Positive Garside-factor tokenizer

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.