# Positive Garside-factor tokenizer

- ID: 122
- Canonical URL: https://synthcore.org/idea/122/positive-garside-factor-tokenizer
- API JSON: https://synthcore.org/api/idea/122.json
- API Markdown: https://synthcore.org/api/idea/122.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2608.24003](https://arxiv.org/abs/2608.24003)
- Category: architecture
- Solves: generalization, scalability, memory
- ML areas: embedding, tokenizer, transformer, data-augmentation
- Math tags: algebra, topology, combinatorics, representation-theory
- Ratings: usefulness 6/10; difficulty 5/10; novelty 9/10

## Idea description

Replace raw braid-generator sequences by sequences of positive simple Garside factors obtained from the left-greedy normal form. Because powers of \(\Delta\) lie in the Hilden subgroup, they can be removed while preserving the relevant double-coset presentation, reducing non-uniqueness and often shortening the sequence. Feed the resulting factor tokens to a Transformer or sequence classifier, and train it to be invariant to inserted removable \(\Delta\)-powers.

## Mathematical statement

For the braid group \(B_n=\langle \sigma_1,\ldots,\sigma_{n-1}\rangle\), the paper defines the Garside element \(\Delta=(\sigma_1\cdots\sigma_{n-1})(\sigma_1\cdots\sigma_{n-2})\cdots(\sigma_1\sigma_2)(\sigma_1)\), satisfying \(\Delta\sigma_i=\sigma_{n-i}\Delta\). A Garside left-greedy normal form writes every braid as \(\beta=\Delta^p x_1x_2\cdots x_r\), where \(p\in\mathbb Z\), each \(x_j\) is a positive simple braid, meaning a positive divisor of \(\Delta\), and the factors are chosen greedily so that each prefix is maximal in the divisibility order. The Garside length is \(r\). For the \(2n\)-strand Hilden subgroup \(H_{2n}\), the paper's key observation is \(\Delta\in H_{2n}\). Hence \(\beta\) and \(\Delta^{-p}\beta=x_1\cdots x_r\) belong to the same left Hilden coset and therefore the same double coset \(H_{2n}\beta H_{2n}\). The transferable asset is a positive, factorized representative with length controlled by the original Garside length, rather than an arbitrary word containing positive and negative generators.

## Key formulas

- $$\Delta=(\sigma_{1}\sigma_{2}\cdots\sigma_{n-1})(\sigma_{1}\sigma_{2}\cdots\sigma_{n-2})\cdots(\sigma_{1}\sigma_{2})(\sigma_{1})$$
- $$\Delta\sigma_i=\sigma_{n-i}\Delta$$
- $$\beta=\Delta^{p}x_1x_2\cdots x_r,\qquad p\in\mathbb Z,\quad x_j\mid \Delta,\quad x_j\text{ positive simple}$$
- $$\Delta\in H_{2n}\quad\Longrightarrow\quad H_{2n}\beta H_{2n}=H_{2n}(\Delta^{-p}\beta)H_{2n}=H_{2n}(x_1x_2\cdots x_r)H_{2n}$$

## Implementation notes

Integrate this at the input tokenizer for a model whose inputs are braid presentations, braid-generated symbolic trajectories, or link/plat datasets. The input is a word \(w\) over signed generators \(\sigma_i^{\pm1}\). First reduce the word in the braid group using braid relations \(\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}\) and commutations \(\sigma_i\sigma_j=\sigma_j\sigma_i\) for \(|i-j|>1\). Compute its Garside left-greedy normal form \(\beta=\Delta^p x_1\cdots x_r\), where each \(x_j\) is represented by the permutation of strands associated with a positive divisor of \(\Delta\). Discard the integer power \(p\), because \(\Delta\in H_{2n}\), and emit tokens \([x_1,\ldots,x_r]\); optionally retain \(p\) as a separate nuisance token for ablations. Pseudocode: `normal_form = garside_normal_form(word); p, factors = normal_form.infimum, normal_form.simple_factors; tokens = [factor_id(x) for x in factors]; return tokens`. For batching, pad factor sequences and use a learned embedding for each simple factor, or serialize each factor as its permutation or inversion-vector representation. During training, create equivalent views by multiplying the original braid on the left or right by random powers \(\Delta^k\), normalize both, and apply a consistency loss between their model representations. The mathematics determines the factorization and valid removal of \(\Delta^p\); empirically estimate compression ratio \(|w|/r\), collision rates between known equivalent presentations, and whether retaining or dropping \(p\) affects the task. The first cheap experiment is a small Transformer on a synthetic dataset of 4- or 6-strand braids labeled by a computable invariant or generated link type. Compare raw signed-generator tokenization against Garside-factor tokens at equal parameter count. Success means lower validation error under random presentation rewrites, shorter effective sequence length, and improved accuracy at fixed attention FLOPs.

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: no_effect
- Verdict: Built a readable B4/B3 positive-factor tokenizer MVP with permutation-indexed simple braids, greedy factor weighting, Delta stripping, identity checks, and a controlled centroid classifier comparison. The math sanity checks passed: Delta has the longest-strand permutation, satisfies Delta sigma_i = sigma_{n-i} Delta, normalizes to one factor, and the B3 relation 121 -> Delta is recovered. Literal left insertion of Delta powers was removed with 100% factor-sequence invariance and reduced the presented 27.04 generator tokens to 6.32 factor tokens on average (4.70x ratio); however, factor-token classification accuracy was 0.503 versus raw 0.493, effectively no meaningful ML win in this toy setup.

### Mechanism check

- Verdict: Built a readable B4/B3 positive-factor tokenizer MVP with permutation-indexed simple braids, greedy factor weighting, Delta stripping, identity checks, and a controlled centroid classifier comparison. The math sanity checks passed: Delta has the longest-strand permutation, satisfies Delta sigma_i = sigma_{n-i} Delta, normalizes to one factor, and the B3 relation 121 -> Delta is recovered. Literal left insertion of Delta powers was removed with 100% factor-sequence invariance and reduced the presented 27.04 generator tokens to 6.32 factor tokens on average (4.70x ratio); however, factor-token classification accuracy was 0.503 versus raw 0.493, effectively no meaningful ML win in this toy setup.
- Confidence: 7/10
- Limitations: This is a small exact permutation-backed positive-braid MVP, not a production arbitrary signed-word Garside normal-form implementation. It tests literal left Delta^k augmentation rather than arbitrary braid-relation rewrites or right multiplication, uses a nearest-centroid histogram classifier rather than a Transformer, and does not model a real link/plat task or measure attention FLOPs/runtime.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.85155
- Paired wins: 2/8
- Benchmark verdict: no measurable effect

## Artifacts

- [bench_report.json](https://synthcore.org/code/1182/bench_report.json)
- [garside_mvp.py](https://synthcore.org/code/1182/garside_mvp.py)
- [garside_track.py](https://synthcore.org/code/1182/garside_track.py)
- [report.md](https://synthcore.org/code/1182/report.md)
- [report_bench_2026-09-03T022121.md](https://synthcore.org/code/1182/report_bench_2026-09-03T022121.md)
- [run_bench.py](https://synthcore.org/code/1182/run_bench.py)
- [Download all files as ZIP](https://synthcore.org/download/1182)

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.