# Cold Review Formula Catalog
*Standalone mathematical formulas for independent verification*

---

## 📚 **SLUQ State Machine (SLUQ.lean)**

### **1. State Evaluation Function**
**Mathematical Expression:**
```
state(acc) = 
  Stable    if acc ∈ [0, 0x4000)
  Rising    if acc ∈ [0x4000, 0x8000)  
  Unstable  if acc ∈ [0x8000, 0xC000)
  Reset     if acc ∈ [0xC000, 0xFFFF]
```

**Implementation:** `evaluateState` in SLUQ.lean:51  
**Domain:** State Machines, Control Theory  
**Validation:** ✅ Mathematically verified

---

### **2. State Transition Dynamics**
**Mathematical Expression:**
```
a_{t+1} = a_t + (value × phi)
```

**Implementation:** `updateNode` in SLUQ.lean:68  
**Where:**
- `a_t` = current accumulator (UInt16)
- `value` = input value (UInt8)  
- `phi` = learning rate parameter (UInt8)
- `a_{t+1}` = new accumulator

**Special Case:**
```
If state(a_{t+1}) = Reset:
  a_{t+1} = 0
  selectionCount += 1
```

**Domain:** State Machines, Reinforcement Learning  
**Validation:** ✅ Computationally tested

---

### **3. Q16.16 Fixed-Point Conversion** ⚠️ **CRITICAL ISSUE**
**Mathematical Expression:**
```
tempQ16(acc) = (acc << 16) as UInt32
```

**Implementation:** `tempQ16` in SLUQ.lean:80  
**Issue Found:** Current implementation `acc.toUInt32` is **WRONG**  
**Correct Implementation:** `acc.toUInt32 << 16`  
**Impact:** Numerical values off by factor of 65536  
**Domain:** Fixed-Point Arithmetic  
**Validation:** ❌ **BROKEN - Requires immediate fix**

---

### **4. Cost Function**
**Mathematical Expression:**
```
cost(nodeA, nodeB) = |nodeA.acc - nodeB.acc|
```

**Implementation:** `sluqCost` in SLUQ.lean:89  
**Returns:** Q16_16 fixed-point value  
**Domain:** Optimization, Metric Spaces  
**Validation:** ✅ Mathematically verified

---

## 📊 **DSP Signal Processing (flac_dsp_node.py)**

### **5. Fast Fourier Transform**
**Mathematical Expression:**
```
X(k) = Σ_{n=0}^{N-1} x(n) · e^{-j(2πkn/N)}  for k = 0,1,...,N-1
```

**Implementation:** `process_flac_chunk` in flac_dsp_node.py:206  
**Parameters:**
- N = 4096 (FFT size)
- window = Hanning window
- hop = 2048 (50% overlap)

**Domain:** Signal Processing, Frequency Analysis  
**Validation:** ✅ Computationally verified

---

### **6. Spectral Centroid**
**Mathematical Expression:**
```
C = Σ(f × |S(f)|) / Σ|S(f)|
```

**Implementation:** `process_flac_chunk` in flac_dsp_node.py:222  
**Where:**
- f = frequency
- S(f) = magnitude spectrum
- C = spectral centroid in Hz

**Domain:** Audio Analysis, Psychoacoustics  
**Validation:** ✅ Mathematically verified

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### **7. Root Mean Square (RMS)**
**Mathematical Expression:**
```
RMS = √(1/N Σ_{n=0}^{N-1} x²(n))
```

**Implementation:** `process_flac_chunk` in flac_dsp_node.py:226  
**Converted to dB:** `dB = 20 × log10(RMS + ε)`  
**Domain:** Audio Level Measurement  
**Validation:** ✅ Computationally verified

---

## 🔧 **MorphicDSP (MorphicDSP.lean)**

### **8. OEPI Weighted Sum**
**Mathematical Expression:**
```
output = Σ(w_i × input_i) + bias
```

**Domain:** Adaptive Filtering, Neural Networks  
**Status:** Documented in module header  
**Validation:** Pending

---

### **9. Acoustic Impedance**
**Mathematical Expression:**
```
Z = |∇f|
```

**Domain:** Signal Processing, Physics  
**Status:** Documented in module header  
**Validation:** Pending

---

## 📈 **Cross-Domain Mathematical Models**

### **10. Constraint Satisfaction**
**Mathematical Framework:**
```
Find basis A ⊆ ℕ such that:
- Order r: every large integer is sum of ≤ r elements from A
- Exact order k: every large integer is sum of = k elements from A
```

**Application:** Erdős #336 problem  
**Domain:** Number Theory, Additive Basis  
**Status:** Referenced in Research_enhancments.md  
**Validation:** Pending

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## 📋 **Validation Status Summary** *(Updated 2026-06-28)*

| Formula ID | Status | Confidence | Review Needed |
|------------|--------|------------|---------------|
| 🔴 3 | Q16.16 Conversion | **CRITICAL ERROR** | **Immediate Fix** |
| ⚠️ 2 | State Transition | High | Add overflow handling |
| ⚠️ 8 | OEPI Weighted Sum | Medium | Documentation update |
| ✅ 1,4,5,6,7 | Verified | High | None |
| 🟡 9,10 | Pending | Medium | Detailed verification |

**🚨 CRITICAL FINDING:** Formula 3 requires immediate correction from `acc.toUInt32` to `acc.toUInt32 << 16`

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## 🎯 **Cold Review Instructions**

1. **Verify Mathematical Correctness** - Check all formulas against literature
2. **Validate Implementations** - Confirm code matches mathematical expressions
3. **Cross-Domain Testing** - Test applicability across domains
4. **Report Discrepancies** - Flag any mismatches or errors

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## 📝 **References**

- SLUQ.lean: `/home/allaun/Research Stack/0-Core-Formalism/lean/Semantics/Semantics/SLUQ.lean`
- flac_dsp_node.py: `/home/allaun/Research Stack/4-Infrastructure/shim/flac_dsp_node.py`
- MorphicDSP.lean: `/home/allaun/Research Stack/0-Core-Formalism/lean/Semantics/Semantics/MorphicDSP.lean`
- Research_enhancments.md: `/home/allaun/Research Stack/6-Documentation/docs/research/Research_enhancments.md`

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**Catalog Generated:** 2026-06-28  
**Review Status:** Ready for Cold Review  
**Confidence Level:** High