Research-Stack/6-Documentation/docs/research/COLD_REVIEW_FORMULA_CATALOG.md

Cold Review Formula Catalog

Standalone mathematical formulas for independent verification

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📚 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

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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

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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

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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

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📊 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

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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

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🔧 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

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9. Acoustic Impedance

Mathematical Expression:

Z = |∇f|

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

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📈 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