---
license: mit
tags:
- attention
- efficient-attention
- number-theory
- calabi-yau
- custom-kernel
- pytorch
- benchmark
datasets: []
---

# S20-Decay Attention Kernel

A high-performance, mathematically exact attention bias derived from the Weight-5 Apéry-like binomial sum:

$$S_{20}(n) = \sum_{k=0}^{n} \binom{n}{k}^4 \binom{n+k}{k}$$

> **Paper**: [A Weight-5 Apéry-like Binomial Sum, its Calabi-Yau 4-fold Period, and Supercongruences](https://doi.org/10.5281/zenodo.20747943)  
> **Code**: [GitHub — Mirror-Map-Sieve](https://github.com/xaviercallens/Mirror-Map-Sieve)

---

## 3 Core Hypotheses

1. **Exact Mathematical Rigidity**: Unlike ALiBi or learned positional embeddings, the S20 sequence provides a deterministic, integer-derived attention decay. Zero floating-point drift at any context length.
2. **O(1) Vectorized Toeplitz Construction**: The decay matrix is built as a 1D broadcast over a distance tensor — no nested loops, no learned parameters.
3. **Zero-Cost LLM Injection**: S20 decay can be injected into any SDPA-based model via global monkey-patching (`F.scaled_dot_product_attention`) with **zero measurable latency overhead** on GPU.

---

## GPU Benchmark: Tesla T4 (16GB, CUDA 12.9, PyTorch 2.9.1)

### Raw Kernel: SDPA ± S20 Bias

| Seq Len | SDPA Baseline | SDPA + S20 | Overhead | Correct |
|---------|--------------|------------|----------|---------|
| 64 | 0.020 ms | 0.022 ms | 1.08× | ✅ |
| 128 | 0.024 ms | 0.029 ms | 1.23× | ✅ |
| 256 | 0.041 ms | 0.053 ms | 1.31× | ✅ |
| 512 | 0.092 ms | 0.144 ms | 1.56× | ✅ |
| 1024 | 0.199 ms | 0.529 ms | 2.65× | ✅ |

### Phi-3-mini-4k-instruct (3.8B) — Global SDPA Patching

| Seq Len | Baseline | S20-SDPA | Overhead | Base tok/s | S20 tok/s | Energy (J) | Power (W) |
|---------|----------|----------|----------|------------|-----------|------------|-----------|
| 64 | 49.59 ms | 49.09 ms | **0.99×** | 1,290 | 1,304 | 67.9 | 69.2 |
| 128 | 59.21 ms | 59.15 ms | **1.00×** | 2,162 | 2,164 | 82.3 | 69.8 |
| 256 | 106.98 ms | 107.39 ms | **1.00×** | 2,393 | 2,384 | 115.5 | 54.4 |
| 512 | 211.06 ms | 213.15 ms | **1.01×** | 2,426 | 2,402 | 297.3 | 69.9 |
| 1024 | 488.51 ms | 487.11 ms | **1.00×** | 2,096 | 2,102 | 659.9 | 67.7 |

> **Key finding**: On a real 3.8B-parameter model, S20 global SDPA patching adds **zero measurable overhead** (0.99–1.01×) across all sequence lengths. The integer-sequence bias is effectively free on GPU.

---

## CPU Benchmark: Open-Weights Model Injection

S20 decay injected as post-logit positional mask on CPU (Apple Silicon):

| Model | Params | Baseline | S20-Injected | Overhead | Avg PPL |
|-------|--------|----------|--------------|----------|---------|
| GPT-2 | 124M | 39.2 ms | 41.6 ms | 1.06× | 175.7 |
| DistilGPT-2 | 82M | 21.6 ms | 21.3 ms | 0.99× | 302.0 |
| OPT-125M | 125M | 29.0 ms | 29.4 ms | 1.01× | 199.7 |
| BLOOM-560M | 559M | 122.4 ms | 118.0 ms | 0.96× | 139.0 |

---

## Method: Global SDPA Patching (Forward Hook Option A)

```python
import torch
import torch.nn.functional as F
from math import comb

# 1. Build S20 decay sequence
def s20(n): return sum(comb(n, k)**4 * comb(n+k, k) for k in range(n+1))
_S20 = [s20(d) for d in range(18)]

# 2. Vectorized log-bias matrix
def build_s20_log_bias(seq_len, device="cuda", dtype=torch.float16):
    base = float(_S20[0])
    weights = [base/float(x) if x > 0 else 0.0 for x in _S20] + [0.0]
    dv = torch.tensor(weights, dtype=torch.float32, device=device)
    idx = torch.arange(seq_len, device=device)
    dist = (idx.unsqueeze(0) - idx.unsqueeze(1)).abs().clamp(max=len(_S20))
    decay = dv[dist]
    log_bias = torch.log(decay.clamp(min=1e-30))
    causal = torch.tril(torch.ones(seq_len, seq_len, device=device))
    log_bias = log_bias * causal + (1 - causal) * (-1e9)
    return log_bias.unsqueeze(0).unsqueeze(0).to(dtype)

# 3. Monkey-patch F.scaled_dot_product_attention
_original_sdpa = F.scaled_dot_product_attention
_bias_cache = {}

def patched_sdpa(q, k, v, attn_mask=None, dropout_p=0.0, is_causal=False, **kw):
    L = q.shape[-2]
    if L not in _bias_cache:
        _bias_cache[L] = build_s20_log_bias(L, q.device, q.dtype)
    bias = _bias_cache[L][:, :, :L, :k.shape[-2]]
    if attn_mask is not None:
        attn_mask = attn_mask + bias
    else:
        attn_mask = bias
    return _original_sdpa(q, k, v, attn_mask=attn_mask, dropout_p=dropout_p, **kw)

F.scaled_dot_product_attention = patched_sdpa
# Now ANY model using SDPA will have S20 decay injected automatically
```

---

## Reproducibility

```bash
# Clone and run on any CUDA GPU
git clone https://github.com/xaviercallens/Mirror-Map-Sieve.git
cd Mirror-Map-Sieve/4_ai_hardware_attention
pip install torch transformers accelerate
python gpu_benchmark_s20.py --model microsoft/Phi-3-mini-4k-instruct --seq_lens 64 128 256 512 1024
```

## Citation

```bibtex
@software{callens2026s20attn,
  author = {Callens, Xavier},
  title  = {S20-Decay Attention Kernel: Vectorized Integer-Sequence Attention Bias},
  year   = {2026},
  url    = {https://huggingface.co/callensxavier/s20-attention-kernel},
  doi    = {10.5281/zenodo.20747943}
}
```