AI 中文总结
本文提出将有限知识存储为精确周期潜在生成器的相位，通过离散傅里叶算子保证循环闭合，在图像和视频基准上实现高保真压缩，验证了以动态存储替代独立实例的可行性。
AI 中文摘要
有限知识通常以外延方式存储，每个项目对应一个代码或向量。我们探究有限集合能否以内涵方式存储，即作为一条紧凑法则的解码轨道，该法则精确地返回到其起点。对于X个对象，我们将项目i编码为学习潜在空间中固定旋转的第i个相位，并用共享网络解码所有相位；潜在表示通过整数谐波周期上的旋转组（一个实数离散傅里叶算子）前进，使得R^X等于恒等映射，精确闭合是保证的而非学习得到的。图像是受控载体；循环视频则是相位顺序为内容自身时间结构的情形。固定解码器而仅改变算子，一般学习算子会发散，保范但非周期的算子会在循环周围退化，而精确周期算子则保持平坦；在真实图像上差距更大。容量是解码器的预算：密集解码器对每个清晰图像承担结构开销，任何规模都无法与压缩调和，而小型卷积解码器在共享流形的对象上能达到清晰且压缩的效果。码本对照显示，生成法则在重构方面是自由的，同时将潜在存储倍增数倍。在七个基准片段上，与匹配的帧索引基线相比，循环在相同参数下达到同等或更好的保真度，并以机器精度闭合，而基线留下可见接缝；将基线频率固定到循环谐波也能消除其接缝，确认精确周期性是有效约束。有限循环知识可以存储为动态而非独立实例，精确重复由代数提供，内容由共享解码器提供。
英文摘要
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.
Comments9 pages, 2 figures
