Certified L∞-Bounded Compression for AI Models, I: The Contract and the Virtual-Band Index

A trained network is a set of matrices. Prevailing compression of AI models fixes the numberof bits and accepts whatever error results; error-bounded lossy compression, established forscientific data and applied to network parameters, fixes the error instead and lets the rate follow.This series follows the second route and makes it certified, verifiable and executable for AI models:the user fixes a tolerance τ before compression, every restored entry is proved, and checked inthe number format of the deployed arithmetic, to lie within τ of the checkpoint, and the packedform is executed directly by the kernels, without being restored to full precision. The first paperof this series [1] introduced this contract and realised it by sorting each group of weights, fittingthe monotone sorted curve with piecewise Chebyshev polynomials (the Tau-Collocation Method,in the form of the approximant of the Generalized Finite Integration Method) and storing thesort permutation. Here the same contract is realised by virtual bands. The sorted curve ismonotone, so a staircase of bands of width 2τ meets the bound, and the band of a weight is amonotone function of its value; the band index is the variable of the band equation itself. Aweight stores only its index ir,c, at its own column, and its restored value isˆ︁w r,c = μr + ir,c srwith two numbers μr and sr per row; the index is both the stored datum and the argument ofthe equation, so no mapping is applied, neither a sort permutation that carries a sorted positionback to a column nor a codebook that carries an index to a value. The sort permutation,the polynomial coefficients and the search for pieces and degrees are not needed: calibrationis one min/max pass and decoding is one fused multiply-add. We prove the grid certificate,the index monotonicity that removes the permutation, the certificate under fp16 restoration, aper-row information floor and the rate of the virtual grid, and describe the acceptance test, theexact escape pool and the independent verifier that re-checks a stored artifact. The same codec,unmodified, stores dense and mixture-of-experts language models, embedding and rerankingmodels, image and video diffusion transformers, text encoders, convolutional decoders and aUNet: at τ = 10−3 a 27B language model occupies 6.22 bits per weight (2.58× below itscheckpoint), and the MiniMax-H3 text-to-video model (transformer, text encoder and videodecoder) shrinks from 129 to 54 GiB, its transformer from 61 to 24 GiB at a tolerance of 10−2chosen from the scale of its weights. Parts II and III apply the contract to language-modelinference and to diffusion and video generation.