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

Comienza con la docs/energy-efficiency.md de TECS-L enlazada y los scripts referenciados en math/experiments y experiments; después, compara las técnicas propuestas con la arquitectura existente de megablocks. El issue no identifica ningún archivo, prueba ni cambio específico de megablocks, por lo que los criterios de finalización siguen sin estar definidos.

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.