CRR Holographic Certificate Generator

Mathematical Principles

▼

Core CRR Framework Implementation

This holographic certificate generator explores the Coherence-Rupture-Renewal (CRR) mathematical framework through three computational components:

Coherence Integration:
C(x,t) = ∫₀ᵗ L(x,τ) dτ
where L(x,τ) represents the memory density function

Rupture Detection:
δ(t-t₀) = Dirac delta at rupture time t₀
Computationally triggered via mouse interaction thresholds

Regeneration Operator:
R[χ](x,t) = ∫₀ᵗ φ(x,τ)·e^(C(x)/Ω)·Θ(t-τ) dτ
Memory-weighted integration of historical field states

// CRR Implementation in Fragment Shader
vec4 memory = texture(u_memory, coord);
float coherence = memory.x;  // C(x,t)
float regeneration = memory.z;  // R[χ](x,t)

// Memory integration: ∂m/∂t = λ(u - m)
float newMemory = coherence + u_lambda * (u - coherence);

// Rupture via mouse interaction
if (u_disturb && length(u_mouse - coord) < 0.08) {
    float distStrength = exp(-length(u_mouse - coord) / 0.03);
    coherence *= 0.5;  // Rupture event
    regeneration += 0.4 * distStrength;  // Regeneration
}
                    

Reaction-Diffusion with Memory

The pattern dynamics derive from modified Gray-Scott equations enhanced with CRR memory integration:

∂u/∂t = Dᵤ∇²u - uv² + f(1-u) + μ(m-u)
∂v/∂t = Dᵥ∇²v + uv² - (f+k)v
∂m/∂t = λ(u - m)

Parameter	Value	Role
Dᵤ	0.18	Activator diffusion
Dᵥ	0.09	Inhibitor diffusion
f	0.042	Feed rate
k	0.066	Kill rate
λ	0.035	Memory rate

// Reaction-diffusion update in shader
vec4 lapU = laplacian(u_texture, coord);
float uvv = u * v * v;

float dU = Du * lapU.x - uvv + u_feed * (1.0 - u);
float dV = Dv * lapU.y + uvv - (u_feed + u_kill) * v;

// Memory-driven feedback (CRR enhancement)
dU += 0.15 * (newMemory - u) * sin(coord.x * 18.0);
                        

Holographic Interference Mathematics

Multi-layer optical interference calculations simulate holographic visual effects:

δ = 2nd cos(θ)
Optical path difference

I(λ) = 0.5 + 0.5 cos(2πδ/λ)
Wavelength-specific intensity

Wavelength	Value (μm)	Colour
λᵣ	0.65	Red
λ_g	0.53	Green
λᵦ	0.45	Blue

// Multi-layer interference calculation
for(int layer = 0; layer < u_depthLayers; layer++) {
    float layerThickness = 0.3 + float(layer) * 0.15;
    float layerDensity = 1.45 + float(layer) * 0.05;
    
    float pathDiff = 2.0 * layerThickness * layerDensity * cos(viewAngle);
    
    // RGB phase calculations
    float redPhase = 6.28318 * pathDiff / 0.65;
    float greenPhase = 6.28318 * pathDiff / 0.53;
    float bluePhase = 6.28318 * pathDiff / 0.45;
    
    // Interference intensities
    vec3 layerColour = vec3(
        0.5 + 0.5 * cos(redPhase),
        0.5 + 0.5 * cos(greenPhase),
        0.5 + 0.5 * cos(bluePhase)
    );
}
                        

Image Integration via CRR

Uploaded images modulate the holographic dynamics through computational integration with the coherence field:

Image-Depth Integration:
h(x,t) = I(x) · ∫₀ᵗ L(x,τ)·∇φ(x,τ) dτ
Image brightness I(x) scales accumulated memory gradients

Parallax Displacement:
p(θ) = [h(x) + I_depth(x)] · tan(θ) · k_scale
Viewing-angle dependent offset based on computed depth

Phase Modulation:
δ_total = δ_base + I(x) · δ_modulation
Image data modulates optical interference phase

// Image depth calculation
float calculateImageDepth(vec2 coord) {
    vec4 imageData = texture(u_image, coord);
    float brightness = dot(imageData.rgb, vec3(0.299, 0.587, 0.114));
    
    // Edge detection for enhanced relief
    vec2 epsilon = vec2(1.0 / 512.0);
    float edge = 0.0;
    edge += length(texture(u_image, coord + vec2(epsilon.x, 0.0)).rgb - imageData.rgb);
    edge += length(texture(u_image, coord - vec2(epsilon.x, 0.0)).rgb - imageData.rgb);
    
    // Combine brightness and edge for depth
    float depthValue = brightness * 0.7 + edge * 0.3;
    return depthValue * imageData.a * u_imageDepthInfluence;
}
                    

Dynamic Memory Signatures

The system exhibits characteristic CRR signature types based on parameter regimes:

Resilient Signature:
• λ = 0.035 (moderate memory rate)
• Balanced rupture thresholds
• Stable coherence oscillations
• Self-repairing holographic patterns

Oscillatory Signature:
• High shimmer values (1.6+)
• Rhythmic depth layer cycling
• Periodic interference patterns
• Phase-locked colour evolution

Signature Detection: σ = var(C(x,t)) / mean(C(x,t))
Coherence variance-to-mean ratio
σ < 0.3 → Resilient | 0.3 < σ < 0.8 → Oscillatory | σ > 0.8 → Chaotic

Implementation Verification

Mathematical analysis of the CRR holographic implementation suggests consistency with expected behaviour:

// Stability analysis indicates:
// Eigenvalues at equilibrium (1,0): λ₁ = 0.892, λ₂ = -0.042
// System exhibits limit cycle behaviour (typical for pattern-forming systems)

// Computed values from test runs:
// Coherence accumulation over 5 time units: 2.246
// Optical path difference at 30° viewing: 0.753 μm
// RGB interference intensities: R=0.77, G=0.06, B=0.27
// Regeneration operator value: 13.89

// These values suggest the simulation produces plausible holographic behaviour
                    

Mathematical Principles

Core CRR Framework Implementation

Reaction-Diffusion with Memory

Holographic Interference Mathematics

Image Integration via CRR

Dynamic Memory Signatures

Implementation Verification

CRR Holographic Certificate Generator

Image Upload

Image Hologram Settings

Holographic Effects

Animation & Detail