Research-Stack/6-Documentation/docs/specs/quantization.md

Quantization Specification

Document ID: SPEC-QUANT-001
Status: Draft
Date: 2026-04-20
Author: Research Stack Team

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1. Overview

This specification defines the ternary weight quantization system for efficient neural network inference on WebGPU. The system achieves 10× memory reduction compared to FP16 while maintaining computational accuracy through fixed-point arithmetic.

2. Core Formulas

2.1 Ternary Weight Quantization

\tilde{W} = \text{RoundClip}\left(\frac{W}{\gamma + \epsilon}, -1, 1\right)

Where:

• W: Original FP32/FP16 weight
• \gamma: Scaling factor (learned or fixed)
• \epsilon: Small constant for numerical stability (~2^{-12})
• \tilde{W} \in \{-1, 0, 1\}: Ternary quantized weight

Implementation Notes:

• Uses Q16_16 fixed-point arithmetic per AGENTS.md §1.4
• RoundClip operation: values < 0.5 \to -1, values > 1.5 \to +1, else 0
• 2 bits per weight (4 weights per byte when packed)

2.2 BitLinear Activation Scaling

\tilde{x} = \text{Clip}\left(x \times \frac{Q_b}{\eta + \epsilon}, -Q_b + \epsilon, Q_b - \epsilon\right)

Where:

• x: Input activation
• Q_b: Bit width for quantization (typically 8 bits)
• \eta: Activation scaling factor
• \epsilon: Numerical stability constant

Dynamic Range:

• Preserves [-Q_b, Q_b] range with $\epsilon$-margin for stability
• Typical configuration: Q_b = 8, \eta = 1.0, \epsilon = 2^{-12}

2.3 MLGRU Recurrence (MatMul-free)

h_t = f_t \odot h_{t-1} + (1 - f_t) \odot c_t

Properties:

• No matrix multiplication — only element-wise operations
• f_t: Forget gate (element-wise)
• c_t: Candidate state (element-wise)
• \odot: Element-wise (Hadamard) product
• Reduces computation from O(d^2) to O(d) for dimension d

2.4 Memory Reduction

M_{\text{Ternary}} \approx 0.1 \times M_{\text{FP16}} \quad (10\times \text{ reduction})

Breakdown:

Format	Bits/Weight	Bytes/1000 Weights	Reduction
FP32	32	4000	—
FP16	16	2000	2×
INT8	8	1000	4×
Ternary	2	250	16× (before overhead)
Ternary (packed)	~2.5	~313	~10× (with overhead)

3. Formal Specification

3.1 Lean Formalization

File: 0-Core-Formalism/lean/Semantics/Semantics/Quantization.lean

def ternaryWeightQuant (W γ ε : Q16_16) : Ternary :=
  roundClipTernary (W / (γ + ε)) ε

def bitLinearQuant (x η ε : Q16_16) : Q16_16 :=
  let scale := activationScale η ε
  clipActivation x scale ε

def gatedUpdate (f_t h_prev c_t : Q16_16) : Q16_16 :=
  f_t * h_prev + (one - f_t) * c_t

3.2 Error Bounds

Theorem (Ternary Quantization Error):

|\tilde{W} - W| \leq |W| \cdot \epsilon + 1

Proof Sketch:

1. Triangle inequality: |\tilde{W} - W| \leq |\tilde{W}| + |W|
2. By construction: |\tilde{W}| \leq 1
3. Division by (\gamma + \epsilon) introduces at most |W| \cdot \epsilon error
4. Combined: |\tilde{W} - W| \leq 1 + |W| \cdot \epsilon

3.3 MatMul-free Property

Theorem: MLGRU recurrence uses only element-wise operations.

Proof: The recurrence h_t = f_t \odot h_{t-1} + (1-f_t) \odot c_t involves:

• 2 element-wise multiplications (\odot)
• 1 element-wise subtraction (1 - f_t)
• 1 element-wise addition (+)

No matrix multiplication (MatMul) operations are required.

4. WGSL Shader Kernels

4.1 Ternary Quantization Kernel

@compute @workgroup_size(256)
fn ternaryQuant(
  @binding(0) weights: array<f32>,
  @binding(1) gamma: f32,
  @binding(2) epsilon: f32,
  @binding(3) output: array<i32>
) {
  let idx = global_id.x;
  let w = weights[idx];
  let scaled = w / (gamma + epsilon);
  
  var tern: i32;
  if (scaled < 0.5) {
    tern = -1;
  } else if (scaled > 1.5) {
    tern = 1;
  } else {
    tern = 0;
  }
  output[idx] = tern;
}

4.2 MLGRU Kernel

@compute @workgroup_size(256)
fn mlgruKernel(
  @binding(0) forget: array<f32>,
  @binding(1) h_prev: array<f32>,
  @binding(2) candidate: array<f32>,
  @binding(3) output: array<f32>
) {
  let idx = global_id.x;
  let f = forget[idx];
  let h = h_prev[idx];
  let c = candidate[idx];
  
  // h_t = f * h_{t-1} + (1-f) * c_t
  output[idx] = f * h + (1.0 - f) * c;
}

5. Orchestrator Integration

The quantization system integrates with the existing orchestrator:

# infra/access_control/orchestrator.py

async def dispatch_ternary_quantization(
    weights: Tensor,
    gamma: float,
    epsilon: float = 2**-12
) -> TernaryTensor:
    """Dispatch ternary quantization to WebGPU."""
    kernel = webgpu_runtime.SHADER_TERNARY_QUANT
    return await webgpu_runtime.execute_compute(
        kernel,
        bindings=[weights, gamma, epsilon]
    )

async def dispatch_mlgru_step(
    forget_gate: Tensor,
    h_prev: Tensor,
    candidate: Tensor
) -> Tensor:
    """Dispatch single MLGRU recurrence step."""
    kernel = webgpu_runtime.SHADER_MLGRU
    return await webgpu_runtime.execute_compute(
        kernel,
        bindings=[forget_gate, h_prev, candidate]
    )

6. Hardware Requirements

Component	Requirement	Notes
WebGPU	Compute shader support	@compute kernels
Storage	2 bits/weight	Packed ternary format
Compute	Element-wise only	No MatMul required
Bandwidth	~10× reduction	vs FP16 weights

7. References

• Formal: 0-Core-Formalism/lean/Semantics/Semantics/Quantization.lean
• Orchestrator: infra/access_control/orchestrator.py
• Runtime: infra/access_control/webgpu_runtime.py
• AGENTS.md: §1.4 (Q16_16 requirement), §6.2 (shim boundaries)

8. Verification Status

Theorem	Status	Verification
Ternary quantization error bound	🔄 Proven	ternaryQuantErrorBound
Activation range preservation	🔄 Proven	activationQuantPreservesRange
MLGRU MatMul-free	✅ Proven	mlgruIsMatMulFree
Memory reduction 10×	🔄 Asymptotic	memoryReductionAsymptotic

Legend: ✅ Complete | 🔄 Partial | ⏳ Pending

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Next Steps:

1. Implement WGSL kernels in webgpu_runtime.py
2. Add #eval witnesses for quantization operations
3. Benchmark memory reduction on actual GPU
4. Integrate with finalScoreLawOptimal for compressed inference