# Adaptive Bi-Coupled Coherence Recovery (ABCR) Algorithm
## Mathematical Formulation and Algorithmic Specification

### 1. Core State Variables

**Coherence State Vector**:
```
κ(t) = [κ_δ(t), κ_θ(t), κ_α(t), κ_β(t), κ_γ(t)]
```

**Phase State Vector**:
```
φ(t) = [φ_δ(t), φ_θ(t), φ_α(t), φ_β(t), φ_γ(t)]
```

**Invariant Field** (long-term memory):
```
Π = [Π_δ, Π_θ, Π_α, Π_β, Π_γ]
```

### 2. Dual-Stream Architecture

The system maintains two complementary processing streams:

**Stream A (Excitatory)**: Detects hypo-coherence
```
S_A(κ) = {b ∈ Bands | κ_b < τ_low}
```

**Stream B (Inhibitory)**: Detects hyper-coherence  
```
S_B(κ) = {b ∈ Bands | κ_b > τ_high}
```

### 3. Adaptive Threshold System

**Context-Aware Thresholds**:
```
τ_low(t) = τ_base * (1 - σ(t))
τ_high(t) = 1 - τ_base * (1 - σ(t))
```

Where σ(t) is the system stress indicator:
```
σ(t) = min(1, Σ|dκ_b/dt| / Σκ_b)
```

**Adaptive Alpha (Renewal Rate)**:
```
α(t) = α_base + α_mod * σ(t)
α_base ∈ [0.5, 0.65]
α_mod ∈ [0, 0.15]
```

### 4. Spatial Encoding with Bi-Coupling

**Forward Capsule** (standard encoding):
```
C_F[m,n,b] = G_r(m,n) * κ_b * exp(iφ_b - ik_b*r)
```

**Mirror Capsule** (complementary encoding):
```
C_M[m,n,b] = G_r(m,n) * (1-κ_b) * exp(i(π-φ_b) + ik_b*r)
```

Where:
- `G_r(m,n) = exp(-r/r_0)` is the spatial envelope
- `k_b = 2πf_b/c` is the wave number
- `r = √(m²+n²) * Δx` is the radius

### 5. Broken Chain Detection Algorithm

```python
FUNCTION DetectBrokenChains(κ, C_F, C_M):
    broken_A = []  # Hypo-coherent chains
    broken_B = []  # Hyper-coherent chains
    intact = {}
    
    FOR each band b:
        κ_b = κ[b]
        
        # Dual-mode detection
        IF κ_b < τ_low:
            phase_coherence = ComputePhaseCoherence(C_F[b])
            IF phase_coherence < τ_phase:
                broken_A.append(ChainComponent(b, κ_b, "hypo"))
        
        ELIF κ_b > τ_high:
            phase_coherence = ComputePhaseCoherence(C_M[b])
            IF phase_coherence < τ_phase:
                broken_B.append(ChainComponent(b, κ_b, "hyper"))
        
        ELSE:
            intact[b] = κ_b
    
    RETURN broken_A, broken_B, intact
```

### 6. Bi-Coupled Reconstruction

**Hamiltonian for Reconstruction**:

For hypo-coherent chains (Stream A):
```
H_A[b] = h_b^F + Σ_j J_{bj}^F * κ_j + λ * h_b^M
```

For hyper-coherent chains (Stream B):
```
H_B[b] = h_b^M + Σ_j J_{bj}^M * (1-κ_j) + λ * h_b^F
```

Where:
- `h_b^F` = forward capsule field bias
- `h_b^M` = mirror capsule field bias  
- `J_{bj}` = coupling strength between bands
- `λ` = cross-stream coupling coefficient

**Iterative Reconstruction**:
```python
FUNCTION ReconstructChains(broken_A, broken_B, intact, H_A, H_B):
    κ_recon = intact.copy()
    
    FOR iteration in 1..MAX_ITER:
        converged = True
        
        # Reconstruct hypo-coherent chains
        FOR chain in broken_A:
            field = H_A[chain.band]
            κ_new = sigmoid(field)
            
            IF |κ_new - κ_recon[chain.band]| > ε:
                converged = False
            κ_recon[chain.band] = κ_new
        
        # Reconstruct hyper-coherent chains
        FOR chain in broken_B:
            field = H_B[chain.band]
            κ_new = 1 - sigmoid(field)
            
            IF |κ_new - κ_recon[chain.band]| > ε:
                converged = False
            κ_recon[chain.band] = κ_new
        
        IF converged:
            BREAK
    
    RETURN κ_recon
```

### 7. Integrity Audit with Bi-Stream Validation

**Dual-Stream Residuals**:
```
s_A = R*τ_R - (Δκ_A + D_ω^A + D_C^A)  # Stream A residual
s_B = R*τ_R - (Δκ_B + D_ω^B + D_C^B)  # Stream B residual
```

**Composite Audit Score**:
```
s_composite = w_A * s_A + w_B * s_B
```

Where weights depend on which streams were active:
```
w_A = |broken_A| / (|broken_A| + |broken_B|)
w_B = |broken_B| / (|broken_A| + |broken_B|)
```

**Seam Classification**:
```
IF |s_composite| < ε_audit AND |Δκ_avg| < ε_type1:
    seam = TYPE_I  # Perfect recovery
ELIF |s_composite| < ε_audit:
    seam = TYPE_II  # Acceptable loss
ELSE:
    seam = TYPE_III  # Failed recovery
```

### 8. Cognitive Renewal with Mode Selection

**Mode-Aware Renewal**:
```python
FUNCTION AdaptiveRenewal(κ_fragmented, Π, mode):
    IF mode == "conservative":
        ρ = 0.8  # High reliance on invariant
        novelty = 0.05
        baseline = 0.5
    
    ELIF mode == "moderate":
        ρ = 0.7
        novelty = 0.1
        baseline = 0.6
    
    ELIF mode == "aggressive":
        ρ = 0.6
        novelty = 0.15
        baseline = 0.65
    
    FOR each band b:
        ξ = Normal(0, novelty)
        κ_renewed[b] = ρ*Π[b] + (1-ρ)*baseline + ξ
        κ_renewed[b] = clip(κ_renewed[b], 0, 1)
    
    RETURN κ_renewed
```

### 9. Main Algorithm Flow

```python
ALGORITHM AdaptiveBiCoupledCoherenceRecovery:
    
    INPUT: κ(t), φ(t), Π, mode, context
    OUTPUT: κ_recovered(t+1)
    
    # Step 1: Compute adaptive thresholds
    σ = ComputeSystemStress(κ, dκ/dt)
    τ_low, τ_high = ComputeAdaptiveThresholds(σ, context)
    α = ComputeAdaptiveAlpha(σ, mode)
    
    # Step 2: Encode dual capsules
    C_F = EncodeForwardCapsule(κ, φ)
    C_M = EncodeMirrorCapsule(κ, φ)
    
    # Step 3: Detect broken chains in both streams
    broken_A, broken_B, intact = DetectBrokenChains(κ, C_F, C_M)
    
    # Step 4: Check for emergency conditions
    IF EmergencyCondition(broken_A, broken_B):
        RETURN EmergencyDecouple()
    
    # Step 5: Compute bi-coupled Hamiltonians
    H_A = ComputeStreamAHamiltonian(broken_A, intact, C_F, C_M)
    H_B = ComputeStreamBHamiltonian(broken_B, intact, C_F, C_M)
    
    # Step 6: Reconstruct broken chains
    κ_recon = ReconstructChains(broken_A, broken_B, intact, H_A, H_B)
    
    # Step 7: Perform integrity audit
    audit = BiStreamAudit(κ, κ_recon, broken_A, broken_B)
    
    # Step 8: Apply renewal if audit passes
    IF audit.pass:
        κ_final = AdaptiveRenewal(κ_recon, Π, mode)
        UpdateInvariantField(Π, κ_final)
        RETURN κ_final
    ELSE:
        RETURN FallbackRecovery(κ, Π)
```

### 10. System Modes

The algorithm supports multiple operational modes:

**1. Standard Mode**:
- Balanced thresholds
- Moderate alpha (0.6)
- Equal stream weighting

**2. High-Sensitivity Mode**:
- Lower τ_low (0.2)
- Higher τ_high (0.8)
- Faster response

**3. Stability Mode**:
- Conservative thresholds
- Lower alpha (0.5)
- Prioritize invariant field

**4. Recovery Mode**:
- Aggressive alpha (0.65)
- Relaxed thresholds
- Higher novelty injection

### 11. Key Innovations

1. **Dual-Stream Processing**: Simultaneous detection of hypo- and hyper-coherence
2. **Mirror Capsules**: Complementary encoding for robust reconstruction
3. **Adaptive Parameters**: Context-aware threshold and renewal rate adjustment
4. **Bi-Coupled Hamiltonians**: Cross-stream information sharing
5. **Composite Auditing**: Validation across both processing streams
6. **Mode Selection**: Configurable behavior for different applications

### 12. Computational Complexity

- **Spatial Encoding**: O(M*N*B) where M,N are grid dimensions, B is number of bands
- **Chain Detection**: O(B*P) where P is positions per band
- **Reconstruction**: O(I*B²) where I is iterations
- **Total**: O(M*N*B + I*B²) ≈ O(n²) for typical parameters

### 13. Convergence Guarantees

The algorithm converges under the following conditions:
1. |J_{ij}| < 1 for all coupling terms
2. Learning rate α ∈ (0, 1)
3. Sigmoid activation ensures bounded outputs
4. Maximum iterations prevent infinite loops

### 14. Applications Mapping

**Trading Systems**:
- Use high-sensitivity mode for volatile markets
- Hypo-coherence → missed opportunities
- Hyper-coherence → overtrading

**Mental Health Device**:
- Hypo-coherence → depression/dissociation
- Hyper-coherence → anxiety/mania
- Adaptive mode based on biometric feedback

**Consciousness Modeling**:
- Full dual-stream for complete state space
- Mirror capsules represent complementary awareness
- Invariant field as persistent self-model