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LINGUISTICS BOROUGH · INVARIANT GRAMMAR BUREAU · LLL LOFT

The LLL Loft is the Bureau’s lattice-normalization studio. It receives a basis written in an arbitrary coordinate dialect and returns a shorter, nearly orthogonal basis for the same lattice. It changes the description, not the object.

Jurisdiction: Green Line structural geometry · Blue Line executable verification · Gold Monorail reproducibility.

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Canonical object

Given basis vectors $b_1,\ldots,b_n\in\mathbb R^n$, define

$\mathcal L(B)=\left\{\sum_{i=1}^n z_i b_i:z_i\in\mathbb Z\right\}.$

LLL produces a reduced basis $B_{\mathrm{LLL}}$ satisfying

$\boxed{\mathcal L(B)=\mathcal L(B_{\mathrm{LLL}})}$

while improving basis length, orthogonality, numerical conditioning, and computational tractability.

The loft engine

1. Gram–Schmidt chamber

$b_i^=b_i-\sum_{j<i}\mu_{ij}b_j^,\qquad \mu_{ij}=\frac{\langle b_i,b_j^\rangle}{\langle b_j^,b_j^*\rangle}.$

The chamber exposes overlap and redundancy without changing the lattice.

2. Size-reduction bench

For $j=k-1,\ldots,1$, whenever $|\mu_{kj}|>1/2$,

$b_k\leftarrow b_k-\operatorname{round}(\mu_{kj})b_j.$

This applies an integer unimodular basis transformation and preserves the lattice exactly.

3. Lovász gate

For a fixed $\delta\in(1/4,1)$, usually $\delta=3/4$,

$\|b_k^\|^2\geq\left(\delta-\mu_{k,k-1}^2\right)\|b_{k-1}^\|^2.$