Energy Efficiency: 10 Mathematical Techniques for 60-70% AI Energy Reduction (Phi6Simple, FFT-Mix, Phi MoE)

リンクされている TECS-L docs/energy-efficiency.md と、math/experiments および experiments にある参照スクリプトから始め、提案されている手法を megablocks の既存アーキテクチャと比較します。この issue では megablocks のファイル、テスト、具体的な変更が特定されていないため、完了基準は未定義のままです。

AI Energy Efficiency: 10 Mathematical Techniques for 60-70% Energy Reduction

TECS-L Research Group | 2026-03-27 (Updated)
Full documentation: github.com/need-singularity/TECS-L/docs/energy-efficiency.md

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Executive Summary

We discovered ten techniques for reducing AI model energy consumption, derived from the mathematical properties of the number 6 (the smallest perfect number). All are empirically validated with reproducible code.

#	Discovery	Energy Saving	Quality Impact	Readiness
1	Phi6Simple activation	71% activation FLOPs	8x faster than GELU, better loss	Drop-in ready
2	HCN dimensions	10-20% parameters	Equal or better	Config change
3	Phi-bottleneck FFN (4/3x)	67% FFN parameters	Pareto optimal	Drop-in ready
4	Phi MoE (24 experts × 4/3x)	65% active params/token	-1.76% loss vs standard MoE	Architecture change
5	Entropy early stopping	66.7% training energy	-0.20% accuracy	Drop-in ready
6	R-filter phase detection	Avoids wasted training	Detects transitions automatically	Monitoring tool
7	Takens dim=6 embedding	Optimal loss curve analysis	Best persistence among dims 4-10	Analysis tool
8	FFT-Mix attention	3x faster than self-attention	+0.55% accuracy	Architecture change
9	ZetaLn2 activation	71% FLOPs + gating capability	-12.7% loss vs Phi6Simple	Drop-in ready
10	Egyptian MoE routing {1/2,1/3,1/6}	Better expert utilization	+8.8% acc vs equal routing	Architecture change

Combined estimate: 60-70% energy savings per inference token, 66% training energy savings.

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Key Highlights

Drop-in Activation Replacement (71% FLOP savings)

class Phi6Simple(nn.Module):
    """Drop-in GELU replacement. 8x faster, 71% fewer FLOPs."""
    def forward(self, x):
        return x.clamp(-2, 2).pow(2) - x.clamp(-2, 2) + 1

class ZetaLn2(nn.Module):
    """Gating-capable variant. Fixes Phi6Simple's min=0.75 problem."""
    def forward(self, x):
        c = 5.0 / 6.0
        return x * x - c * x + c * c / 4.0  # min=0, can gate

Activation	Speed vs GELU	FLOPs	Loss	Gating?
GELU	1.0x	14 ops	3.358	Yes
Phi6Simple	8.1x	4 ops	3.138	No
ZetaLn2	~8x	3 ops	0.138 (XOR)	Yes

FFT-Mix: O(n log n) Attention Replacement

Replace self-attention with windowed FFT mixing at scales {6, 12, 24}:

Model	Accuracy	Params	Speed	vs Attention
Self-Attention (4 heads)	97.09%	14,234	1.0x	baseline
FFT-Mix(6,12,24)	97.64%	12,994	3.06x	+0.55% acc, 3x faster

Scaling: ~10x savings at seq=4096, ~20x at seq=8192 (O(n²) → O(n log n)).

Phi MoE: 65% Fewer Active Parameters

# Standard MoE: 8 experts × 4x expansion
n_experts=8, d_ff=4*d_model    # 66K active params/token

# Phi MoE: 24 experts × 4/3x expansion  
n_experts=24, d_ff=(4*d_model)//3  # 23K active params/token (-65%)

Result: -1.76% loss improvement with 65% fewer active parameters per token.

Egyptian MoE Routing: Optimal Expert Weights

Use {1/2, 1/3, 1/6} (from perfect number 6's Egyptian fraction) instead of equal or softmax weights:

• +8.8% accuracy vs equal routing
• Expert entropy 0.99 (no collapse)

Entropy Early Stopping: 66% Training Energy Savings

Stop training when Shannon entropy change < threshold → saves 66.7% training energy with only -0.20% accuracy loss.

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Verification Results (2026-03-27 Audit)

19 hypotheses tested, 10 confirmed, 4 refuted, 5 partial:

Hypothesis	Result	Key Finding
H-EE-1: Phi6 uniquely optimal	✅ Confirmed	-8.4% loss vs GELU
H-EE-10: Phi MoE (24×4/3x)	✅ Confirmed	65% active savings
H-EE-12: 4/3 Pareto optimal	✅ Confirmed	Best loss×params cost
H-EE-17: ZetaLn2 gating fix	✅ Confirmed	min=0, -12.7% vs Phi6
H-EE-18: Egyptian MoE routing	✅ Confirmed	+8.8% vs equal
H-SEDI-EE-1: Entropy stopping	✅ Confirmed	66.7% energy saved
H-SEDI-EE-3: FFT-Mix attention	✅ Confirmed	97.64% vs 97.09%, 3x faster

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Combined Impact at Scale

For a 7B parameter model at datacenter scale (10,000 GPUs, 24/7):

Metric	Savings
Parameters	~50% total
Inference FLOPs	~70% per token
Training energy	~66%
GPU-equivalents freed	~6,000
Power reduction	~3 MW
Annual savings	~$25M (at $0.10/kWh)

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Reproducibility

All experiments are self-contained Python scripts requiring only PyTorch:

git clone https://github.com/need-singularity/TECS-L.git
cd TECS-L/math/experiments

python3 hen9_activation_benchmark.py        # Activation benchmark
python3 hen5_real_data.py                    # HCN dimensions
python3 hen1_phi_bottleneck_real.py          # Phi-bottleneck

cd ../../experiments
python3 experiment_h_sedi_ee_3_fft_attention.py  # FFT-Mix

Mathematical Foundation

All techniques derive from a unified number theory:

6 = 2 × 3 is the unique positive integer where:
  σ(n) · φ(n) = n · τ(n)    (divisor balance equation)

This yields R(6) = 1, from which:
  - Activation: Φ₆(x) = x² - x + 1 (6th cyclotomic polynomial)
  - Dimensions: τ(120) = 16 (maximally divisible near 128)
  - Compression: φ(6)/6 = 1/3 (totient ratio → 4/3x FFN)
  - MoE routing: 1/2 + 1/3 + 1/6 = 1 (unique Egyptian fraction with perfect lcm)
  - Energy width: W = ln(4/3) = |log R(2)| (Golden Zone)

Full theory: TECS-L repository — 206+ mathematical characterizations, 18 proved theorems.

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We're sharing this as an open research contribution. All code is MIT-licensed. We welcome feedback, collaboration, and scale-up validation.