# Exact ReLU realization of binary affine refinement iterates via reflection folding and cone switching

arXiv: https://arxiv.org/abs/2609.11962 · HTML: https://mlsanity.com/paper/2609.11962/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 13 references that are in the map.
- First posted on arXiv 12 Aug 2026.

## Builds on (most cited first, 4 of 4)

- [Benefits of depth in neural networks](https://mlsanity.com/paper/1602.04485/) (2016)
- [Neural Network Approximation of Refinable Functions](https://mlsanity.com/paper/2107.13191/) (2021)
- [Exact Loop Controllers for ReLU Realization of Homogeneous Curve Refinements](https://mlsanity.com/paper/2605.01655/) (2026)
- [Exact ReLU realization of affine one-dimensional refinement iterates via residual memory and offset frames](https://mlsanity.com/paper/2607.20586/) (2026)

## Abstract (arXiv)

We study vector-valued binary affine refinement operators with finitely supported matrix masks and compactly supported continuous piecewise linear input and forcing data. We prove that every finite refinement iterate admits an exact ReLU realization of fixed width and depth linear in the number of iterations. No separation of the forcing profile from the binary cell seams is required. The main mechanism is universal reflection doubling. Pairing each residual profile with its reflection replaces the two binary transition matrices by one fixed block matrix together with a fixed swap involution. The cell-seam identity makes the two branch candidates agree at the tent fold, while their swap-odd component is bounded linearly by the distance to the fold. This permits exact branch selection by a fixed continuous piecewise linear cone switch, without multiplication by a variable selector. The resulting primal recursion requires the residual orbit in reverse order. We obtain exact backward replay from the residual memory controller developed previously for affine refinement, interpreted here through the reflection quotient of circle doubling. The construction propagates the full vectorized profiles rather than decomposing the input and forcing into reference atoms. We also treat stage-dependent forcing from a fixed finite-dimensional family and show that genuine reflection equivariance reduces the doubled cascade to a single parity sector.