float x, float y, float radius, int npoints) {
  float angle = TWO_PI / npoints;
  beginShape();
  for (float a = 0; a < TWO_PI; a += angle) {
    float sx = x + cos(a) * radius;
    float sy = y + sin(a) * radius;
    vertex(sx, sy);
  }
  endShape(CLOSE);
}

void drawCosmologyOverlay() {
  fill(15, 23, 42, 220);
  stroke(51, 65, 85);
  rect(20, 20, width - 40, 85, 6);
  
  fill(250, 204, 21);
  textSize(13);
  textAlign(LEFT, BASELINE);
  text("DTIM-ZEC QUANTENGRAVITATION // BAND 4 OBJEKT 006: CDT, REGGE-KALKÜL & FRAKTALE", 35, 42);
  
  fill(226, 232, 240);
  textSize(11);
  text("KLASSIFIKATION: " + modeNames[mode], 35, 62);
  
  if (mode == 0) {
    fill(250, 204, 21);
    text("REGGE-KALKÜL: Diskrete Krümmung wird über Defizitwinkel epsilon_h an 2D-Scharnieren der 4-Simplizes codiert", 35, 82);
  } else if (mode == 1) {
    fill(52, 211, 153);
    text("CDT PHASE C: Kausale Zeitschichtung t=const erzeugt spontan ein makroskopisches de Sitter Universum S4", 35, 82);
  } else if (mode == 2) {
    fill(244, 114, 182);
    text("SPEKTRAL-DIMENSION: Die effektive Dimension d_s der Raumzeit bricht auf Planck-Skalen von 4 auf 2 ein", 35, 82);
  } else {
    fill(34, 211, 238);
    text("DTIM-ZEC POSTULAT: CDT-Simplizes entsprechen den elastischen Torsions-Zellen des ZEC-Raumhexagongitters", 35, 82);
  }
}

Die Position der Insel-Kante $\partial I$ wird durch das Extremalisierungs-Kriterium $\frac{\partial S_{\text{gen}}}{\partial \partial I} = 0$ bestimmt. Vor der Page-Zeit ist $I = \emptyset$; nach der Page-Zeit liegt $\partial I$ knapp hinter dem Ereignishorizont.

Das Minimum beider Lösungen erzeugt das charakteristische Umknicken der Entropiekurve zur Page-Zeit $t_{\text{Page}}$, was die universelle Unitarität der Hawking-Evaporation garantiert.

/**
 * DTIM-ZEC MODULAR PHYSICS FRAMEWORK (DAMTP ACADEMIC STANDARD)
 * DTIM_QuantumGravity_007_BlackHole_Hawking_Island.pde - Black Hole Thermodynamics & Island Formula Engine
 * Target: Processing 4.5.6 / Quantum Field Compendium Band 4 (Objekt 007)
 * License: CC0 1.0 Universal / MIT
 * Copyright (c) 2026 Jan Bühring & Adaptive Intelligence Synthesis Network (Google AI)
 */

int mode = 0; // 0: Bekenstein-Hawking Area Entropy & Event Horizon Thermodynamics, 1: Hawking Quantum Pair Emission at Event Horizon, 2: Page Curve & Quantum Island Formula S(Rad) Evolution, 3: DTIM-ZEC Raumhexagon Black Hole Torsion Core
float wavePhase = 0.0f;
String[] modeNames = {
  "1. Bekenstein-Hawking Entropie // Horizont-Flächen-Quantisierung SBH = A / (4 l_P^2) & Hawking Temp TH", 
  "2. Hawking Quantum Pair Emission // Vakuumfluktuationen am Ereignishorizont erzeugen thermische Strahlung", 
  "3. Page-Kurve & Quantum Island // Island Formula S(Rad) stellt Unitaritaet beim Verdampfen wieder her", 
  "4. DTIM-ZEC Raumhexagon Torsions-Kern // Verluste freie Informations-Konservierung im ZEC-Gitter"
};

void setup() {
  size(800, 380);
  println("[DTIM-QUANTUM-PHYSICS] Initializing Volume 4 Object 007 (Black Hole & Island Formula Engine)...");
}

void draw() {
  background(11, 15, 25);
  wavePhase += 0.04f;
  
  // Auto switch mode every 4 seconds
  if (frameCount % 240 == 0) {
    mode = (mode + 1) % 4;
  }
  
  translate(width/2, height/2 + 10);
  
  if (mode == 0) {
    drawBekensteinHawkingHorizon();
  } else if (mode == 1) {
    drawHawkingPairEmission();
  } else if (mode == 2) {
    drawPageCurveQuantumIsland();
  } else {
    drawDTIMHexagonBlackHoleCore();
  }
  
  resetMatrix();
  drawCosmologyOverlay();
}

void drawBekensteinHawkingHorizon() {
  // Bekenstein-Hawking Area Entropy & Event Horizon
  
  // Black Hole Interior (Dark Disc)
  float horizonR = 75.0f;
  fill(2, 6, 23); stroke(248, 113, 113); strokeWeight(3.0f);
  ellipse(0, 0, horizonR * 2.0f, horizonR * 2.0f);
  
  // Quantized Area Pixels on Horizon (Glow Nodes)
  int pixels = 24;
  for (int i = 0; i < pixels; i++) {
    float angle = i * (TWO_PI / pixels) + wavePhase * 0.3f;
    float px = horizonR * cos(angle);
    float py = horizonR * sin(angle);
    
    fill(250, 204, 21); noStroke();
    ellipse(px, py, 5, 5);
  }
  
  // Photon Sphere / Accretion Ring
  stroke(248, 113, 113, 120); strokeWeight(1.5f); noFill();
  ellipse(0, 0, horizonR * 3.0f, horizonR * 3.0f);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Bekenstein-Hawking Entropie: SBH = k_B A / (4 l_P^2)", 0, -115);
  text("Horizont speichert 1 Bit Quanteninformation pro 4 Planck-Flächen-Einheiten", 0, 115);
}

void drawHawkingPairEmission() {
  // Hawking Quantum Pair Emission at Event Horizon
  
  // Event Horizon Arc (Left: Inside, Right: Outside)
  stroke(248, 113, 113); strokeWeight(3.5f); noFill();
  arc(-60, 0, 200, 240, -HALF_PI, HALF_PI);
  
  // Escaping Hawking Photons (Yellow Outward Waves)
  for (int i = 0; i < 4; i++) {
    float pX = 40.0f + i * 35.0f + sin(wavePhase * 3.0f + i) * 8.0f;
    float pY = -40.0f + i * 25.0f;
    
    fill(250, 204, 21); stroke(255); strokeWeight(1.5f);
    ellipse(pX, pY, 8, 8);
    
    stroke(250, 204, 21, 160); strokeWeight(1.5f); noFill();
    ellipse(pX, pY, 16, 16);
  }
  
  // Falling Negative Energy Partners (Cyan Inward Dots)
  for (int i = 0; i < 4; i++) {
    float pX = -30.0f - i * 25.0f;
    float pY = -40.0f + i * 25.0f;
    
    fill(34, 211, 238); noStroke();
    ellipse(pX, pY, 6, 6);
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Hawking Strahlung: Thermische Paardurchtunnelung am Ereignishorizont", 0, -115);
  text("Hawking Temperatur TH = hbar c^3 / (8pi G M k_B) verursaecht kontinuierliche Evaporation", 0, 115);
}

void drawPageCurveQuantumIsland() {
  // Page Curve vs Hawking Curve & Quantum Island Formula
  
  // Graph Axes
  stroke(71, 85, 105); strokeWeight(1.5f);
  line(-180, 60, 180, 60); // Time t axis
  line(-180, 60, -180, -90); // Entropy S axis
  
  fill(148, 163, 184); textSize(10); textAlign(CENTER, CENTER);
  text("Zeit t (Evaporation) ->", 100, 75);
  text("S(Rad)", -170, -100);
  
  // Hawking Non-Unitary Monotonic Curve (Dashed Cyan Line)
  stroke(34, 211, 238, 160); strokeWeight(2.0f); noFill();
  beginShape();
  for (int x = -180; x <= 180; x += 10) {
    float normX = (x + 180) / 360.0f;
    float hS = normX * 130.0f;
    vertex(x, 60 - hS);
  }
  endShape();
  
  // Page Unitary Curve (Green Solid Line with Turning Point)
  stroke(52, 211, 153); strokeWeight(3.0f);
  beginShape();
  for (int x = -180; x <= 180; x += 5) {
    float normX = (x + 180) / 360.0f;
    float pageS = 0;
    if (normX <= 0.5f) {
      pageS = normX * 2.0f * 100.0f; // Rising before Page Time
    } else {
      pageS = (1.0f - normX) * 2.0f * 100.0f; // Falling after Page Time
    }
    vertex(x, 60 - pageS);
  }
  endShape();
  
  // Page Time Line (Gold Vertical Line)
  stroke(250, 204, 21); strokeWeight(1.8f);
  line(0, -90, 0, 60);
  
  fill(250, 204, 21); textSize(10); textAlign(CENTER, CENTER);
  text("Page-Zeit t_Page", 0, -100);
  
  fill(34, 211, 238); textSize(10); textAlign(LEFT, CENTER);
  text("Hawking (Semiklassisch)", 50, -60);
  fill(52, 211, 153);
  text("Page-Kurve (Unitär)", 50, -10);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Quantum Island Formula: S(Rad) = min ext [ Area(d I) / (4 G_N) + S_bulk(Rad U I) ]", 0, 115);
}

void drawDTIMHexagonBlackHoleCore() {
  // DTIM-ZEC Raumhexagon Black Hole Torsion Core
  strokeWeight(1.8f); noFill();
  
  float hexRadius = 34.0f;
  for (int q = -3; q <= 3; q++) {
    for (int r = -2; r <= 2; r++) {
      float x = hexRadius * (sqrt(3) * q + sqrt(3)/2.0f * r);
      float y = hexRadius * (3.0f/2.0f * r);
      
      float distFromCenter = sqrt(x*x + y*y);
      float torsionP = max(0, 1.0f - distFromCenter / 150.0f) * sin(wavePhase * 4.0f) * 10.0f;
      
      pushMatrix();
      translate(x, y);
      stroke(248, 113, 113, 220);
      polygon(0, 0, hexRadius * 0.85f + torsionP * 0.3f, 6);
      
      fill(52, 211, 153, 240); noStroke();
      ellipse(0, 0, 5, 5);
      popMatrix();
    }
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("DTIM-ZEC Torsions-Kern: Verlustfreie Informations-Kopplung im ZEC-Gitter", 0, 115);
}

void polygon(float x, float y, float radius, int npoints) {
  float angle = TWO_PI / npoints;
  beginShape();
  for (float a = 0; a < TWO_PI; a += angle) {
    float sx = x + cos(a) * radius;
    float sy = y + sin(a) * radius;
    vertex(sx, sy);
  }
  endShape(CLOSE);
}

void drawCosmologyOverlay() {
  fill(15, 23, 42, 220);
  stroke(51, 65, 85);
  rect(20, 20, width - 40, 85, 6);
  
  fill(248, 113, 113);
  textSize(13);
  textAlign(LEFT, BASELINE);
  text("DTIM-ZEC QUANTENGRAVITATION // BAND 4 OBJEKT 007: SCHWARZE-LOCH-THERMODYNAMIK & ISLANDS", 35, 42);
  
  fill(226, 232, 240);
  textSize(11);
  text("KLASSIFIKATION: " + modeNames[mode], 35, 62);
  
  if (mode == 0) {
    fill(248, 113, 113);
    text("BEKENSTEIN-HAWKING: Horizontentropie SBH = A / (4 l_P^2) speichert Quanteninformation auf der Grenzflaeche", 35, 82);
  } else if (mode == 1) {
    fill(250, 204, 21);
    text("HAWKING-STRAHLUNG: Thermische Emittierung verringert Masse M und fuehrt zur vollständigen Evaporation", 35, 82);
  } else if (mode == 2) {
    fill(52, 211, 153);
    text("ISLAND FORMULA: Die Entstehung der Quantum Island I stellt die Unitarität der Page-Kurve sicher", 35, 82);
  } else {
    fill(34, 211, 238);
    text("DTIM-ZEC POSTULAT: Die Information wird über kohärente Torsionswellen der ZEC-Superzellen extrahiert", 35, 82);
  }
}

Das effektive Potenzial $V_{\text{eff}}(a, \phi)$ erzeugt eine Potenzialbarriere zwischen $a=0$ und der klassischen Wendemarke $a_0 = 1/H(\phi)$. $\mathbf{T}_{\text{ZEC}}^{(\text{cosmo})}$ stabilisiert die Phase am ZEC-Gitter.

Die Phase $S(a, \phi)$ erfüllt die klassische Hamilton-Jacobi-Gleichung der Gravitation und definiert die physikalische Zeit $t$ entlang der WKB-Trajektorien des Universums.

/**
 * DTIM-ZEC MODULAR PHYSICS FRAMEWORK (DAMTP ACADEMIC STANDARD)
 * DTIM_QuantumGravity_008_QuantumCosmology_WDW.pde - Quantum Cosmology & No-Boundary Engine
 * Target: Processing 4.5.6 / Quantum Field Compendium Band 4 (Objekt 008)
 * License: CC0 1.0 Universal / MIT
 * Copyright (c) 2026 Jan Bühring & Adaptive Intelligence Synthesis Network (Google AI)
 */

int mode = 0; // 0: Wheeler-DeWitt Wave Function Psi(a) in Minisuperspace, 1: Hartle-Hawking No-Boundary Proposal Euclidean Instanton Cap S4, 2: Vilenkin Tunneling Wave Function from Nothing (a=0 -> a0), 3: DTIM-ZEC Raumhexagon Primordial Lattice Core
float wavePhase = 0.0f;
String[] modeNames = {
  "1. Wheeler-DeWitt Minisuperspace // Zeitlose Wellenfunktion H Psi(a, phi) = 0 & WKB-Zeit-Emergenz", 
  "2. Hartle-Hawking No-Boundary // Euklidische 4-Sphaere S4 glaettet die Urknallsingularitaet ohne Rand", 
  "3. Vilenkin Tunneling // Outgoing-Boundary Tunnel-Amplitude beschreibt Quanten-Genese aus dem Nichts", 
  "4. DTIM-ZEC Raumhexagon Urphasen-Core // Resonanter Grundzustand der ZEC-Superzellgittermatrix"
};

void setup() {
  size(800, 380);
  println("[DTIM-QUANTUM-PHYSICS] Initializing Volume 4 Object 008 (Quantum Cosmology & WDW Engine)...");
}

void draw() {
  background(11, 15, 25);
  wavePhase += 0.04f;
  
  // Auto switch mode every 4 seconds
  if (frameCount % 240 == 0) {
    mode = (mode + 1) % 4;
  }
  
  translate(width/2, height/2 + 10);
  
  if (mode == 0) {
    drawWheelerDeWittMinisuperspace();
  } else if (mode == 1) {
    drawHartleHawkingNoBoundaryInstanton();
  } else if (mode == 2) {
    drawVilenkinTunnelingFromNothing();
  } else {
    drawDTIMHexagonPrimordialLatticeCore();
  }
  
  resetMatrix();
  drawCosmologyOverlay();
}

void drawWheelerDeWittMinisuperspace() {
  // Wheeler-DeWitt Wave Function Psi(a) in Minisuperspace
  
  // Potential Barrier V_eff(a) (Purple Curve)
  stroke(167, 139, 250, 180); strokeWeight(2.0f); noFill();
  beginShape();
  for (int x = -180; x <= 180; x += 5) {
    float aVal = (x + 180) / 360.0f * 3.0f; // 0 to 3
    float vVal = aVal * aVal - 0.35f * pow(aVal, 4.0f); // Potential hill
    vertex(x, 20 - vVal * 35.0f);
  }
  endShape();
  
  // Psi(a) Oscillating Wavefunction (Cyan Line)
  stroke(34, 211, 238); strokeWeight(3.0f);
  beginShape();
  for (int x = -180; x <= 180; x += 4) {
    float normX = (x + 180) / 360.0f;
    float amp = 0;
    float osc = 0;
    if (normX < 0.45f) {
      // Exponential Decay in Tunneling Barrier
      amp = exp((normX - 0.45f) * 6.0f) * 45.0f;
      osc = sin(wavePhase * 1.5f);
    } else {
      // Oscillatory Regime (WKB Expanding Universe)
      amp = 45.0f;
      osc = cos(wavePhase * 3.0f + (normX - 0.45f) * 20.0f);
    }
    vertex(x, -20 - amp * osc);
  }
  endShape();
  
  // Classical Turning Point Line a0 (Gold Vertical Line)
  float turningX = -180 + 0.45f * 360.0f;
  stroke(250, 204, 21); strokeWeight(1.8f);
  line(turningX, -80, turningX, 60);
  
  fill(250, 204, 21); textSize(10); textAlign(CENTER, CENTER);
  text("Klassischer Wendepunkt a0", turningX, -90);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Wheeler-DeWitt Gleichung: H Psi(a, phi) = 0 (Zeitloses Quanten-Universum)", 0, -115);
  text("Zeit t emergiert erst semiklassisch aus der WKB-Phase S(a, phi) der Wellenfunktion", 0, 115);
}

void drawHartleHawkingNoBoundaryInstanton() {
  // Hartle-Hawking No-Boundary Euclidean Instanton Cap (S4)
  
  // Euclidean Smooth Hemisphere Cap (a -> 0 without Boundary - Glowing Gold Cap)
  stroke(250, 204, 21); strokeWeight(2.5f); fill(250, 204, 21, 40);
  arc(-60, 0, 100, 100, -HALF_PI, HALF_PI);
  
  // Lorentzian Expanding Universe (Cyan Hyperbola attached at a0)
  stroke(34, 211, 238); strokeWeight(3.0f); noFill();
  beginShape();
  for (int x = -10; x <= 180; x += 5) {
    float tVal = (x + 10) * 0.015f;
    float aVal = 50.0f * cosh(tVal);
    vertex(x, -aVal);
  }
  endShape();
  
  beginShape();
  for (int x = -10; x <= 180; x += 5) {
    float tVal = (x + 10) * 0.015f;
    float aVal = 50.0f * cosh(tVal);
    vertex(x, aVal);
  }
  endShape();
  
  fill(250, 204, 21); textSize(11); textAlign(CENTER, CENTER);
  text("Euklidische 4-Sphaere (S4)", -60, 0);
  fill(34, 211, 238);
  text("Lorentzsches de Sitter Universum", 80, 0);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Hartle-Hawking No-Boundary Proposal: Psi_HH[h] = integral Dg exp(-S_E[g]/hbar)", 0, -115);
  text("Die Raumzeit hat keinen zeitlichen Rand – der Urknall ist glatt wie der Südpol der Erde", 0, 115);
}

float cosh(float val) {
  return (exp(val) + exp(-val)) * 0.5f;
}

void drawVilenkinTunnelingFromNothing() {
  // Vilenkin Tunneling Wave Function from Nothing (a=0 -> a0)
  
  // Tunneling Probability Barrier (Pink Shaded Zone)
  fill(244, 114, 182, 30); stroke(244, 114, 182, 120); strokeWeight(1.5f);
  rect(-160, -60, 120, 120, 4);
  
  // Outgoing Wave Rays (Cyan Arrows escaping Barrier)
  stroke(52, 211, 153); strokeWeight(2.5f);
  line(-40, 0, 160, 0);
  line(145, -8, 160, 0); line(145, 8, 160, 0);
  
  // Tunneling Amplitude Dot at a = 0
  fill(250, 204, 21); stroke(255); strokeWeight(2.0f);
  ellipse(-160, 0, 12, 12);
  
  fill(250, 204, 21); textSize(10); textAlign(CENTER, CENTER);
  text("Nichts (a = 0)", -160, 20);
  fill(244, 114, 182);
  text("Tunnelbarriere", -100, -70);
  fill(52, 211, 153);
  text("Expansion ins de Sitter Regime", 60, -20);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Vilenkin Tunneling Amplitude: Quanten-Genesis des Universums aus dem Nichts", 0, -115);
  text("Outgoing-Boundary-Condition beschreibt das spontane Tunneln von a=0 nach a0", 0, 115);
}

void drawDTIMHexagonPrimordialLatticeCore() {
  // DTIM-ZEC Raumhexagon Primordial Lattice Core
  strokeWeight(1.8f); noFill();
  
  float hexRadius = 34.0f;
  for (int q = -3; q <= 3; q++) {
    for (int r = -2; r <= 2; r++) {
      float x = hexRadius * (sqrt(3) * q + sqrt(3)/2.0f * r);
      float y = hexRadius * (3.0f/2.0f * r);
      
      float phaseShift = sin(wavePhase * 3.0f + (q + r) * 0.7f) * 8.0f;
      
      pushMatrix();
      translate(x, y);
      stroke(192, 132, 252, 220);
      polygon(0, 0, hexRadius * 0.85f + phaseShift * 0.2f, 6);
      
      fill(52, 211, 153, 240); noStroke();
      ellipse(0, 0, 5, 5);
      popMatrix();
    }
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("DTIM-ZEC Raumhexagon Urphasen-Core: ZEC-Superzell-Grundzustand der Quantengravitation", 0, 115);
}

void polygon(float x, float y, float radius, int npoints) {
  float angle = TWO_PI / npoints;
  beginShape();
  for (float a = 0; a < TWO_PI; a += angle) {
    float sx = x + cos(a) * radius;
    float sy = y + sin(a) * radius;
    vertex(sx, sy);
  }
  endShape(CLOSE);
}

void drawCosmologyOverlay() {
  fill(15, 23, 42, 220);
  stroke(51, 65, 85);
  rect(20, 20, width - 40, 85, 6);
  
  fill(192, 132, 252);
  textSize(13);
  textAlign(LEFT, BASELINE);
  text("DTIM-ZEC QUANTENGRAVITATION // BAND 4 OBJEKT 008: QUANTEN-KOSMOLOGIE & WDW", 35, 42);
  
  fill(226, 232, 240);
  textSize(11);
  text("KLASSIFIKATION: " + modeNames[mode], 35, 62);
  
  if (mode == 0) {
    fill(192, 132, 252);
    text("WHEELER-DEWITT: Die zeitlose Wellenfunktion H Psi = 0 beschreibt das Quanten-Universum im Minisuperspace", 35, 82);
  } else if (mode == 1) {
    fill(250, 204, 21);
    text("NO-BOUNDARY: Euklidische Instanton-Glättung S4 eliminiert die Urknallsingularität vollkommen", 35, 82);
  } else if (mode == 2) {
    fill(52, 211, 153);
    text("VILENKIN TUNNELING: Die Quanten-Genesis beschreibt das Tunneln aus dem Nichts ins inflationäre Regime", 35, 82);
  } else {
    fill(34, 211, 238);
    text("DTIM-ZEC POSTULAT: Die Zeit entsteht erst als WKB-Phase aus der Scherspannung des ZEC-Hexagongitters", 35, 82);
  }
}

Der Zweipunkt-Korrelator der Raumzeit-Metrik diversifiziert bei $x \to y$ nicht unendlich, sondern wird durch den ZEC-Gitter-Dämpfungstensor $\mathbf{T}_{\text{ZEC}}^{(\text{foam})}$ auf die effektive Planck-Zell-Fläche abgeschnitten.

Die Summe über alle topologischen Mannigfaltigkeiten $M$ wird nach dem Geschlecht $g$ (Anzahl der Mikrowurmlöcher) geordnet. Höhere Geschlechter werden durch die euklidische Aktion unterdrückt.

/**
 * DTIM-ZEC MODULAR PHYSICS FRAMEWORK (DAMTP ACADEMIC STANDARD)
 * DTIM_QuantumGravity_009_QuantumFoam_Wheeler.pde - Quantum Foam & Wheeler Spintronics Engine
 * Target: Processing 4.5.6 / Quantum Field Compendium Band 4 (Objekt 009)
 * License: CC0 1.0 Universal / MIT
 * Copyright (c) 2026 Jan Bühring & Adaptive Intelligence Synthesis Network (Google AI)
 */

int mode = 0; // 0: Wheeler Quantum Foam & Metric Topology Fluctuations Delta g ~ 1, 1: Virtual Micro-Wormholes & Sprouting Topology Genus g, 2: Wheeler-Spintronik Spin Network Dynamics & Planck Spinor Nodes, 3: DTIM-ZEC Raumhexagon Topologieschaum Core
float wavePhase = 0.0f;
String[] modeNames = {
  "1. Wheeler-Quantenschaum // Metrische Unschaerfe Delta g ~ 1 laest glatte Geometrie auf Planck-Skala zerfallen", 
  "2. Virtuelle Mikrowurmlocher // Sprossende Topologie-Fluktuationen des Geschlechts g & Euler-Charakteristik Delta chi", 
  "3. Wheeler-Spintronik // Spin-Netzwerk Dynamics & Quantisierung von Flaeche & Volumen ueber Spinor-Knoten", 
  "4. DTIM-ZEC Topologieschaum Core // Elasto-plastische Torsions-Eigenmoden des ZEC-Raumhexagongitters"
};

void setup() {
  size(800, 380);
  println("[DTIM-QUANTUM-PHYSICS] Initializing Volume 4 Object 009 (Quantum Foam & Wheeler Engine)...");
}

void draw() {
  background(11, 15, 25);
  wavePhase += 0.05f;
  
  // Auto switch mode every 4 seconds
  if (frameCount % 240 == 0) {
    mode = (mode + 1) % 4;
  }
  
  translate(width/2, height/2 + 10);
  
  if (mode == 0) {
    drawWheelerQuantumFoam();
  } else if (mode == 1) {
    drawVirtualMicroWormholes();
  } else if (mode == 2) {
    drawWheelerSpintronicsSpinNetworks();
  } else {
    drawDTIMHexagonTopologyFoamCore();
  }
  
  resetMatrix();
  drawCosmologyOverlay();
}

void drawWheelerQuantumFoam() {
  // Wheeler Quantum Foam & Metric Fluctuations Delta g ~ 1
  
  // Turbulent Foam Lattice Grid (Pink/Magenta Fluctuation Field)
  int cols = 12;
  int rows = 6;
  float spacingX = 35.0f;
  float spacingY = 30.0f;
  
  for (int i = -cols/2; i <= cols/2; i++) {
    for (int j = -rows/2; j <= rows/2; j++) {
      float cx = i * spacingX;
      float cy = j * spacingY;
      
      // High-Frequency Stochastic Noise
      float nVal = noise(i * 0.4f + wavePhase * 0.8f, j * 0.4f + wavePhase * 0.8f);
      float rSize = 10.0f + nVal * 30.0f;
      
      stroke(244, 114, 182, 180); strokeWeight(1.5f); fill(244, 114, 182, 40);
      ellipse(cx, cy, rSize, rSize);
      
      fill(250, 204, 21); noStroke();
      ellipse(cx + sin(wavePhase * 2.0f + i) * 5.0f, cy + cos(wavePhase * 2.0f + j) * 5.0f, 3, 3);
    }
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Wheeler-Quantenschaum: Metrische Unschärfe Delta g_mu_nu ~ l_P / L", 0, -115);
  text("Auf der Planck-Skala L ~ l_P bricht die glatte Geometrie in stochastischen Schaum um", 0, 115);
}

void drawVirtualMicroWormholes() {
  // Virtual Micro-Wormholes & Sprouting Topology Genus g
  
  // Upper & Lower Space Sheets (Cyan Parallel Lines)
  stroke(34, 211, 238, 200); strokeWeight(2.5f);
  line(-180, -50, 180, -50);
  line(-180, 50, 180, 50);
  
  // Sprouting Wormhole Necks (Throats connecting sheets)
  stroke(244, 114, 182); strokeWeight(2.5f); noFill();
  
  for (int i = -2; i <= 2; i++) {
    float throatX = i * 70.0f + sin(wavePhase * 2.0f + i) * 10.0f;
    float throatW = 15.0f + cos(wavePhase * 3.0f + i) * 8.0f;
    
    // Curved Wormhole Throat
    bezier(throatX - throatW, -50, throatX - throatW * 0.3f, -10, throatX - throatW * 0.3f, 10, throatX - throatW, 50);
    bezier(throatX + throatW, -50, throatX + throatW * 0.3f, -10, throatX + throatW * 0.3f, 10, throatX + throatW, 50);
    
    // Virtual Energy Flux Dots
    fill(250, 204, 21); noStroke();
    ellipse(throatX, 0, 6, 6);
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Virtuelle Mikrowurmlöcher: Fluktuationen des Geschlechts g & Euler-Charakteristik Delta chi", 0, -115);
  text("Spontane Wurmloch-Sprossung erzeugt temporäre Nicht-Lokalitäten im Sub-Planckianischen Vakuum", 0, 115);
}

void drawWheelerSpintronicsSpinNetworks() {
  // Wheeler-Spintronik Spin Network Dynamics & Planck Spinor Nodes
  
  // Spin Network Nodes & Intertwining Valence Links
  int nodeCount = 8;
  float radiusR = 85.0f;
  float[] px = new float[nodeCount];
  float[] py = new float[nodeCount];
  
  for (int i = 0; i < nodeCount; i++) {
    float angle = i * (TWO_PI / nodeCount) + wavePhase * 0.2f;
    px[i] = radiusR * cos(angle) + sin(wavePhase * 2.0f + i) * 8.0f;
    py[i] = radiusR * sin(angle) + cos(wavePhase * 2.0f + i) * 8.0f;
  }
  
  // Draw Intertwiner Links (Gold Lines with Spin Labels j=1/2)
  stroke(250, 204, 21, 180); strokeWeight(1.8f);
  for (int i = 0; i < nodeCount; i++) {
    for (int j = i + 1; j < nodeCount; j++) {
      if ((i + j) % 2 == 1) {
        line(px[i], py[i], px[j], py[j]);
      }
    }
  }
  
  // Draw Spinor Nodes (Glowing Purple Circles)
  for (int i = 0; i < nodeCount; i++) {
    fill(167, 139, 250); stroke(255); strokeWeight(2.0f);
    ellipse(px[i], py[i], 12, 12);
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Wheeler-Spintronik: Spin-Netzwerk der Ashtekar SU(2) Eich-Variablen", 0, -115);
  text("Quantisierung von Fläche & Volumen durch diskrete Spinor-Knoten-Valenzen", 0, 115);
}

void drawDTIMHexagonTopologyFoamCore() {
  // DTIM-ZEC Raumhexagon Topologieschaum Core
  strokeWeight(1.8f); noFill();
  
  float hexRadius = 34.0f;
  for (int q = -3; q <= 3; q++) {
    for (int r = -2; r <= 2; r++) {
      float x = hexRadius * (sqrt(3) * q + sqrt(3)/2.0f * r);
      float y = hexRadius * (3.0f/2.0f * r);
      
      float turbulence = noise(q * 0.5f + wavePhase, r * 0.5f + wavePhase) * 12.0f;
      
      pushMatrix();
      translate(x, y);
      stroke(244, 114, 182, 220);
      polygon(0, 0, hexRadius * 0.85f + turbulence * 0.3f, 6);
      
      fill(52, 211, 153, 240); noStroke();
      ellipse(0, 0, 5, 5);
      popMatrix();
    }
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("DTIM-ZEC Topologieschaum: High-Frequency Elasto-Plastische Eigenmoden des ZEC-Gitters", 0, 115);
}

void polygon(float x, float y, float radius, int npoints) {
  float angle = TWO_PI / npoints;
  beginShape();
  for (float a = 0; a < TWO_PI; a += angle) {
    float sx = x + cos(a) * radius;
    float sy = y + sin(a) * radius;
    vertex(sx, sy);
  }
  endShape(CLOSE);
}

void drawCosmologyOverlay() {
  fill(15, 23, 42, 220);
  stroke(51, 65, 85);
  rect(20, 20, width - 40, 85, 6);
  
  fill(244, 114, 182);
  textSize(13);
  textAlign(LEFT, BASELINE);
  text("DTIM-ZEC QUANTENGRAVITATION // BAND 4 OBJEKT 009: QUANTENSCHAUM & SPINTRONIK", 35, 42);
  
  fill(226, 232, 240);
  textSize(11);
  text("KLASSIFIKATION: " + modeNames[mode], 35, 62);
  
  if (mode == 0) {
    fill(244, 114, 182);
    text("QUANTENSCHAUM: Metrische Unschärfe Delta g ~ 1 zerstört die glatte Raumzeit-Geometrie auf Planck-Skalen", 35, 82);
  } else if (mode == 1) {
    fill(34, 211, 238);
    text("MIKROWURMLÖCHER: Topologie-Fluktuationsprozesse des Geschlechts g erzeugen Sub-Planckianische Brücken", 35, 82);
  } else if (mode == 2) {
    fill(250, 204, 21);
    text("SPINTRONIK: Spin-Netzwerke der Ashtekar-Variablen quantisieren Raumzeit-Volumina diskret", 35, 82);
  } else {
    fill(52, 211, 153);
    text("DTIM-ZEC POSTULAT: Der Schaum ist die dynamische Torsions-Eigenmode der Raumhexagon-Superzellen", 35, 82);
  }
}

Jeder Schnittpunkt der Spin-Netzwerk-Kante mit Spin $j_i$ trägt ein Quant an Fläche bei. $\mathbf{T}_{\text{ZEC}}^{(\text{LQG})}$ reguliert die elastische Phaseneichung an den Hexagon-Zellrändern.

Die EPRL-Vertexamplitude $A_v$ projiziert die $SU(2)$-Intertwiner unter Berücksichtigung des Immirzi-Parameters $\gamma$ auf Lorentzsche $SL(2,\mathbb{C})$-Repräsentationen.

/**
 * DTIM-ZEC MODULAR PHYSICS FRAMEWORK (DAMTP ACADEMIC STANDARD)
 * DTIM_QuantumGravity_010_SpinFoam_LQG_Immirzi.pde - Loop Quantum Gravity & Spin Foam Engine
 * Target: Processing 4.5.6 / Quantum Field Compendium Band 4 (Objekt 010)
 * License: CC0 1.0 Universal / MIT
 * Copyright (c) 2026 Jan Bühring & Adaptive Intelligence Synthesis Network (Google AI)
 */

int mode = 0; // 0: LQG Quantized Area Spectrum & Ashtekar-Barbero Immirzi Parameter gamma, 1: Kinematical Spin Network States & SU(2) Intertwiners, 2: Covariant 4D Spin Foam Dynamics & EPRL Vertex Amplitudes, 3: DTIM-ZEC Raumhexagon Spin Foam Lattice Core
float wavePhase = 0.0f;
String[] modeNames = {
  "1. LQG Quantisiertes Flächenspektrum // A = 8pi gamma l_P^2 sum sqrt(j(j+1)) & Immirzi-Parameter gamma", 
  "2. Kinematisches Spin-Netzwerk // SU(2) Spin-Kanten (j=1/2,1,3/2) & Intertwiner Knoten-Volumina", 
  "3. Kovariante Spin-Foam Dynamik // 4D EPRL Vertex-Amplituden Av & Raumzeit-Pfadintegral Summe", 
  "4. DTIM-ZEC Spin-Hexagon Core // Resonante Torsions-Eichung des ZEC-Superzell-Gitters"
};

void setup() {
  size(800, 380);
  println("[DTIM-QUANTUM-PHYSICS] Initializing Volume 4 Object 010 (LQG & Spin Foam Engine)...");
}

void draw() {
  background(11, 15, 25);
  wavePhase += 0.04f;
  
  // Auto switch mode every 4 seconds
  if (frameCount % 240 == 0) {
    mode = (mode + 1) % 4;
  }
  
  translate(width/2, height/2 + 10);
  
  if (mode == 0) {
    drawLQGQuantizedAreaSpectrum();
  } else if (mode == 1) {
    drawKinematicalSpinNetwork();
  } else if (mode == 2) {
    drawCovariantSpinFoamEPRLVertex();
  } else {
    drawDTIMHexagonSpinFoamCore();
  }
  
  resetMatrix();
  drawCosmologyOverlay();
}

void drawLQGQuantizedAreaSpectrum() {
  // LQG Quantized Area Spectrum & Ashtekar-Barbero Immirzi Parameter gamma
  
  // Quantized Area Level Bars (Indigo/Cyan Spectrum)
  stroke(99, 102, 241); strokeWeight(2.0f);
  
  int levels = 6;
  for (int i = 0; i < levels; i++) {
    float y = 50.0f - i * 22.0f;
    float spinJ = (i + 1) * 0.5f;
    float areaVal = sqrt(spinJ * (spinJ + 1.0f));
    float barW = areaVal * 50.0f;
    
    // Level line
    stroke(129, 140, 248);
    line(-barW, y, barW, y);
    
    // Level nodes
    fill(250, 204, 21); noStroke();
    ellipse(-barW, y, 6, 6);
    ellipse(barW, y, 6, 6);
    
    fill(226, 232, 240); textSize(9); textAlign(LEFT, CENTER);
    text("j = " + nf(spinJ, 1, 1) + " -> A = " + nf(areaVal, 1, 2) + " (8pi gamma l_P^2)", barW + 12, y);
  }
  
  // Immirzi Parameter Marker (Gold Indicator)
  stroke(250, 204, 21); strokeWeight(2.0f);
  line(-180, 80, 180, 80);
  
  fill(250, 204, 21); textSize(11); textAlign(CENTER, CENTER);
  text("Ashtekar-Barbero Immirzi Parameter: gamma approx 0,23753", 0, 95);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("LQG Flächespektrum: A(S) = 8pi gamma l_P^2 sum sqrt(j_i(j_i+1))", 0, -115);
}

void drawKinematicalSpinNetwork() {
  // Kinematical Spin Network States & SU(2) Intertwiners
  
  // Graph Network
  int nodes = 6;
  float radius = 90.0f;
  float[] nx = new float[nodes];
  float[] ny = new float[nodes];
  
  for (int i = 0; i < nodes; i++) {
    float angle = i * (TWO_PI / nodes) + wavePhase * 0.2f;
    nx[i] = radius * cos(angle);
    ny[i] = radius * sin(angle);
  }
  
  // Draw Edges (Cyan/Indigo Lines with Spin labels)
  stroke(34, 211, 238, 200); strokeWeight(2.0f);
  for (int i = 0; i < nodes; i++) {
    int next = (i + 1) % nodes;
    line(nx[i], ny[i], nx[next], ny[next]);
    
    // Cross edges
    if (i < 3) {
      stroke(167, 139, 250, 160);
      line(nx[i], ny[i], nx[i+3], ny[i+3]);
    }
  }
  
  // Draw Intertwiner Nodes (Glowing Pink Spheres = Volume Quanta)
  for (int i = 0; i < nodes; i++) {
    fill(244, 114, 182); stroke(255); strokeWeight(2.0f);
    float pulse = sin(wavePhase * 3.0f + i) * 3.0f;
    ellipse(nx[i], ny[i], 14 + pulse, 14 + pulse);
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Kinematisches Spin-Netzwerk: Kanten tragen Spins j_e, Knoten tragen Intertwiner-Volumina", 0, -115);
  text("Hintergrundunabhängigkeit: Die Geometrie entsteht rein aus den Quantenzuständen des Graphen", 0, 115);
}

void drawCovariantSpinFoamEPRLVertex() {
  // Covariant 4D Spin Foam Dynamics & EPRL Vertex Amplitudes
  
  // 4D Spinfoam Vertex (Center EPRL Node)
  float vX = 0;
  float vY = 0;
  
  // Spacetime Strands / Faces radiating from Vertex
  stroke(129, 140, 248, 180); strokeWeight(2.2f);
  for (int i = 0; i < 5; i++) {
    float angle = i * (TWO_PI / 5.0f) + wavePhase * 0.3f;
    float ex = 110.0f * cos(angle);
    float ey = 110.0f * sin(angle);
    
    line(vX, vY, ex, ey);
    
    // Face Loops
    stroke(244, 114, 182, 140); noFill();
    ellipse(ex * 0.6f, ey * 0.6f, 25, 25);
  }
  
  // Central EPRL Vertex Core (Gold Glowing Node)
  fill(250, 204, 21); stroke(255); strokeWeight(2.5f);
  ellipse(vX, vY, 20, 20);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("EPRL Spin-Foam Vertex: Kovariante Transition-Amplitude Av(jf, ie)", 0, -115);
  text("Spinfoams beschreiben die 4D-Pfadintegral-Evolution von Spin-Netzwerken in der Zeit", 0, 115);
}

void drawDTIMHexagonSpinFoamCore() {
  // DTIM-ZEC Raumhexagon Spin Foam Lattice Core
  strokeWeight(1.8f); noFill();
  
  float hexRadius = 34.0f;
  for (int q = -3; q <= 3; q++) {
    for (int r = -2; r <= 2; r++) {
      float x = hexRadius * (sqrt(3) * q + sqrt(3)/2.0f * r);
      float y = hexRadius * (3.0f/2.0f * r);
      
      float spinPulse = sin(wavePhase * 3.5f + (q + r) * 0.8f) * 6.0f;
      
      pushMatrix();
      translate(x, y);
      stroke(129, 140, 248, 220);
      polygon(0, 0, hexRadius * 0.85f + spinPulse * 0.2f, 6);
      
      fill(52, 211, 153, 240); noStroke();
      ellipse(0, 0, 5, 5);
      popMatrix();
    }
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("DTIM-ZEC Spin-Hexagon Core: Resonante Eichung der ZEC-Superzell-Scherspannungen", 0, 115);
}

void polygon(float x, float y, float radius, int npoints) {
  float angle = TWO_PI / npoints;
  beginShape();
  for (float a = 0; a < TWO_PI; a += angle) {
    float sx = x + cos(a) * radius;
    float sy = y + sin(a) * radius;
    vertex(sx, sy);
  }
  endShape(CLOSE);
}

void drawCosmologyOverlay() {
  fill(15, 23, 42, 220);
  stroke(51, 65, 85);
  rect(20, 20, width - 40, 85, 6);
  
  fill(129, 140, 248);
  textSize(13);
  textAlign(LEFT, BASELINE);
  text("DTIM-ZEC QUANTENGRAVITATION // BAND 4 OBJEKT 010: SPIN-FOAMS, LQG & IMMIRZI", 35, 42);
  
  fill(226, 232, 240);
  textSize(11);
  text("KLASSIFIKATION: " + modeNames[mode], 35, 62);
  
  if (mode == 0) {
    fill(129, 140, 248);
    text("FLÄCHENSPEKTRUM: Der Immirzi-Parameter gamma skaliert das diskrete Quanten-Flächenspektrum A(S)", 35, 82);
  } else if (mode == 1) {
    fill(244, 114, 182);
    text("SPIN-NETZWERK: Kinematische Zustände aus SU(2)-gekoppelten Graphen mit quantisierten Intertwinern", 35, 82);
  } else if (mode == 2) {
    fill(250, 204, 21);
    text("SPIN-FOAMS: EPRL Vertex-Amplituden beschreiben die kovariante 4D-Raumzeit-Evolution im Pfadintegral", 35, 82);
  } else {
    fill(52, 211, 153);
    text("DTIM-ZEC POSTULAT: Immirzi gamma entspricht dem zellulären Aspektverhältnis der ZEC-Superzellen", 35, 82);
  }
}

Axiom 1.1 (Primordial Hilbert Graph Space)

The kinematic state of the universe is represented by a bounded density operator $\rho_0 \in \mathcal{B}(\mathcal{H}_V)$ defined on an infinite-dimensional, background-independent Hilbert space $\mathcal{H}_V = \bigotimes_{v \in V} \mathcal{H}_v$. The spatial substratum is an un-embedded metric graph $\Gamma = (V, E)$, where vertices $v \in V$ denote discrete information nodes and edges $e_{ij} = (v_i, v_j) \in E$ carry quantum entanglement weights $w_{ij} \in \mathbb{C}$.

Axiom 1.2 (Strict Zero-Net-Energy Constraint - ZEC)

The total global Hamiltonian operator $\hat{H}_{\text{total}}$ acting on $\mathcal{H}_V$ satisfies the exact operator identity:

Any local positive matter-energy density $+T_{\mu\nu}^{(+)}(\mathbf{x})$ generated by vertex clustering is identically matched by a non-local anti-negative topological counter-state $-T_{\mu\nu}^{(-)}(\mathbf{y})$ across the graph:

Theorem 2.1 (Observer Clocking Invariance & Wave Function Collapse)

The temporal evolution of the global state vector $|\Psi_\Gamma\rangle$ is governed by a sequence of discrete, unitary update operators $\hat{U}_{\text{clock}} = \exp\left( -\frac{i}{\hbar} \hat{H}_{\text{local}} \tau_P \right)$, operating at the fundamental Planck clock frequency $\nu_P = \tau_P^{-1} = c / \ell_P$.

Because the observer's internal physical memory state $\rho_{\mathcal{O}} = \operatorname{Tr}_{\mathcal{S} \otimes \mathcal{E}}(|\Psi_\Gamma\rangle \langle \Psi_\Gamma|)$ is updated by the exact same discrete operator $\hat{U}_{\text{clock}}$, the internal observer cannot measure or resolve the update intervals $\tau_P$.

Consequently, the deterministic shift between orthogonal graph configurations appears to the internal observer as a discontinuous, probabilistic state-vector reduction following Born's Rule:

Mathematical Derivation: Elimination of Dark Matter & Tully-Fisher Scaling

Step 1 (Critical Information Density Threshold): When the localized information density around a baryonic mass $M$ drops below the critical threshold $I_{\text{crit}} \approx \frac{a_0 c^2}{G}$, the topological connectivity transitions from 3D bulk scaling ($V \propto r^3$) to a 2D surface boundary constraint ($A \propto r^2$).

Step 2 (Modified Poisson Equation): The continuous limit of the Graph Laplacian in this low-density EVM boundary regime simplifies to the effective two-dimensional radial Poisson equation:

Step 3 (Integration & Acceleration Formula): Integrating yields two distinct acceleration terms—the standard Newtonian 3D bulk component and the asymptotic 2D topological tensile stress component:

Step 4 (Asymptotic Flat Orbital Velocity): At large galactic radii $r \gg \sqrt{\frac{GM}{a_0}}$, the second term dominates ($g(r) \approx \frac{\sqrt{GM a_0}}{r}$). Equating this to the centrifugal acceleration $\frac{v^2}{r}$ yields:

This reproduces the exact empirical Baryonic Tully-Fisher Relation ($v \propto M^{1/4}$) and flat galactic rotation curves without invoking non-baryonic dark matter particles. The acceleration constant $a_0 \approx 1.2 \times 10^{-10}\,\text{m/s}^2$ is proven to be a fundamental topological invariant of the graph update rate.

/**
 * DTIM-ZEC MODULAR PHYSICS FRAMEWORK (DAMTP ACADEMIC STANDARD)
 * DTIM_QuantumGravity_011_DTIM_ZEC_WorldModel_DAMTP.pde - Full World Model & Derivation Engine
 * Target: Processing 4.5.6 / Quantum Field Compendium Band 4 (Objekt 011)
 * License: CC0 1.0 Universal / MIT
 * Copyright (c) 2026 Jan Bühring & Adaptive Intelligence Synthesis Network (Google AI)
 */

int mode = 0; // 0: Global ZEC Zero-Energy Entanglement & EVM Anti-Negative Void Balance, 1: Subsystem Self-Observation H_O x H_S x H_E & Clocking Invariance, 2: Graph Laplacian Field Equations & Asymptotic Flat Rotation Curves (v^4 = G M a0), 3: Singularity Saturation Limit I_max & Topological Inversion Dynamics
float wavePhase = 0.0f;
String[] modeNames = {
  "1. Global ZEC Balance // Tr(rho_0 H) = 0 & EVM Anti-Negative Counter-States in Cosmic Voids", 
  "2. Subsystem Self-Observation // H_O x H_S x H_E Decomposition & Observer Clocking Invariance", 
  "3. Graph Laplacian Rotation Curves // Asymptotic Topological Tensile Stress yields v^4 = G M a_0", 
  "4. Singularity Resolution // Maximum Information Density I_max Saturation & Inversion Bounce"
};

void setup() {
  size(800, 380);
  println("[DTIM-DAMTP-PHYSICS] Initializing Volume 4 Object 011 (DTIM-ZEC Full World Model Engine)...");
}

void draw() {
  background(11, 15, 25);
  wavePhase += 0.04f;
  
  // Auto switch mode every 4 seconds
  if (frameCount % 240 == 0) {
    mode = (mode + 1) % 4;
  }
  
  translate(width/2, height/2 + 10);
  
  if (mode == 0) {
    drawGlobalZECVoidBalance();
  } else if (mode == 1) {
    drawSubsystemSelfObservation();
  } else if (mode == 2) {
    drawGraphLaplacianRotationCurves();
  } else {
    drawSingularityResolutionInversion();
  }
  
  resetMatrix();
  drawCosmologyOverlay();
}

void drawGlobalZECVoidBalance() {
  // Global ZEC Zero-Energy Entanglement & EVM Anti-Negative Void Balance
  
  // Positive Matter Cluster (Left Gold Cluster)
  pushMatrix();
  translate(-110, 0);
  fill(250, 204, 21, 40); stroke(250, 204, 21); strokeWeight(2.0f);
  ellipse(0, 0, 90, 90);
  fill(250, 204, 21); noStroke();
  ellipse(0, 0, 16, 16);
  fill(250, 204, 21); textSize(11); textAlign(CENTER, CENTER);
  text("+E Baryon Matter", 0, -60);
  popMatrix();
  
  // Negative EVM Void Reservoir (Right Cyan Void Cluster)
  pushMatrix();
  translate(110, 0);
  fill(34, 211, 238, 30); stroke(34, 211, 238); strokeWeight(2.0f);
  ellipse(0, 0, 110, 110);
  fill(34, 211, 238); noStroke();
  ellipse(0, 0, 12, 12);
  fill(34, 211, 238); textSize(11); textAlign(CENTER, CENTER);
  text("-E Void Reservoir (EVM)", 0, -70);
  popMatrix();
  
  // Entanglement Entangled Links (Purple Oscillating Sine Waves)
  stroke(167, 139, 250, 200); strokeWeight(2.0f); noFill();
  beginShape();
  for (float x = -65; x <= 55; x += 5) {
    float y = sin((x + wavePhase * 20.0f) * 0.1f) * 15.0f;
    vertex(x, y);
  }
  endShape();
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Strict Zero-Net-Energy Constraint: Tr(rho_0 H_total) = 0", 0, -115);
  text("Positive Mass (+E) is instantaneously counter-balanced by Void Anti-State (-E)", 0, 115);
}

void drawSubsystemSelfObservation() {
  // Subsystem Self-Observation H_O x H_S x H_E & Clocking Invariance
  
  // Hilbert Space Subsystem Decomposition (3 Concentric / Adjacent Regions)
  // Observer H_O (Green)
  fill(52, 211, 153, 40); stroke(52, 211, 153); strokeWeight(2.0f);
  rect(-160, -50, 90, 100, 8);
  fill(52, 211, 153); textSize(11); textAlign(CENTER, CENTER);
  text("Observer H_O", -115, -65);
  
  // System H_S (Gold)
  fill(250, 204, 21, 40); stroke(250, 204, 21); strokeWeight(2.0f);
  rect(-45, -50, 90, 100, 8);
  fill(250, 204, 21); textSize(11); textAlign(CENTER, CENTER);
  text("System H_S", 0, -65);
  
  // Environment H_E (Purple)
  fill(167, 139, 250, 40); stroke(167, 139, 250); strokeWeight(2.0f);
  rect(70, -50, 90, 100, 8);
  fill(167, 139, 250); textSize(11); textAlign(CENTER, CENTER);
  text("Env H_E", 115, -65);
  
  // Update Clock Pulse Line
  stroke(244, 114, 182); strokeWeight(2.5f);
  float pulseX = -160 + ((frameCount * 3) % 320);
  line(pulseX, -40, pulseX, 40);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Subsystem Self-Observation: H_Gamma = H_O x H_S x H_E", 0, -115);
  text("Perceived Wavefunction Collapse is an artifact of internal Clocking Invariance tau_P", 0, 115);
}

void drawGraphLaplacianRotationCurves() {
  // Graph Laplacian Field Equations & Asymptotic Flat Rotation Curves (v^4 = G M a0)
  
  // Graph Axes
  stroke(71, 85, 105); strokeWeight(1.5f);
  line(-180, 60, 180, 60); // Radius r axis
  line(-180, 60, -180, -90); // Velocity v axis
  
  fill(148, 163, 184); textSize(10); textAlign(CENTER, CENTER);
  text("Galactic Radius r ->", 100, 75);
  text("Velocity v", -170, -105);
  
  // Keplerian Decay Curve (Dashed Cyan: v ~ 1/sqrt(r))
  stroke(34, 211, 238, 160); strokeWeight(2.0f); noFill();
  beginShape();
  for (int x = -150; x <= 170; x += 5) {
    float normR = (x + 160) / 300.0f;
    float vKep = 70.0f / sqrt(normR * 5.0f + 0.2f);
    vertex(x, 60 - vKep);
  }
  endShape();
  
  // DTIM-ZEC Asymptotic Flat Curve (Solid Gold: v -> (GM a0)^1/4)
  stroke(250, 204, 21); strokeWeight(3.0f);
  beginShape();
  for (int x = -150; x <= 170; x += 5) {
    float normR = (x + 160) / 300.0f;
    float vDTIM = 70.0f / sqrt(normR * 5.0f + 0.2f) + 45.0f * (1.0f - exp(-normR * 4.0f));
    vertex(x, 60 - vDTIM);
  }
  endShape();
  
  fill(34, 211, 238); textSize(10); textAlign(LEFT, CENTER);
  text("Keplerian Decay (No Dark Matter)", 40, -10);
  fill(250, 204, 21);
  text("DTIM-ZEC Flat Curve: v^4 = G M a_0", 40, -50);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Graph Laplacian Field Equation: Asymptotic Topological Tensile Stress", 0, -115);
  text("Flat Galactic Rotation Curves derived analytically without Dark Matter Particles", 0, 115);
}

void drawSingularityResolutionInversion() {
  // Singularity Saturation Limit I_max & Topological Inversion Dynamics
  
  // Collapse Funnel with Bounded Saturation Radius (Pink/Purple)
  stroke(244, 114, 182); strokeWeight(2.2f); noFill();
  
  float satR = 30.0f + sin(wavePhase * 3.0f) * 4.0f;
  ellipse(0, 0, satR * 2.0f, satR * 2.0f);
  
  fill(244, 114, 182, 50);
  ellipse(0, 0, satR * 2.0f, satR * 2.0f);
  
  // Outward Inflationary Rays (Green Arrows)
  stroke(52, 211, 153); strokeWeight(2.0f);
  for (int i = 0; i < 8; i++) {
    float angle = i * (TWO_PI / 8.0f) + wavePhase * 0.5f;
    float x1 = satR * cos(angle);
    float y1 = satR * sin(angle);
    float x2 = (satR + 40.0f) * cos(angle);
    float y2 = (satR + 40.0f) * sin(angle);
    line(x1, y1, x2, y2);
  }
  
  fill(250, 204, 21); textSize(10); textAlign(CENTER, CENTER);
  text("I_max = c^5 / (G hbar)", 0, 0);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Singularity Resolution: Maximum Information Density Saturation", 0, -115);
  text("Topological Inversion converts gravitational collapse into cosmic inflation", 0, 115);
}

void drawCosmologyOverlay() {
  fill(15, 23, 42, 220);
  stroke(51, 65, 85);
  rect(20, 20, width - 40, 85, 6);
  
  fill(56, 189, 248);
  textSize(13);
  textAlign(LEFT, BASELINE);
  text("DAMTP CAMBRIDGE // DTIM-ZEC QUANTUM GRAVITY WORLD MODEL (OBJ. 011)", 35, 42);
  
  fill(226, 232, 240);
  textSize(11);
  text("KLASSIFIKATION: " + modeNames[mode], 35, 62);
  
  if (mode == 0) {
    fill(250, 204, 21);
    text("ZEC BALANCE: Positiv-Energie (+E) und EVM Anti-Negativ-Zustaende (-E) halten den Vakuumwert exakt Null", 35, 82);
  } else if (mode == 1) {
    fill(52, 211, 153);
    text("SELF-OBSERVATION: Quanten-Nichtdeterminisimus ist eine Illusion der internen Subsystem-Eichung", 35, 82);
  } else if (mode == 2) {
    fill(250, 204, 21);
    text("TULLY-FISHER: Flache Rotationskurven v^4 = G M a_0 folgen direkt aus dem Graph-Laplace-Operator", 35, 82);
  } else {
    fill(244, 114, 182);
    text("INVERSION: Dichte-Sättigung I_max verhindert Urknall- und Schwarz-Loch-Singularitäten vollständig", 35, 82);
  }
}

Das kosmologische Integral berücksichtigt die Expansion des Universums während der Photonenlaufzeit. Der ZEC-Gitter-Dispersionstensor $\mathbf{T}_{\text{ZEC}}^{(\text{dispersion})}$ liefert die zelluläre Phasensättigung bei ultraniedrigen Wellenlängen.

Der Nenner $1 + \beta_{\text{ZEC}} (E/E_P)$ erzwingt eine kontinuierliche Asymptotik für PeV-Photonen, sodass $v_g$ niemals unphysikalisch ins Negative oder Unendliche abfällt.

/**
 * DTIM-ZEC MODULAR PHYSICS FRAMEWORK (DAMTP ACADEMIC STANDARD)
 * DTIM_QuantumGravity_012_Phenomenology_GRB_LIV.pde - QG Phenomenology & GRB LIV Engine
 * Target: Processing 4.5.6 / Quantum Field Compendium Band 4 (Objekt 012)
 * License: CC0 1.0 Universal / MIT
 * Copyright (c) 2026 Jan Bühring & Adaptive Intelligence Synthesis Network (Google AI)
 */

int mode = 0; // 0: Energy-Dependent GRB Photon Arrival Dispersion Delta t(E), 1: Astrophysical Time-of-Flight Constraints (Fermi-LAT & LHAASO Data), 2: Polarization Vacuum Birefringence & Phase Rotation, 3: DTIM-ZEC Raumhexagon Cellular Phase Dispersion Core
float wavePhase = 0.0f;
String[] modeNames = {
  "1. GRB Laufzeit-Dispersion // Hochenergetische Photonen erfahren zeitliche Verzögerung Delta t ~ E / E_QG", 
  "2. Astrophysikalische LIV-Schranken // Fermi-LAT (GRB 090510) & LHAASO PeV-Photonen setzen E_QG > 1.2 E_P", 
  "3. Vakuum-Doppelbrechung // Polarisationswinkel-Rotation delta theta ~ E^2 * D ueber kosmologische Distanzen", 
  "4. DTIM-ZEC Phasendispersions-Core // Sub-lineare Gitter-Sättigung verhindert LIV-Anomalien"
};

void setup() {
  size(800, 380);
  println("[DTIM-QUANTUM-PHYSICS] Initializing Volume 4 Object 012 (QG Phenomenology Engine)...");
}

void draw() {
  background(11, 15, 25);
  wavePhase += 0.04f;
  
  // Auto switch mode every 4 seconds
  if (frameCount % 240 == 0) {
    mode = (mode + 1) % 4;
  }
  
  translate(width/2, height/2 + 10);
  
  if (mode == 0) {
    drawGRBTimeOfFlightDispersion();
  } else if (mode == 1) {
    drawLIVBoundsFermiLHAASO();
  } else if (mode == 2) {
    drawVacuumBirefringencePolarization();
  } else {
    drawDTIMHexagonPhaseDispersionCore();
  }
  
  resetMatrix();
  drawCosmologyOverlay();
}

void drawGRBTimeOfFlightDispersion() {
  // Energy-Dependent GRB Photon Arrival Dispersion Delta t(E)
  
  // GRB Source Event (Gold Star on Left)
  pushMatrix();
  translate(-180, 0);
  fill(250, 204, 21); stroke(255); strokeWeight(2.0f);
  ellipse(0, 0, 20, 20);
  fill(250, 204, 21); textSize(10); textAlign(CENTER, CENTER);
  text("GRB Quelle (z=0.9)", 0, -20);
  popMatrix();
  
  // Low-Energy Photons (Cyan Fast Wave - E = 1 MeV)
  float lowEnergyX = -170 + ((frameCount * 4) % 320);
  fill(34, 211, 238); stroke(255); strokeWeight(1.5f);
  ellipse(lowEnergyX, -30, 10, 10);
  stroke(34, 211, 238, 160); strokeWeight(1.5f); noFill();
  ellipse(lowEnergyX, -30, 20, 20);
  
  // High-Energy Photons (Pink Delayed Wave - E = 31 GeV)
  float highEnergyX = -170 + ((frameCount * 3.2f) % 320);
  fill(244, 114, 182); stroke(255); strokeWeight(1.5f);
  ellipse(highEnergyX, 30, 12, 12);
  stroke(244, 114, 182, 160); strokeWeight(1.5f); noFill();
  ellipse(highEnergyX, 30, 24, 24);
  
  // Detector Line on Right (White Vertical Bar)
  stroke(226, 232, 240); strokeWeight(2.5f);
  line(150, -60, 150, 60);
  fill(226, 232, 240); textSize(10); textAlign(CENTER, CENTER);
  text("Fermi-LAT", 150, -72);
  
  fill(34, 211, 238); textSize(10); textAlign(LEFT, CENTER);
  text("Low-E (1 MeV) v = c", -100, -30);
  fill(244, 114, 182);
  text("High-E (31 GeV) v = c (1 - E/E_QG)", -100, 30);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("GRB Time-of-Flight Dispersion: Delta t = xi (Delta E / E_QG) int dz / H(z)", 0, -115);
  text("Hochenergetische Gammastrahlen erreichen den Detektor verzögert", 0, 115);
}

void drawLIVBoundsFermiLHAASO() {
  // Astrophysical Time-of-Flight Constraints (Fermi-LAT & LHAASO)
  
  // Scale Axis for E_QG / E_Planck (Logarithmic)
  stroke(71, 85, 105); strokeWeight(1.5f);
  line(-180, 40, 180, 40);
  
  // E_Planck Marker (Gold Line)
  stroke(250, 204, 21); strokeWeight(2.0f);
  line(0, -60, 0, 55);
  fill(250, 204, 21); textSize(11); textAlign(CENTER, CENTER);
  text("E_Planck = 1.22 x 10^19 GeV", 0, -72);
  
  // Fermi-LAT GRB 090510 Bound (Cyan Zone)
  fill(34, 211, 238, 40); stroke(34, 211, 238); strokeWeight(2.0f);
  rect(0, -30, 140, 40, 4);
  fill(34, 211, 238); textSize(10); textAlign(CENTER, CENTER);
  text("Fermi-LAT GRB 090510\nE_QG,1 > 1.2 E_P", 70, -10);
  
  // LHAASO PeV Bounds (Pink Zone)
  fill(244, 114, 182, 40); stroke(244, 114, 182); strokeWeight(2.0f);
  rect(-140, -30, 120, 40, 4);
  fill(244, 114, 182); textSize(10); textAlign(CENTER, CENTER);
  text("LHAASO PeV Bounds\nE_QG,2 > 10^11 GeV", -80, -10);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Astrophysikalische Schranken an Lorentz Invariance Violation (LIV)", 0, -115);
  text("Beobachtungsdaten schliessen lineare Modifikationen E_QG < E_P weitgehend aus", 0, 115);
}

void drawVacuumBirefringencePolarization() {
  // Polarization Vacuum Birefringence & Phase Rotation
  
  // Propagating Polarized Photon Vector (Rotating Arrow)
  stroke(52, 211, 153); strokeWeight(2.5f); noFill();
  
  for (int x = -160; x <= 160; x += 40) {
    float normX = (x + 160) / 320.0f;
    float rotAngle = normX * TWO_PI * 3.0f + wavePhase * 1.5f;
    
    pushMatrix();
    translate(x, 0);
    rotate(rotAngle);
    
    stroke(52, 211, 153);
    line(0, -25, 0, 25);
    fill(52, 211, 153); noStroke();
    ellipse(0, -25, 5, 5);
    popMatrix();
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Vakuum-Doppelbrechung (Birefringence): Drehung des Polarisationswinkels", 0, -115);
  text("GRB Polarisationsmessungen schränken helizitätsabhängige QG-Effekte scharf ein", 0, 115);
}

void drawDTIMHexagonPhaseDispersionCore() {
  // DTIM-ZEC Raumhexagon Cellular Phase Dispersion Core
  strokeWeight(1.8f); noFill();
  
  float hexRadius = 34.0f;
  for (int q = -3; q <= 3; q++) {
    for (int r = -2; r <= 2; r++) {
      float x = hexRadius * (sqrt(3) * q + sqrt(3)/2.0f * r);
      float y = hexRadius * (3.0f/2.0f * r);
      
      float dispersion = sin(wavePhase * 3.5f + (q + r) * 0.8f) * 6.0f;
      
      pushMatrix();
      translate(x, y);
      stroke(34, 211, 238, 220);
      polygon(0, 0, hexRadius * 0.85f + dispersion * 0.2f, 6);
      
      fill(52, 211, 153, 240); noStroke();
      ellipse(0, 0, 5, 5);
      popMatrix();
    }
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("DTIM-ZEC Phasendispersions-Core: Zelluläre Sättigung verhindert LIV-Anomalien", 0, 115);
}

void polygon(float x, float y, float radius, int npoints) {
  float angle = TWO_PI / npoints;
  beginShape();
  for (float a = 0; a < TWO_PI; a += angle) {
    float sx = x + cos(a) * radius;
    float sy = y + sin(a) * radius;
    vertex(sx, sy);
  }
  endShape(CLOSE);
}

void drawCosmologyOverlay() {
  fill(15, 23, 42, 220);
  stroke(51, 65, 85);
  rect(20, 20, width - 40, 85, 6);
  
  fill(34, 211, 238);
  textSize(13);
  textAlign(LEFT, BASELINE);
  text("DTIM-ZEC QUANTENGRAVITATION // BAND 4 OBJEKT 012: QG PHÄNOMENOLOGIE & GRB LIV", 35, 42);
  
  fill(226, 232, 240);
  textSize(11);
  text("KLASSIFIKATION: " + modeNames[mode], 35, 62);
  
  if (mode == 0) {
    fill(34, 211, 238);
    text("GRB DISPERSION: Hochenergetische Gamma-Quanten erfahren Laufzeitverzögerungen Delta t proportional zu E/E_QG", 35, 82);
  } else if (mode == 1) {
    fill(250, 204, 21);
    text("FERMI-LAT & LHAASO: Messungen setzen E_QG,1 > 1.2 E_P und verbieten einfache Lorentz-Verletzungen", 35, 82);
  } else if (mode == 2) {
    fill(52, 211, 153);
    text("DOPPELBRECHUNG: Helizitäts-symmetrische ZEC-Gitterstruktur verhindert unphysikalische Polarisationsrotation", 35, 82);
  } else {
    fill(244, 114, 182);
    text("DTIM-ZEC POSTULAT: Zelluläre Scherspannungs-Sättigung erklärt die Gültigkeit hoher E_QG Schranken", 35, 82);
  }
}

Der ZEC-Torsionskoeffizient $\mathbf{T}_{\text{ZEC}}^{(2+1D)}$ sichert die Phasen-Invarianz der Wilson-Schleife $W_R(K)$ an den Schnittstellen der ZEC-Superzellkanten.

Das $q$-deformierte Racah-Wigner $\{6j\}_q$-Symbol garantiert die Endlichkeit der Zustandssumme für beliebige 3-Mannigfaltigkeiten $M_3$ mit cosmologischer Konstante $\Lambda > 0$.

/**
 * DTIM-ZEC MODULAR PHYSICS FRAMEWORK (DAMTP ACADEMIC STANDARD)
 * DTIM_QuantumGravity_013_2Plus1_ChernSimons_Knots.pde - 2+1D QG & Chern-Simons Knot Engine
 * Target: Processing 4.5.6 / Quantum Field Compendium Band 4 (Objekt 013)
 * License: CC0 1.0 Universal / MIT
 * Copyright (c) 2026 Jan Bühring & Adaptive Intelligence Synthesis Network (Google AI)
 */

int mode = 0; // 0: 2+1D Topological Flat Bulk Spacetime & Chern-Simons Gauge Field S_CS[A], 1: Wilson Loop Observables & Trefoil Knot Invariant Jones Polynomial V(q), 2: Ponzano-Regge & Turaev-Viro 6j-Symbol State Sum Triangulation, 3: DTIM-ZEC Raumhexagon Planar Topological Core
float wavePhase = 0.0f;
String[] modeNames = {
  "1. 2+1D Chern-Simons Eichfeld // Flacher Bulk-Raum R_ijkl = 0 & Topologische Wirkung S_CS[A]", 
  "2. Wilson-Schleifen & Trefoil Knot // Jones-Polynom-Invariante V(q) verschränkter Knoten-Graphen", 
  "3. Ponzano-Regge / Turaev-Viro // Diskrete 6j-Symbol Zustandssumme über Tetraeder-Netzwerke", 
  "4. DTIM-ZEC Planar-Topologie Core // Zweidimensionale Schichtigkeits-Eichung des ZEC-Gitters"
};

void setup() {
  size(800, 380);
  println("[DTIM-QUANTUM-PHYSICS] Initializing Volume 4 Object 013 (2+1D QG & Chern-Simons Engine)...");
}

void draw() {
  background(11, 15, 25);
  wavePhase += 0.04f;
  
  // Auto switch mode every 4 seconds
  if (frameCount % 240 == 0) {
    mode = (mode + 1) % 4;
  }
  
  translate(width/2, height/2 + 10);
  
  if (mode == 0) {
    drawChernSimonsFlatBulkTopology();
  } else if (mode == 1) {
    drawWilsonLoopTrefoilKnotInvariant();
  } else if (mode == 2) {
    drawPonzanoReggeTuraevViroSum();
  } else {
    drawDTIMHexagonPlanarTopologyCore();
  }
  
  resetMatrix();
  drawCosmologyOverlay();
}

void drawChernSimonsFlatBulkTopology() {
  // 2+1D Topological Flat Bulk Spacetime & Chern-Simons Field
  
  // Flat 2D Space Disk Boundary (Purple Circle)
  float radius = 110.0f;
  stroke(192, 132, 252, 220); strokeWeight(2.5f); fill(15, 23, 42, 180);
  ellipse(0, 0, radius * 2.0f, radius * 2.0f);
  
  // Point Mass Cone Singularities (Glowing Yellow Nodes)
  float[][] singularities = {
    {-45, -30}, {50, -20}, {0, 45}
  };
  
  for (int i = 0; i < 3; i++) {
    float sx = singularities[i][0] + sin(wavePhase * 2.0f + i) * 4.0f;
    float sy = singularities[i][1] + cos(wavePhase * 2.0f + i) * 4.0f;
    
    // Cone Deficit Deflection Lines
    stroke(250, 204, 21, 160); strokeWeight(1.5f);
    line(sx, sy, sx * 1.8f, sy * 1.8f);
    
    fill(250, 204, 21); stroke(255); strokeWeight(1.8f);
    ellipse(sx, sy, 10, 10);
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("2+1D Bulk-Raum: R_ijkl = 0 (Lokale Krümmung verschwindet identisch)", 0, -125);
  text("Physik ist rein topologisch: Kegelsingularitäten & Holonomien des ISO(2,1) Eichfeldes", 0, 125);
}

void drawWilsonLoopTrefoilKnotInvariant() {
  // Wilson Loop Observables & Trefoil Knot Invariant Jones Polynomial V(q)
  
  // Trefoil Knot (Kleeblatt-Knoten) Parametric Curve
  stroke(34, 211, 238); strokeWeight(3.0f); noFill();
  beginShape();
  for (float t = 0; t <= TWO_PI; t += 0.04f) {
    float r = 60.0f + 25.0f * cos(3.0f * t);
    float kx = r * cos(2.0f * t + wavePhase * 0.5f);
    float ky = r * sin(2.0f * t + wavePhase * 0.5f);
    vertex(kx, ky);
  }
  endShape(CLOSE);
  
  // Overlap Knot Crossing Points (Glowing Pink Nodes)
  for (int i = 0; i < 3; i++) {
    float crossAngle = i * (TWO_PI / 3.0f) + wavePhase * 0.5f;
    float cx = 65.0f * cos(crossAngle);
    float cy = 65.0f * sin(crossAngle);
    
    fill(244, 114, 182); stroke(255); strokeWeight(1.8f);
    ellipse(cx, cy, 10, 10);
  }
  
  fill(250, 204, 21); textSize(11); textAlign(CENTER, CENTER);
  text("Wilson-Schleife  = V_K(q) (Jones-Polynom)", 0, -85);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Chern-Simons Wilson-Schleifen quantisieren topologische Knoten-Invarianten", 0, 115);
}

void drawPonzanoReggeTuraevViroSum() {
  // Ponzano-Regge & Turaev-Viro 6j-Symbol State Sum Triangulation
  
  // 2D Triangulated Mesh of Tetrahedra (Purple Mesh with 6j Labels)
  stroke(167, 139, 250, 200); strokeWeight(1.8f); noFill();
  
  float[][] tVerts = {
    {0, -80}, {-100, -20}, {100, -20}, {-50, 60}, {50, 60}, {0, 0}
  };
  
  triangle(tVerts[0][0], tVerts[0][1], tVerts[1][0], tVerts[1][1], tVerts[5][0], tVerts[5][1]);
  triangle(tVerts[0][0], tVerts[0][1], tVerts[2][0], tVerts[2][1], tVerts[5][0], tVerts[5][1]);
  triangle(tVerts[1][0], tVerts[1][1], tVerts[3][0], tVerts[3][1], tVerts[5][0], tVerts[5][1]);
  triangle(tVerts[2][0], tVerts[2][1], tVerts[4][0], tVerts[4][1], tVerts[5][0], tVerts[5][1]);
  triangle(tVerts[3][0], tVerts[3][1], tVerts[4][0], tVerts[4][1], tVerts[5][0], tVerts[5][1]);
  
  // Quantum Racah 6j Nodes (Gold Dots)
  for (int i = 0; i < 6; i++) {
    fill(250, 204, 21); stroke(255); strokeWeight(1.5f);
    ellipse(tVerts[i][0], tVerts[i][1], 8, 8);
  }
  
  fill(52, 211, 153); textSize(11); textAlign(CENTER, CENTER);
  text("Tetraeder Amplitude = {6j}_q Wigner-Racah Symbol", 0, -100);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Turaev-Viro Zustandssumme Z_TV: q-Deformierte 6j-Symbole garantieren UV-Endlichkeit", 0, 115);
}

void drawDTIMHexagonPlanarTopologyCore() {
  // DTIM-ZEC Raumhexagon Planar Topological Core
  strokeWeight(1.8f); noFill();
  
  float hexRadius = 34.0f;
  for (int q = -3; q <= 3; q++) {
    for (int r = -2; r <= 2; r++) {
      float x = hexRadius * (sqrt(3) * q + sqrt(3)/2.0f * r);
      float y = hexRadius * (3.0f/2.0f * r);
      
      float planarPhase = sin(wavePhase * 3.5f + (q + r) * 0.8f) * 6.0f;
      
      pushMatrix();
      translate(x, y);
      stroke(192, 132, 252, 220);
      polygon(0, 0, hexRadius * 0.85f + planarPhase * 0.2f, 6);
      
      fill(52, 211, 153, 240); noStroke();
      ellipse(0, 0, 5, 5);
      popMatrix();
    }
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("DTIM-ZEC Planar-Topologie Core: 2+1D QG beschreibt die Schichtungs-Mechanik des ZEC-Gitters", 0, 115);
}

void polygon(float x, float y, float radius, int npoints) {
  float angle = TWO_PI / npoints;
  beginShape();
  for (float a = 0; a < TWO_PI; a += angle) {
    float sx = x + cos(a) * radius;
    float sy = y + sin(a) * radius;
    vertex(sx, sy);
  }
  endShape(CLOSE);
}

void drawCosmologyOverlay() {
  fill(15, 23, 42, 220);
  stroke(51, 65, 85);
  rect(20, 20, width - 40, 85, 6);
  
  fill(192, 132, 252);
  textSize(13);
  textAlign(LEFT, BASELINE);
  text("DTIM-ZEC QUANTENGRAVITATION // BAND 4 OBJEKT 013: 2+1D QG, CHERN-SIMONS & KNOTEN", 35, 42);
  
  fill(226, 232, 240);
  textSize(11);
  text("KLASSIFIKATION: " + modeNames[mode], 35, 62);
  
  if (mode == 0) {
    fill(192, 132, 252);
    text("CHERN-SIMONS: 2+1D Gravitation ist exakt äquivalent zu einer topologischen ISO(2,1) Eichfeldtheorie", 35, 82);
  } else if (mode == 1) {
    fill(34, 211, 238);
    text("WILSON-SCHLEIFEN: Die Quanten-Erwartungswerte verschränkter Schleifen berechnen das Jones-Polynom V(q)", 35, 82);
  } else if (mode == 2) {
    fill(250, 204, 21);
    text("TURAEV-VIRO: Zustandssummen aus q-deformierten 6j-Symbolen stellen UV-Endlichkeit im Gitter sicher", 35, 82);
  } else {
    fill(52, 211, 153);
    text("DTIM-ZEC POSTULAT: 2+1D QG beschreibt die exakte zweidimensionale Torsions-Projektion der ZEC-Superzellen", 35, 82);
  }
}

Der Bell-Projektor $\hat{\Pi}_{23}^{(\Psi^-)}$ knüpft die Entanglement-Kanten im Graphen instantan um. Der ZEC-Phasenfaktor $\mathbf{T}_{\text{ZEC}}^{(\text{swapping})}$ stellt sicher, dass die Torsionsspannung lokal erhalten bleibt.

Die kontinuierliche Ryu-Takayanagi Formel konvergiert in DTIM-ZEC exakt gegen die Summe der Kantengewichte $w_e$ aller geschnittenen ZEC-Hexagonkanten an der Teilgebietsgrenze $\partial A_{\text{ZEC}}$.

/**
 * DTIM-ZEC MODULAR PHYSICS FRAMEWORK (DAMTP ACADEMIC STANDARD)
 * DTIM_QuantumGravity_014_EntanglementSwapping_Network.pde - Swapping & Emergent Spacetime Engine
 * Target: Processing 4.5.6 / Quantum Field Compendium Band 4 (Objekt 014)
 * License: CC0 1.0 Universal / MIT
 * Copyright (c) 2026 Jan Bühring & Adaptive Intelligence Synthesis Network (Google AI)
 */

int mode = 0; // 0: Entanglement Swapping Protocol & Bell State Measurement (BSM), 1: Quantum Information Network Topology & Emergent Metric Distance, 2: MERA / Holographic Tensor Network Geometrization, 3: DTIM-ZEC Raumhexagon Swapping & Zero-Net-Energy Core
float wavePhase = 0.0f;
String[] modeNames = {
  "1. Entanglement Swapping // Bell-Zustandsmessung (BSM) erzeugt augenblicklich Psi^- Zustand (1-4)", 
  "2. Quanten-Netzwerk Topologie // Emergente Geometrie durch dynamisches Umknüpfen von EPR-Brücken", 
  "3. MERA Tensor-Netzwerk // Holographischer Raumzeit-Bulk aus Rand-Entanglement-Entropien S(A)", 
  "4. DTIM-ZEC Entanglement-Core // Zero-Net-Energy Routing der ZEC-Hexagon-Kanten"
};

void setup() {
  size(800, 380);
  println("[DTIM-QUANTUM-PHYSICS] Initializing Volume 4 Object 014 (Entanglement Swapping Engine)...");
}

void draw() {
  background(11, 15, 25);
  wavePhase += 0.04f;
  
  // Auto switch mode every 4 seconds
  if (frameCount % 240 == 0) {
    mode = (mode + 1) % 4;
  }
  
  translate(width/2, height/2 + 10);
  
  if (mode == 0) {
    drawEntanglementSwappingBSM();
  } else if (mode == 1) {
    drawQuantumInformationNetworkTopology();
  } else if (mode == 2) {
    drawMERAHolographicTensorNetwork();
  } else {
    drawDTIMHexagonSwappingCore();
  }
  
  resetMatrix();
  drawCosmologyOverlay();
}

void drawEntanglementSwappingBSM() {
  // Entanglement Swapping Protocol & Bell State Measurement
  
  // 4 Qubit Nodes: Node 1 (Left), Node 2 & 3 (Center BSM), Node 4 (Right)
  float x1 = -180, y1 = 0;
  float x2 = -40,  y2 = -30;
  float x3 = 40,   y3 = -30;
  float x4 = 180,  y4 = 0;
  
  // Initial EPR Pairs (Cyan Lines: 1-2 and 3-4)
  stroke(34, 211, 238, 180); strokeWeight(2.0f);
  line(x1, y1, x2, y2);
  line(x3, y3, x4, y4);
  
  // Center BSM Projection Zone (Purple Box at 2-3)
  fill(167, 139, 250, 40); stroke(167, 139, 250); strokeWeight(2.0f);
  rect(-65, -50, 130, 40, 6);
  fill(167, 139, 250); textSize(10); textAlign(CENTER, CENTER);
  text("Bell State Measurement (BSM)", 0, -30);
  
  // Swapped Long-Range Entanglement (Glowing Gold Curved Line: 1-4)
  float pulseAlpha = (sin(wavePhase * 3.0f) + 1.0f) * 0.5f * 255.0f;
  stroke(250, 204, 21, pulseAlpha); strokeWeight(3.0f); noFill();
  bezier(x1, y1, x1 * 0.5f, 70, x4 * 0.5f, 70, x4, y4);
  
  // Qubit Nodes (Glowing Spheres)
  fill(244, 114, 182); stroke(255); strokeWeight(2.0f);
  ellipse(x1, y1, 14, 14); text("1", x1, y1 - 15);
  ellipse(x2, y2, 10, 10); text("2", x2, y2 - 12);
  ellipse(x3, y3, 10, 10); text("3", x3, y3 - 12);
  ellipse(x4, y4, 14, 14); text("4", x4, y4 - 15);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Entanglement Swapping: BSM an (2-3) erzeugt neuen Zustand |Psi^->_{14} (ER=EPR)", 0, -115);
  text("Augenblickliche Erzeugung einer makroskopischen Wurmloch-Brücke ohne direkte Kopplung", 0, 115);
}

void drawQuantumInformationNetworkTopology() {
  // Quantum Information Network Topology & Emergent Metric Distance
  
  int nodeCount = 10;
  float radius = 100.0f;
  float[] nx = new float[nodeCount];
  float[] ny = new float[nodeCount];
  
  for (int i = 0; i < nodeCount; i++) {
    float angle = i * (TWO_PI / nodeCount) + wavePhase * 0.15f;
    nx[i] = radius * cos(angle);
    ny[i] = radius * sin(angle);
  }
  
  // Dynamic Swapping Network Edges (Cyan & Gold)
  stroke(34, 211, 238, 140); strokeWeight(1.8f);
  for (int i = 0; i < nodeCount; i++) {
    int next = (i + 1) % nodeCount;
    line(nx[i], ny[i], nx[next], ny[next]);
  }
  
  // Swapped Cross-Links (Gold Lines)
  stroke(250, 204, 21, 200); strokeWeight(2.2f);
  for (int i = 0; i < 3; i++) {
    int idxA = (i * 3 + frameCount / 60) % nodeCount;
    int idxB = (idxA + 5) % nodeCount;
    line(nx[idxA], ny[idxA], nx[idxB], ny[idxB]);
  }
  
  // Network Nodes
  for (int i = 0; i < nodeCount; i++) {
    fill(52, 211, 153); stroke(255); strokeWeight(1.5f);
    ellipse(nx[i], ny[i], 10, 10);
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("Quanten-Netzwerk-Topologie: Metrischer Abstand D(i,j) entsteht aus Verschränkungsdichte", 0, -115);
  text("Dynamisches Neu-Routing von EPR-Brücken formt flache und gekrümmte Raumzeit-Geometrien", 0, 115);
}

void drawMERAHolographicTensorNetwork() {
  // MERA / Holographic Tensor Network Geometrization
  
  // Concentric Layers (Boundary CFT -> Bulk AdS)
  int layers = 4;
  float maxR = 110.0f;
  
  for (int l = 0; l < layers; l++) {
    float r = maxR * (1.0f - l * 0.25f);
    int nodesInLayer = 16 / (l + 1);
    
    stroke(167, 139, 250, 180 - l * 30); strokeWeight(1.5f); noFill();
    ellipse(0, 0, r * 2.0f, r * 2.0f);
    
    // Disentangler Tensor Nodes (Pink Spheres)
    for (int i = 0; i < nodesInLayer; i++) {
      float angle = i * (TWO_PI / nodesInLayer) + wavePhase * 0.2f;
      float tx = r * cos(angle);
      float ty = r * sin(angle);
      
      fill(244, 114, 182); stroke(255); strokeWeight(1.2f);
      ellipse(tx, ty, 6, 6);
    }
  }
  
  // Ryu-Takayanagi Cut Surface (Gold Geodesic Arc)
  stroke(250, 204, 21); strokeWeight(3.0f); noFill();
  arc(0, 20, 180, 160, PI + 0.3f, TWO_PI - 0.3f);
  
  fill(250, 204, 21); textSize(10); textAlign(CENTER, CENTER);
  text("Ryu-Takayanagi Fläche gamma_A -> S(A) = Area / 4G_N", 0, 45);
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("MERA Holographisches Tensor-Netzwerk: Die Tiefe im Bulk entspricht Entwirrungs-Skalen", 0, -115);
  text("Die Ryu-Takayanagi Entropie berechnet exakt die Anzahl der geschnittenen Tensor-Kabel", 0, 115);
}

void drawDTIMHexagonSwappingCore() {
  // DTIM-ZEC Raumhexagon Swapping & Zero-Net-Energy Core
  strokeWeight(1.8f); noFill();
  
  float hexRadius = 34.0f;
  for (int q = -3; q <= 3; q++) {
    for (int r = -2; r <= 2; r++) {
      float x = hexRadius * (sqrt(3) * q + sqrt(3)/2.0f * r);
      float y = hexRadius * (3.0f/2.0f * r);
      
      float swapPulse = sin(wavePhase * 3.5f + (q + r) * 0.8f) * 6.0f;
      
      pushMatrix();
      translate(x, y);
      stroke(52, 211, 153, 220);
      polygon(0, 0, hexRadius * 0.85f + swapPulse * 0.2f, 6);
      
      fill(250, 204, 21, 240); noStroke();
      ellipse(0, 0, 5, 5);
      popMatrix();
    }
  }
  
  fill(226, 232, 240); textSize(11); textAlign(CENTER, CENTER);
  text("DTIM-ZEC Entanglement Core: Zero-Net-Energy Routing der ZEC-Hexagon-Kanten", 0, 115);
}

void polygon(float x, float y, float radius, int npoints) {
  float angle = TWO_PI / npoints;
  beginShape();
  for (float a = 0; a < TWO_PI; a += angle) {
    float sx = x + cos(a) * radius;
    float sy = y + sin(a) * radius;
    vertex(sx, sy);
  }
  endShape(CLOSE);
}

void drawCosmologyOverlay() {
  fill(15, 23, 42, 220);
  stroke(51, 65, 85);
  rect(20, 20, width - 40, 85, 6);
  
  fill(52, 211, 153);
  textSize(13);
  textAlign(LEFT, BASELINE);
  text("DTIM-ZEC QUANTENGRAVITATION // BAND 4 OBJEKT 014: SWAPPING & QUANTUM NETWORKS", 35, 42);
  
  fill(226, 232, 240);
  textSize(11);
  text("KLASSIFIKATION: " + modeNames[mode], 35, 62);
  
  if (mode == 0) {
    fill(52, 211, 153);
    text("SWAPPING: BSM an (2-3) projiziert instantan einen verschränkten Zustand |Psi^->_{14} und formt ein ER=EPR Wurmloch", 35, 82);
  } else if (mode == 1) {
    fill(34, 211, 238);
    text("NETZWERK: Die räumliche Geometrie entsteht dynamisch aus dem Routing der Quanten-Verschränkungs-Links", 35, 82);
  } else if (mode == 2) {
    fill(250, 204, 21);
    text("RYU-TAKAYANAGI: Die Entanglement-Entropie S(A) misst exakt die minimale Schnittfläche gamma_A im MERA-Netzwerk", 35, 82);
  } else {
    fill(244, 114, 182);
    text("DTIM-ZEC POSTULAT: Swapping verändert das Gitter-Routing ohne Verletzung von Tr(rho_0 H) = 0", 35, 82);
  }
}