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Knowledge as Orbit: Finite Collections as Phases of an Exactly Periodic Latent Generator
Siddharth Pal, Viktoria Rojkova
TL;DR
The paper asks whether finite collections can be stored as decoded orbits of one compact law rather than independent instances. It uses an exactly periodic Fourier rotation with a shared decoder, finding stable recurrence and competitive reconstruction while showing that compression depends on shared structure and that the approach is limited to cyclic collections.
Problem
Finite collections are conventionally stored with one code per item, leaving open whether one law can provide representation, exact recurrence, capacity, and compression.
Method
The method encodes each object as a phase of an integer-harmonic latent Fourier rotation and decodes all phases with a shared network.
Results
The exactly periodic operator remains near 70 dB through 128 images, while the general operator fails beyond X = 64 and the non-periodic unitary drops from 76 to roughly 13 dB.
Takeaways & Limitations
Finite cyclic knowledge can be stored as dynamics rather than independent instances, with exact recurrence supplied algebraically and content supplied by a shared decoder.
Takeaways & Limitations
The representation is limited to cyclically indexed collections; arbitrary graphs, taxonomies, and branching structures may lose native adjacency when forced into one orbit.
Abstract
from arXiv · showhide
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.
1 Finite knowledge as a generated orbit
The paper asks whether finite collections can be represented by one dynamical law rather than independent codes, with exact recurrence, capacity, and compression made explicit. It distinguishes imposed cyclic order for unordered objects from intrinsic temporal order in looping video.
- 1 Finite knowledge as a generated orbit: Finite collections can be represented as decoded orbits of one generator instead of lists with one code per item.The representation uses an initial latent state, a fixed advance operator, and a shared decoder.
- 1 Finite knowledge as a generated orbit: The paper defines finite knowledge operationally as a finite set of objects a model must reproduce, not as propositional truth or reasoning.Its construction requires cyclic indexing, with arbitrary order for image sets and intrinsic temporal order for looping video.
- 1 Finite knowledge as a generated orbit: For unordered image collections, imposed order tests capacity without requiring intermediate phases to be semantically meaningful.For looping video, phase order is the signal’s own time structure and exact closure is required.
- 1 Finite knowledge as a generated orbit: The central challenge is keeping every decoded phase on the object manifold while making the latent orbit close exactly.The construction places periodicity in a latent Fourier rotation and object validity in a learned decoder.
- 1 Finite knowledge as a generated orbit: The scoped claims are that exact periodicity prevents drift, costs nothing against independent latent codes in measured reconstruction settings, and guarantees closure for naturally cyclic content.The representation also supplies deterministic addressing and exact closure while using less latent storage.
2 Method: one law, X phases
The method encodes X objects as phases of a fixed latent rotation and renders them with one shared decoder. Integer-harmonic Fourier rotations guarantee R^X = I, shifting the remaining limit from drift to decoder capacity.
- 2 Method: one law, X phases: Each object is generated as xi ≈ D(zi) from a phase latent zi = R^i z0, with only the initial state and decoder trained.The objects in the experiments are images or video frames.
- 2 Method: one law, X phases: Integer-harmonic plane rotations form a real discrete Fourier operator whose X-th power is identity, guaranteeing exact orbit closure.The rotation acts as both address generator and clock, while the decoder assigns content to each phase.
- 2 Method: one law, X phases: The construction replaces X independent latent addresses with one base state and a known cyclic-group action, storing memory through dynamics and content through the decoder.The orbit specifies item locations; the decoder specifies item identities.
- 2 Method: one law, X phases: Roots-of-unity spectra make the operator both norm-preserving and exactly closing, unlike general operators that explode or vanish and generic norm-preserving operators that fail to return.Exact periodicity, rather than linearity alone, is the operative constraint for finite enumeration.
- 2 Method: one law, X phases: Once closure is guaranteed, the limiting resource is decoder capacity, with latent size requiring on the order of X/2 rotating planes to resolve X phases.The predicted scaling is with the decoder’s rendering budget rather than an error-accumulation ceiling.
- 2 Method: one law, X phases: Beyond X = 2m phases, latent crowding and decoder capacity are entangled, so only the regime up to 2m cleanly isolates decoder capacity.A fixed bank of m planes still produces distinct latents beyond that threshold, but they crowd onto a smoother curve.
3 Relation to prior work
The paper distinguishes its exactly periodic, collection-level latent orbit from prior operators for open-trajectory forecasting, single-object transformations, cyclic topologies, and video indexing. Its narrow novelty claim combines distinct images or frames, exact closure, and capacity/compression accounting.
- 3 Relation to prior work: Unlike Koopman-autoencoder and norm-preserving operator work, this method fixes an exact period and uses the orbit to represent a finite collection rather than forecast an open trajectory.Its eigenvalues are roots of unity, making the operator a discrete Fourier operator.
- 3 Relation to prior work: Prior latent rotations and circular topologies represent one object’s transformed-view orbit, not a collection of distinct images with measured capacity and compression.The paper positions its collection-level accounting as a difference from those approaches.
- 3 Relation to prior work: NeRV-style implicit video representations use frame-index embeddings that are not periodic in loop length, whereas this construction isolates periodicity through seam measurement.The distinction concerns the temporal embedding, not the general idea of storing video in network weights.
- 3 Relation to prior work: The method is complementary to loop-synthesis systems because it fits a given loop and guarantees closure through the latent operator.This places it in the lineage of the classical video-textures problem for neural representations.
- 3 Relation to prior work: The claimed combination is distinct images or frames as phases of one exactly periodic operator, codebook-controlled capacity and compression analysis, and measured closure guarantees.The broader finite-knowledge interpretation explains the representation without expanding the narrow novelty claim.
4 Exact recurrence as the representation constraint
With the decoder held fixed, exact periodicity is the decisive operator constraint: unconstrained dynamics diverge, non-periodic unitary dynamics degrade around the loop, while the periodic operator remains stable as collections grow.
- Synthetic results: On synthetic images, the exactly periodic operator is best and flat, whereas the general operator collapses and the unitary operator degrades because its cycle does not close.Learning frequencies does not help because integer harmonics already form a complete basis; exact periodicity is the operative ingredient.
- Capacity boundary: The periodic orbit reproduces one CIFAR-100 class crisply to roughly one hundred images, establishing a decoder-limited capacity rather than an error-accumulation ceiling.Figure 1 shows real images above decoded Fourier phases below, with reconstruction plotted against the number of held images.
- Real-image results: The exactly periodic operator remains near 70 dB through 128 real CIFAR-100 images, while the general operator overflows beyond X = 64 and the unitary operator falls to roughly 13 dB.At larger collections, the periodic operator still leads by roughly thirty decibels as reconstruction approaches the decoder’s budget.
- Trade-off: At X = 16, the unconstrained operator fits 13 dB better, showing that exact recurrence pays its stability dividend mainly on longer orbits.With few phases, freedom fits better because long-run stability is less relevant.
- Interpretation: A fixed rotation supplies exact recurrence while the shared decoder renders each phase, so the representation remains valid around a finite orbit.The construction treats the cyclic generator as a stable address system for independently rendered items.
5 Capacity and structural sharing
Capacity is governed by decoder expressiveness and shared structure: arbitrary collections expose a crisp-versus-compressed trade-off, while manifold-sharing objects allow both.
- Intermediate phases: Untrained phases become blurry class-average blobs roughly 40 dB below trained phases, and denoising improves them by at most 0.2 dB while costing up to 2.7 dB on trained phases.Cyclic indexing supplies order and closure, not semantic interpolation, unless the content itself provides a manifold.
- Dense-decoder scaling: Dense decoders store about 1,150 parameters per crisp image, or roughly 2.25× the raw-image bytes in half precision, so they cannot be crisp and compressed simultaneously.For the 50,000-image corpus, compression crosses one only beyond 4,000 images, after reconstruction falls below 20 dB.
- Convolutional sharing: A 39K-parameter convolutional decoder reaches 32.2 dB on 128 single-class images while using 1.66× fewer bytes than raw pixels, achieving crisp and compressed storage together.On 128 maximally diverse images, the same decoder reaches 29.3 dB at the same size.
- Interpretation: Compression comes from decoder-shared regularity rather than the cyclic generator alone; collections without shared structure force the decoder toward memorization and lose the economy.The generator removes the independent latent table, while the decoder compresses regularity shared across items.
6 The explicit-storage control: the generative law is free
The explicit codebook control tests whether cyclic structure sacrifices reconstruction: it does not in these studies, while requiring far less latent storage and adding ordering and closure.
- Storage cost: The codebook uses 128 times more latent storage, with 16,000 latent parameters versus 128 for the cycle, and produces a visibly worse compression ratio.Both representations use the same decoders and image sets.
- Reconstruction control: The free codebook never wins reconstruction: it ties at the smaller decoder and loses modestly at the larger decoder on both shared-manifold and diverse image sets.Across three seeds, its mean advantage never exceeds +0.14 dB, while the cycle leads by more than 1 dB at the larger decoder.
- Quantitative comparison: At the larger decoder, the cycle leads by 1.16 ± 0.20 and 1.33 ± 0.25 dB across the two image settings.Negative gaps denote better reconstruction by the cycle.
- Consequence: The cyclic representation therefore preserves reconstruction while replacing independent instances with a generated orbit that provides total ordering, one-step transitions, and exact closure.These properties are not provided by the independent codebook alone.
7 When phase is meaning: looping video
Looping video gives the cyclic representation a natural phase order: time. The exactly periodic embedding matches or exceeds the frame-index baseline while guaranteeing machine-precision closure.
- Setup: The video representation uses temporal phase as the clip’s intrinsic cyclic order, unlike unordered image sets where order is imposed.The experiments use palindromed loops and 128 × 128 RGB clips with one decoder architecture across encoders.
- Benchmark results: 0.7 dB average PSNR advantage over the frame-index baseline across all seven clips, at equal parameter count.The cycle also remains roughly 2.5× smaller than raw frames.
- Benchmark results: 3.3 × 10^-7 maximum seam error for the cycle versus 1.5 × 10^-2 minimum for the frame-index representation, a visible baseline pop.The cycle wraps at numerical precision because frame T equals frame 0 by construction.
- Sampling boundary: Missing-phase evaluation fails with the full harmonic bank because frequencies above the training grid’s Nyquist limit alias between samples; Nyquist capping repairs it.This qualifies interpolation: consecutive video frames provide a usable manifold only when the spectrum respects the sampling grid.
- Periodic control: Harmonically pinning the baseline closes its seam at or below 8.5 × 10^-8, but trails the cycle by 1.9 dB on average across all seven clips.This control isolates exact periodicity as the closure ingredient while showing that the fixed rotation is the stronger carrier.
- Scope: For natively non-looping content played once, the construction offers no advantage over a standard frame-index embedding.Compression is compared with raw frames and a matched per-clip neural baseline, not conventional video codecs.
8 What the finite-knowledge perspective adds
The finite-knowledge perspective separates addressing, rendering, and shared structure: a cyclic group supplies order and recurrence, while the decoder supplies content. Its compression benefit depends on content regularity, and the one-dimensional cycle has clear structural limits.
- Representation principle: A finite collection can be encoded as an orbit when group action supplies addresses and recurrence, the decoder supplies content, and shared structure determines compression or interpolation.This separates the representation’s algebraic, generative, and content-dependent roles.
- Representation principle: For unrelated objects, the orbit provides compact addressing but does not create semantic neighborhoods between adjacent phases.The missing-phase experiment supports this boundary: unordered objects need not have meaningful intermediate phases.
- Natural cyclicity: For naturally cyclic collections such as looping video, the orbit can encode genuine transition structure because the collection already has a cyclic order.Periodic simulations, robotic gaits, biological cycles, and procedural animation are proposed extensions rather than empirically established applications.
- Limits: A single cyclic generator is unsuitable for taxonomies, graphs, permutation spaces, and branching workflows when forcing them into one orbit would destroy native adjacency.The paper limits its established result to one exactly periodic generator, one closed orbit, and one shared decoder.
- Compression: Compression comes from replacing X independent latent entries with a law, while total representation size still depends on the decoder’s ability to exploit shared structure.The work therefore compresses relations—how items are addressed and rendered—rather than only pixels.
9 Scope
Within its stated limits, the paper claims exact periodicity enables drift-free enumeration, preserves reconstruction quality while reducing latent storage, and turns loop seams into guaranteed closure.
- The evidence is small-scale and internally controlled, using grayscale 32 × 32 images, 128 × 128 clips, matched conditions, and limited replication for operator comparisons.
- The capacity ceiling is the decoder budget, so crisp compression requires shared structure in the represented collection.
- The present operator is limited to cyclically indexed finite collections rather than arbitrary graphs, taxonomies, or branching knowledge structures.
- Exact periodicity lets one operator enumerate many objects without accumulated drift and converts a looping video's seam into a guaranteed property of representation.
- In measured reconstruction settings, the periodic constraint costs nothing against independent latent codes while supplying total order, one-step transitions, and exact closure.