---
license: mit
tags:
- attention
- positional-encoding
- benchmark
- negative-result
- calabi-yau
- number-theory
pretty_name: CY-Sieve Attention — GPU Quality/Perf Benchmark (NVIDIA L4)
configs:
- config_name: quality_perplexity
  data_files: quality_perplexity.csv
- config_name: perf_hbm
  data_files: perf_hbm.csv
---

# CY-Sieve Attention — GPU benchmark results (NVIDIA L4, 2026-06-22)

Benchmark artifacts for the **CY-Sieve positional-attention kernel**, a falsifiable
engineering experiment from the [Mirror-Map-Sieve](https://github.com/xaviercallens/Mirror-Map-Sieve)
project. The bias derives from the weight-5 Apéry-like sequence
$S_{20}(n)=\sum_k \binom{n}{k}^4\binom{n+k}{k}$ (a Calabi–Yau **3-fold** period;
the geometry fixes the long-range decay slope $\log\lambda=3.762$ and curvature
$\beta=2$).

## ⚠️ Headline: this is a documented NEGATIVE result

On real WikiText-2, trained from scratch, the positional scheme **failed its
quality gate (KILL, +10.15%)** — a plain sliding window beat every CY-Sieve
variant. The kernel is numerically correct and has a real memory advantage, but a
fast kernel that hurts model quality is a failed kernel. We publish the negative
result deliberately; it is the science working as intended.

## Files

| file | contents |
|---|---|
| `quality_perplexity.csv` | §5 quality gate — val perplexity per positional scheme × context (the decisive result) |
| `perf_hbm.csv` | §6 — kernel latency + bias-path HBM bytes per sequence length |
| `run3_20260622_l4_quality.json` | raw §5 output (corpus, config, verdict) |
| `run3_20260622_l4_perf.json` | raw §6 output |
| `run3_20260622_l4_gpu_phase.json` | orchestrator summary (§4/§5/§6 headline) |
| `run3_20260622_l4.log` | full run log |
| `PHASE3_CYSIEVE_GPU_FINDINGS.md` | the complete findings writeup + redesign directions |

## §5 — Quality gate (the decisive result)

Methodology: train small GPTs **from scratch**, identical arch/data/compute, one
per positional scheme, on real WikiText-2 (`Salesforce/wikitext`), byte-level.
(Zero-shot-swapping the scheme on a *frozen* model was tried and rejected as
invalid — it collapses every scheme equally.) Validation perplexity:

| scheme | @512 (train) | @1024 (2×) | @2048 (4×) |
|---|---|---|---|
| **learned-absolute** | **4.22** | 12.10 | 20.82 |
| ALiBi | 10.74 | 11.73 | 11.35 |
| **sliding-window** | **4.99** | **5.07** | **5.03** |
| CY-Sieve τ-ladder | 11.33 | 12.31 | 12.05 |
| CY-Sieve τ=20 | 16.02 | 16.81 | 16.49 |
| CY-Sieve τ=128 | 6.80 | 7.12 | 7.00 |
| CY-Sieve τ=512 | 4.65 | 6.08 | 10.62 |

**Verdict: KILL.** Best baseline 4.22 (learned-absolute); best CY-Sieve 4.65
(τ=512) → **+10.15%**, past the >5% kill threshold. The geometry-fixed slope is too
steep for a drop-in scheme: no single τ balances absolute quality against
extrapolation, and the τ-ladder lands at ~11–12.

## §6 — Performance + memory (NVIDIA L4, D=64, fp16, causal)

| L | CY-Sieve (ms) | dense SDPA (ms) | bias-HBM reduction |
|---|---|---|---|
| 4096 | 0.26 | 0.06 | 2048× |
| 8192 | 1.08 | 0.29 | 4096× |
| 16384 | 4.16 | 1.02 | **8192×** |
| 32768 | 15.28 | 2.51 | 16384× |

The bias-path HBM claim is **confirmed** — O(L) bytes (recurrence-generated) vs
O(L²) for a materialized table. But the unfused kernel is **~4–6× slower** than
fused dense SDPA: a memory-traffic win, **not** a latency win. Per the project's
honesty rule, with §5 failing these numbers are *not* presented as a contribution.

## Autoresearch follow-up (2026-06-22): can a learnable slope beat the baselines?

A propose→screen→select sweep of 10 hypotheses (`autoresearch_results.csv`,
`raw/autoresearch_*`) testing two fixes to the KILL: **learnable per-head γ
"Holonomic-ALiBi"** ($\text{bias}_h(d)=-\gamma_h\log S_{20}(d)$, γ learnable, O(L)
kept) and a **"Comet" hybrid** (local window + CY tail).

- **Screen (1200 steps): learnable-γ Holonomic-ALiBi BEAT every baseline** —
  holo_ladder **5.89** vs ALiBi 6.15. The mechanism works.
- **Full (6000 steps): the ranking INVERTED** — best CY 12.7 vs best baseline 4.3.
  The setup over-trained (~37 epochs over a 2 MB corpus); the expressive learnable
  bias overfit hardest (train loss 3× lower, val 3× worse) and γ drifted *steeper*,
  not flatter. **Still KILL — but UNCONFIRMED, not refuted.**
- One survivor: the holonomic schemes **extrapolate flat** (12.7→13.3 over
  512→2048) where learned-absolute collapses (4.3→20.6).

A **v2 run** (γ-regularization toward flat + validation early-stopping + larger
corpus) confirmed the margin survives: **+8.1% PASS** at GPU scale. A controlled
attribution experiment then showed the gain is **learnability, not the Calabi–Yau
shape** (`PHASE4_HC_ATTRIBUTION.md`). Lesson: a short screen preferentially crowns
the highest-capacity hypothesis — exactly the one that overfits at scale; validate
the winner at the target budget.

## Final selection (Phases 5–6): plain learnable-ALiBi

Two further GPU bake-offs settled the shipped form (`hetero_pos_extrapolation.csv`,
`raw/hetero_pos_gpu_20260623.*`):

- **Bias-family bake-off** (`PHASE5_FINAL_BAKEOFF.md`): of six learnable families,
  **learnable-ALiBi wins**; log-curvature, Fourier, the fixed CY shape, and a
  local+tail hybrid all lose at scale.
- **Hetero-positional** (`PHASE6_HETERO_POS.md`, NVIDIA L4, 2026-06-23, from
  scratch, ctx 512, 4000 steps): the last orthogonal axis — per-head
  **content↔position balance** — only **ties** the learnable-ALiBi control
  (+0.35% @4×, +0.17% @8×), and adding a per-head **softmax temperature
  *regresses***:

  | mode | 1× | 2× | 4× | 8× |
  |---|---|---|---|---|
  | alibi_fixed | 3.705 | 3.623 | 3.577 | 3.513 |
  | **alibi_learn** (control) | 3.651 | 3.559 | 3.500 | 3.429 |
  | nope | 10.290 | 10.845 | 11.571 | 12.311 |
  | content_balance | 3.642 | 3.548 | 3.488 | 3.428 |
  | cb_softmax_temp | 3.732 | 3.659 | 3.599 | 3.548 |

**Selected hypothesis:** ship **learnable-ALiBi** — one learnable linear slope per
head, no log term, no $S_{20}$ sequence, no content scale, no temperature. Every
extra knob ties within noise or hurts at scale.

## Hardware / reproducibility

NVIDIA L4 (24 GB), PyTorch 2.9.1+cu129, Triton 3.5.1. §4 Triton↔reference parity:
PASS (4/4). Full method: `PHASE3_CYSIEVE_GPU_FINDINGS.md` and
[the repo](https://github.com/xaviercallens/Mirror-Map-Sieve).

## Citation

```bibtex
@misc{callens2026cysieve,
  author = {Callens, Xavier},
  title  = {CY-Sieve Attention: a Calabi--Yau positional bias and its negative quality result},
  year   = {2026},
  url    = {https://github.com/xaviercallens/Mirror-Map-Sieve},
  doi    = {10.5281/zenodo.20747943}
}
```