# Knowledge as Orbit: Finite Collections as Phases of an Exactly Periodic Latent Generator

arXiv: https://arxiv.org/abs/2609.17417 · HTML: https://mlsanity.com/paper/2609.17417/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 17 references that are in the map.
- First posted on arXiv 11 Jul 2026.

## Builds on (most cited first, 13 of 13)

- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [The 2017 DAVIS Challenge on Video Object Segmentation](https://mlsanity.com/paper/1704.00675/) (2017)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [NeRV: Neural Representations for Videos](https://mlsanity.com/paper/2110.13903/) (2021)
- [COIN: COmpression with Implicit Neural representations](https://mlsanity.com/paper/2103.03123/) (2021)
- [Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group](https://mlsanity.com/paper/1901.08428/) (2019)
- [HiNeRV: Video Compression with Hierarchical Encoding-based Neural Representation](https://mlsanity.com/paper/2306.09818/) (2023)
- [Explorations in Homeomorphic Variational Auto-Encoding](https://mlsanity.com/paper/1807.04689/) (2018)
- [Interpretable Transformations with Encoder-Decoder Networks](https://mlsanity.com/paper/1710.07307/) (2017)
- [LoopGaussian: Creating 3D Cinemagraph with Multi-view Images via Eulerian Motion Field](https://mlsanity.com/paper/2404.08966/) (2024)
- [LoopAnimate: Loopable Salient Object Animation](https://mlsanity.com/paper/2404.09172/) (2024)
- [Mobius: Text to Seamless Looping Video Generation via Latent Shift](https://mlsanity.com/paper/2502.20307/) (2025)

## Abstract (arXiv)

Finite knowledge is usually stored extensionally, one code or vector per item. We ask whether a finite collection can instead be stored intensionally, as the decoded orbit of one compact law that returns exactly to its start. For X objects, we encode item i as the i-th phase of a fixed rotation in a learned latent space and decode all phases with a shared network; the latent advances through a bank of rotations at integer harmonics of the cycle, a real discrete Fourier operator, so that R^X equals the identity and exact closure is guaranteed rather than learned. Images are a controlled carrier; looping video is the case where the phase order is the content's own temporal structure. Holding the decoder fixed and varying only the operator, a general learned operator diverges, a norm-preserving but non-periodic one degrades around the loop, and the exactly periodic operator is flat; on real images the gap widens. Capacity is then the decoder's budget: dense decoders carry a structural overhead per crisp image that no size reconciles with compression, while a small convolutional decoder on objects that share a manifold reaches crisp and compressed. A codebook control shows the generative law is free in reconstruction terms while multiplying the latent store many-fold. On seven benchmark clips, against a matched frame-index baseline, the cycle reaches equal or better fidelity at equal parameters while wrapping at machine precision, where the baseline leaves a visible seam; pinning the baseline's frequencies to loop harmonics closes its seam too, confirming that exact periodicity is the operative constraint. Finite cyclic knowledge can be stored as dynamics rather than independent instances, with exact recurrence supplied by algebra and content by a shared decoder.