🐚 NautilusQuant

Deterministic Orthogonal KV-Cache Quantization via Golden Ratio Geometry

Can nature's most irrational number beat Google's random matrices
at compressing the memory of neural networks?

Hypothesis • How It Works • Results • Quick Start • Architecture • Benchmarks • Roadmap

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The Problem

Large Language Models are memory-bound, not compute-bound. During text generation, up to 80% of GPU memory is consumed by the KV-cache (Key-Value cache) — the attention mechanism's working memory. For a 7B model at 128K context, this can reach 64 GB in FP16.

State-of-the-art compression (Google's TurboQuant, ICLR 2026) uses random orthogonal rotation followed by polar quantization to compress the KV-cache to 3 bits. It works — but the rotation angles are random, requiring a PRNG seed and offering no mathematical guarantees about angular distribution quality.

We ask: What if we replace random chaos with nature's optimal geometry?

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Core Hypothesis

NautilusQuant replaces random rotation with a deterministic orthogonal matrix whose angles follow the golden ratio (φ ≈ 1.618) and π:

θₖ = (2π / φ²) × (k + 1) ≈ 137.507764° × (k + 1)

This is the golden angle — the same angle that governs:

• 🌻 Seed arrangement in sunflowers
• 🐚 The spiral of the Nautilus shell
• 🌀 Phyllotaxis patterns in plants

The golden angle is provably the most irrational angle: its continued fraction [1; 1, 1, 1, ...] converges slower than any other, guaranteeing that no two points ever cluster and no angular region is left empty.

Why This Might Work

Property	Random Rotation (TurboQuant)	Golden Rotation (NautilusQuant)
Deterministic	No (seed-dependent)	Yes (φ and π are constants)
Angular uniformity	O(1/√N) statistical	O(1/N) mathematical guarantee
Reproducibility	Depends on PRNG state	100% identical results always
LUT precomputation	Not possible	64 cos/sin values = 512 bytes
Overhead storage	Seed required	Zero (angles computed from formula)

Critical Constraint: Orthogonality

The rotation matrix must be orthogonal (T^T · T = I) to preserve:

• Vector norms: ‖Tv‖ = ‖v‖
• Dot products: ⟨Tq, Tk⟩ = ⟨q, k⟩
• Attention scores: unchanged after transformation

v1 Bug (Fixed): Our initial design included centripetal scaling (φ^(-i/d)) which broke orthogonality and corrupted attention. v2 uses pure Givens rotations — no scaling, no norm changes.

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How It Works

Pipeline: 5 Stages

┌──────────┐   ┌───────────┐   ┌──────────┐   ┌───────────┐   ┌──────────┐
│ 1. Input │──▶│ 2. Rotate │──▶│ 3. Polar │──▶│ 4. Quant  │──▶│ 5. QJL   │
│   FP16   │   │  Golden φ │   │  (r, θ)  │   │ Lloyd-Max │   │  ±1 bit  │
│  16 bit  │   │  T^T·T=I  │   │          │   │   3 bit   │   │  1 bit   │
└──────────┘   └───────────┘   └──────────┘   └───────────┘   └──────────┘
     HBM ────────────── SRAM (fused, single pass) ──────────────▶ HBM

Stage 2: Orthogonal Spiral Matrix (The Core Innovation)

Three layers of non-overlapping Givens rotations:

# Layer 1: Adjacent pairs — golden angle × (k+1)
for k in range(dim // 2):
    θ = GOLDEN_ANGLE * (k + 1)                    # ≈ 137.5° × (k+1)
    givens_rotate(v, 2*k, 2*k+1, θ)

# Layer 2: Shifted pairs — golden angle × (k+1) × φ
for k in range((dim - 1) // 2):
    θ = GOLDEN_ANGLE * (k + 1) * φ
    givens_rotate(v, 2*k+1, 2*k+2, θ)

# Layer 3: Butterfly pairs — golden angle × (k+1) × φ²
# Non-overlapping pairs enforced via used-index tracking

Orthogonality proof:

• Each Givens rotation G(i,j,θ) is orthogonal by construction
• Pairs within each layer are non-overlapping → layer is orthogonal
• Product of orthogonal matrices is orthogonal: T = L₃·L₂·L₁, T^T·T = I ✓

Stage 5: Inverse Transform (Decode)

T⁻¹ = L₁^T · L₂^T · L₃^T  # Reverse layer order, negate angles

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Key Results

⚠️ Research Preview: These are preliminary results on synthetic data with realistic outlier distributions. Real KV-cache validation on Gemma 3 4B is the next milestone.

Orthogonality Verification

Norm preservation error:    < 1e-14
Roundtrip error (T⁻¹·T·v): < 1e-14
Dot product error:          < 1e-14

Compression

Config	Bits/value	Compression	Overhead
FP16 baseline	16	1.0x	—
KIVI (no rotation)	2	2.6x	32 bit/group
TurboQuant (random)	3+1	4.0x	32 bit/group
NautilusQuant (φ)	3+1	4.0x	32 bit/group ¹

¹ Overhead is currently identical to TurboQuant. Achieving 0 overhead requires empirical proof that golden angles produce a predictable enough distribution to eliminate scale/zero-point constants. This is the primary open research question.

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Quick Start

Interactive 3D Visualization (No Install Required)

Open index.html in any browser to explore:

• 3D Simulation — animated 5-stage pipeline, TurboQuant vs NautilusQuant side-by-side
• Formula Lab — edit mathematical formulas live, see results instantly
• MiroFish Lab — AI agents discuss optimal quantization strategies

Validation Scripts

# Install dependencies
pip install torch numpy

# Phase 2: Synthetic validation with realistic outliers (±60 in 6 dimensions)
python validate_real_kv.py --sweep --dim 128 --count 500

# Phase 2: Real KV-cache from Gemma 3
pip install transformers accelerate
python validate_real_kv.py --model google/gemma-3-4b-it --sweep

# Phase 3: GPU kernel benchmark (Triton)
pip install triton
python nautilus_triton.py --dim 128 --n 10000

# Phase 4: All hardware co-design concepts
python nautilus_hardware.py

# Phase 4: Needle-in-a-Haystack (104K tokens)
python benchmark_needle.py --model google/gemma-3-4b-it --method both

# Vector search: GloVe recall@k (no torch needed — pure numpy)
python benchmark_glove.py --profile

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Architecture

Project Structure

NautilusQuant/
├── index.html              # Project portal (browser)
├── quantsim3d.html         # 3D pipeline visualization
├── formula_lab.html        # Live formula editor + data inspector
├── mirofish_lab.html       # Multi-agent discussion (MiroFish)
├── compute_worker.js       # WebWorker compute engine
│
├── validate_real_kv.py     # Phase 2: KV-cache extraction + comparison
├── nautilus_triton.py      # Phase 3: Triton GPU kernel + PyTorch ref
├── nautilus_hardware.py    # Hardware co-design (SRAM, Dataflow, MX, Sub-1-bit)
├── benchmark_needle.py     # Phase 4: Needle-in-a-Haystack + accuracy
├── benchmark_glove.py      # GloVe vector search + KIVI baseline
│
├── NautilusQuant_Model.md  # Full theoretical model (v2)
├── goldt.txt               # Source research on TurboQuant
└── .gitignore

Hardware Co-design Concepts

Concept	Description	Implementation
SRAM-Fused Kernel	All 5 stages in single SRAM pass. No HBM round-trips. ~2.5x energy savings.	nautilus_hardware.py → NautilusFusedKernel
Deterministic Dataflow	Static execution schedule. Zero branches. Compatible with Groq LPU, TPU, Cerebras.	nautilus_hardware.py → NautilusDataflow
MX-Format Fallback	If 0-overhead fails → MXFP4 block scaling (0.25 bit/value overhead). OCP standard.	nautilus_hardware.py → NautilusWithMX
Sub-1-bit + Multimodal	Separate bit allocation for radius vs angle. Adaptive compression per modality.	nautilus_hardware.py → SubBitExperiment

Supported Hardware Targets

Platform	Optimization	Status
NVIDIA H100/H200/B200	Triton kernel + TMEM	🟡 Prototype
NVIDIA RTX 5080	Triton kernel + shared memory	🟡 Prototype
Google TPU v5	OpenXLA + VMEM	📋 Planned
Groq LPU	Static dataflow schedule	📋 Planned
Apple Silicon (M1-M4)	MLX framework	📋 Planned
llama.cpp / GGUF	Custom quant type Q3_PHI	📋 Planned

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Benchmarks

Vector Search (GloVe d=200)

python benchmark_glove.py --profile

Compares: FP16 (baseline) → KIVI (2-bit, no rotation) → TurboQuant (random rotation) → NautilusQuant (golden rotation)

Metrics: Recall@10, MSE, indexing time, memory savings, energy estimation.

Needle-in-a-Haystack

python benchmark_needle.py --model google/gemma-3-4b-it --method both --haystack 4096

Hides a fact in long context. Tests if the model can retrieve it after KV-cache quantization.

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Risk Analysis

Three fundamental things that can break. Full analysis: RISKS.md

Risk	What breaks	Mitigation	File
Structural resonance	Golden angles align with outlier dims → MSE explodes	Add fixed permutation layer before rotation	validate_real_kv.py
0-overhead failure	Angle distribution not predictable enough → need scale/zp	MX-Format fallback (0.25 bit/val overhead)	nautilus_hardware.py
FP16 drift	Roundtrip errors accumulate on 104K contexts	Kahan summation / periodic renormalization	validate_real_kv.py

Worst case: Even if ALL 3 risks materialize, NautilusQuant still wins on: determinism, dataflow-compatibility (Groq/Cerebras), 512-byte LUT, and full auditability.

Plan B: Experimental Modules (plan_b/)

#	Module	Replaces	Potential gain
1	quasicrystal.py	Lloyd-Max (Stage 4)	2-bit quality at 3-bit budget
2	golden_jl.py	QJL (Stage 5)	Kill QJL → 5.3x instead of 4x
3	phinary.py	Scalar quant	φ-base covers outliers naturally
4	fractal_hash.py	Angle encoding	Sub-1-bit per angle
5	groq_dataflow.py	Runtime scheduler	Static LUT for Groq/Cerebras/TPU
6	multimodal_spiral.py	Fixed φ	Adaptive silver/bronze ratio per modality

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The Open Question

The central research question remains empirically unresolved:

Does the golden angle (137.5°) produce a more concentrated angular distribution than random angles after polar decomposition of real KV-cache tensors?

If yes → NautilusQuant can achieve 0-overhead quantization (no scale/zero-point storage), making it fundamentally more efficient than TurboQuant at extreme compression (2-3 bits).

If no → NautilusQuant still offers: determinism, reproducibility, precomputable LUT, and compatibility with static dataflow architectures (Groq, Cerebras). But the compression ratio matches, not exceeds, TurboQuant.

How to Test This

# Run this on a machine with GPU + transformers installed:
python validate_real_kv.py --model google/gemma-3-4b-it --sweep

# Look for:
#   Angle variance (Nautilus) < Angle variance (Turbo)  → φ wins
#   MSE (Nautilus) < MSE (Turbo)                        → φ wins

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Roadmap

• Phase 1: Fix linear algebra (orthogonal matrix, inverse transform, butterfly pairs)
• Phase 2: Validation framework (synthetic outliers, real KV-cache extraction)
• Phase 3: Triton GPU kernel + hardware co-design
• Phase 4: Benchmark suite (Needle-in-a-Haystack, GloVe, KIVI baseline)
• Phase 5: Run validation on real Gemma 3 4B KV-cache tensors
• Phase 6: Prove/disprove 0-overhead hypothesis with empirical data
• Phase 7: Optimize Triton kernel for RTX 5080 (target: < 0.0013s indexing)
• Phase 8: Integration with llama.cpp (GGUF format Q3_PHI)
• Phase 9: Paper submission

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Related Work

Method	Year	Approach	Bits	Paper
GPTQ	2022	Layer-wise Hessian quantization	4	arXiv:2210.17323
AWQ	2023	Activation-aware weight protection	4	arXiv:2306.00978
QuIP#	2023	Hadamard rotation + E8 lattice codebooks	2	arXiv:2402.04396
SqueezeLLM	2023	Dense-and-sparse quantization	3-4	arXiv:2306.07629
KIVI	2024	Per-channel KV-cache quantization	2	—
BitNet b1.58	2024	Ternary weights (-1, 0, +1) from training	1.58	arXiv:2402.17764
TurboQuant	2025	Random rotation + PolarQuant + QJL	3	arXiv:2504.19874
NautilusQuant	2026	Golden ratio rotation + PolarQuant + QJL	3	This work

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Industrial Applications

While NautilusQuant targets LLM inference, the core technology — deterministic signal compression with mathematical guarantees — transfers directly to industrial edge computing.

Maritime and Offshore IoT

Ship power plants generate thousands of sensor readings per second (RPM, exhaust temperature, fuel pressure, lube oil quality, vibration spectra). Satellite bandwidth between vessel and shore is typically 64–512 kbps. 4x deterministic compression enables real-time shore-side condition monitoring without saturating the link.

Challenge	How NautilusQuant Helps
Limited satellite bandwidth	4–5.3x compression reduces data volume to fit within VSAT/Iridium constraints
Edge inference on shipboard hardware	KV-cache quantization enables larger ML models on embedded GPUs (Jetson, Movidius)
Safety-critical determinism	No PRNG seed — IMO/SOLAS auditors can verify identical results every time
Resource-constrained devices	512-byte LUT fits in any PLC/FPGA register file
Real-time processing	SRAM-fused pipeline achieves sub-millisecond latency for sensor stream compression

Hardware Fit

• 512-byte LUT — precomputed cos/sin table stored in constant memory. Compatible with marine-grade embedded controllers.
• SRAM-fused pipeline — single-pass encode/decode eliminates memory bottleneck on constrained devices.
• Static dataflow — no branches, no PRNG, no runtime decisions. Suitable for safety-critical real-time systems with deterministic scheduling requirements.
• ONNX-exportable — quantized inference pipeline can be deployed alongside predictive maintenance models on shipboard edge nodes.

Author Background

The author brings 3+ years of hands-on marine engineering experience — ship power plant maintenance, diesel engine diagnostics, and propulsion system overhaul — bridging the gap between physical machinery domain knowledge and modern ML/signal processing research.

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Citation

@software{nautilusquant2026,
  author    = {ORTODOX1},
  title     = {NautilusQuant: Deterministic Orthogonal KV-Cache Quantization via Golden Ratio Geometry},
  year      = {2026},
  url       = {https://github.com/ORTODOX1/NautilusQuant},
  note      = {Research Preview}
}

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φ = 1.6180339887...
The most irrational number meets the most memory-hungry algorithm.