Geometric Versor Memory and Geometric Attention

A Clifford-Algebra Substrate for One-Shot Binding, Editing, and Order Encoding

Ignacio María Ozcáriz Arraiza · ORCID 0000-0002-9511-016X · anaciber@gmail.com

Reproducible code, data logs, and figures for the paper. The paper studies an associative-memory and attention model whose state is a multivector in a Clifford geometric algebra and whose operations are geometric products by *versors*: binding is one geometric product, recall is multiplication by the versor inverse, and a stored fact is deleted exactly with no retraining. All experiments run on small algebras Cl(n,0), n ≤ 8, with synthetic data.

📄 Paper: paper/paper.pdf

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What is here

Every number in the paper comes from a script in src/ and is logged as JSON in results/; the three figures are regenerated from those logs by paper/figures/make\_figures.py. Nothing is hand-entered.

geometric-versor-memory/
├── README.md
├── LICENSE                      # MIT (code)
├── CITATION.cff                 # citation metadata (software + paper)
├── requirements.txt             # numpy, scipy, matplotlib, jax
├── .gitignore
├── src/                         # all Python (flat: imports are unchanged)
│   ├── cln.py                   # Cl(n,0) geometric-algebra engine  (core)
│   ├── vam.py                   # versor associative memory          (core)
│   ├── encoding.py              # E1 — single-field capacity \& encoding
│   ├── sparse.py                # E2 — sparse addressable memory
│   ├── generalize.py            # E3 — noise, analogy, interpolation
│   ├── attention.py             # E4 — geometric attention
│   ├── level0.py                # E0 — classical baselines (Hebbian/Hopfield/MLP)
│   ├── level2.py                # E6 — versor position binding in transformers
│   └── level3.py                # E7 — sequential knowledge editing (flagship)
├── results/                     # JSON logs behind every reported number
│   ├── encoding\_results.json  capacity\_curves.json
│   ├── sparse\_results.json
│   ├── generalization\_results.json
│   ├── attention\_results.json
│   ├── level0\_results.json    level0\_noise.json
│   ├── level2\_results.json
│   └── level3\_results.json    level3\_load2.json  level3\_stress.json
├── paper/
│   ├── paper.tex   paper.pdf
│   └── figures/  fig\_editing.pdf  fig\_sparse.pdf  fig\_capacity.pdf  make\_figures.py
└── dashboards/                  # optional interactive HTML dashboards

**Note.** Copy your working .py files into src/ and your \*\_results.json into results/ (keep the filenames above so the mapping below holds). The Quantum-Market network simulator and the topological (L1) notes are separate projects and are intentionally not included here.

Experiment → script → result map

Exp.	What it shows	Script	Log
E0	Classical baselines (Hebbian, modern Hopfield, MLP)	src/level0.py	level0\_results.json, level0\_noise.json
E1	Single-field capacity; role of the code family	src/encoding.py	encoding\_results.json, capacity\_curves.json
E2	Sparse addressable memory → linear scaling	src/sparse.py	sparse\_results.json
E3	Generalization: noise, analogy, interpolation	src/generalize.py	generalization\_results.json
E4	Geometric attention (interpolation, analogy, order)	src/attention.py	attention\_results.json
E6	Versor position binding vs RoPE (trained transformer)	src/level2.py	level2\_results.json
E7	Sequential knowledge editing vs ROME (flagship)	src/level3.py	level3\_results.json, level3\_load2.json, level3\_stress.json

Install

python -m venv .venv \&\& source .venv/bin/activate
pip install -r requirements.txt

Python ≥ 3.10. The memory experiments (E0–E4) need only numpy; the trained-transformer experiments (E6–E7) additionally use jax.

Reproduce

Run any experiment from inside src/ (scripts write their JSON to the working directory):

cd src
python encoding.py     # E1
python sparse.py       # E2
python generalize.py   # E3
python attention.py    # E4
python level0.py       # E0
python level2.py       # E6   (jax)
python level3.py       # E7   (jax)

Regenerate the three paper figures from the committed logs:

python paper/figures/make\_figures.py     # writes fig\_editing/sparse/capacity.pdf

Build the paper

cd paper
pdflatex paper.tex \&\& pdflatex paper.tex   # two passes for references

The bibliography is embedded (thebibliography), so no BibTeX run is needed.

Citation

If you use this code or the results, please cite the paper (see also CITATION.cff):

@article{ozcariz2026versor,
  title   = {Geometric Versor Memory and Geometric Attention: A Clifford-Algebra
             Substrate for One-Shot Binding, Editing, and Order Encoding},
  author  = {Ozc\\'ariz Arraiza, Ignacio Mar\\'ia},
  doi     = {10.5281/zenodo.21752907},
  url     = {https://doi.org/10.5281/zenodo.21752907},
  year    = {2026}
}

License

Code is released under the MIT License (LICENSE). The paper text and figures are © the author; on arXiv they are distributed under the license selected at submission (CC BY 4.0 recommended).