Calibration-free post-training quantization of neural networks is needed when calibration data is unavailable, for example because of licensing or privacy restrictions. Existing data-free quantizers also serve only a single bit-width per stored model.

ApexQuant recursively re-quantizes the residual error, applying a fresh random rotation before each pass. A proposition shows that a Haar rotation returns each residual to the uniform distribution on the sphere, so one codebook is optimal at every pass and error decay follows a closed-form law. The method is instantiated with scalar Lloyd–Max, E8 lattice and trellis stages. All Section 3 results except an arithmetic proposition are machine-checked in Lean 4 against Mathlib (154 public theorems, no extra axioms), including Mathlib-based statements about the normalised surface measure.

The predicted decay matches measured values to at least three decimals. On four LLMs, ApexQuant with the trellis stage is close to full precision at 4 bits and gives the best 2-bit results among data-free methods. Every prefix of passes is a valid lower-rate model, enabling elastic multi-precision inference from one artifact.