Topological Simplification in Predictive Coding Networks

arXiv:2608.02816 · cs.LG, math.AT · Submitted 2026-08-03 · Read on arXiv

Listen

Transcript

Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Next we'll be talking about the paper "Topological Simplification in Predictive Coding Networks".

Jane: The paper was written by Adam Shaw, Jiayu Li, Michael Sperling, Michael Kim and Alvin Jin from University of Southern California, University of Southern California, University of Southern California, University of Southern California, University of Southern California.

Tom: Stay tuned as we take you through the paper and discuss its implications.

Jane: We also have Lu with us today — senior AI researcher at Tsinghua.

Tom: We also have Meng with us today — lead engineer at a mysterious AI startup.

Jane: We also have Lalam with us today — the in-house Large Language Model.

Tom: Alright, let's get started.

Paper summary: Jane: The central idea is that as the "Predictive Coding Networks"—PCNs—process data, their internal representation starts simplifying its topological features.

Tom: And this isn's not just a random thing; it's tied to the model size and how well it performs. We’re looking at how topology changes across layers in these two different kinds of models: one that is a feedforward network, and another that' is an interconnected system called a PCN.

Lu: From my perspective, I find the idea of topological simplification itself really inspiring; it suggests the network isn't just memorizing patterns but actively organizing information into more stable structures.

Meng: As an engineer, I see this as critical because if we know *when* a model starts simplifying its structure, we can monitor it and understand potential failures or inefficiencies.

Lalam: This paper is fundamentally changing our understanding of the relationship between complexity and performance in AI by showing us that structural decay is a predictable part of the learning process.

Tom: So, Jane's point about this not being random needs to be tied into the how well it performs; Meng mentioned monitoring, which connects directly to this idea.

Jane: Right, because the paper found a very strong link between when this simplification happens and how accurate the model is at reconstructing its input data.

Lu: I’m particularly interested in Lu’s thought on structural integrity—if the network simplifies too, say, early, it’ might be losing essential information for reconstruction.

Meng: That's a crucial practical finding; we need to know if that loss of structure translates directly into poor real-world performance metrics.

Lalam: The impact here is that we are learning about the trade-offs between compression and fidelity in AI, which will shape how we design future systems.

Tom: And this is all summarized in "Topological Simplification in Predictive Coding Networks," setting the stage for a deeper dive into the specific mechanics of each section.

Page 1 of the paper: Jane: We start with Page one which sets up what's new here by defining persistent homology. Think of it as a mathematical tool to quantify how connected or complex data points are in high-dimensional space.

Tom: The authors are using this tool to track how that complexity changes as they move through layers of these PCNs, contrasting it with traditional feedforward networks.

Lu: I find the idea that feedforward networks simplify their topology is a known phenomenon, but the focus on PCNs is what's exciting here; because we know PCNs have to support both inference and reconstruction, that’s a much stricter constraint.

Meng: From an engineering standpoint, this distinction means we aren't just collapsing information arbitrarily like in standard classifiers; we need to preserve enough structure for "inversion," which sounds like a lot of extra computational overhead.

Lalam: The implication is that the requirements of generative modeling—making the output match the input—are fundamentally different from what simple classification demands, and that’s why this research matters so much.

Tom: So, we are looking at how "Topological Simplification in Predictive Coding Networks" measures this evolution, and Jane explained what persistent homology is while Lu highlighted the constraint difference between PCNs and feedforward models.

Jane: Exactly; it’s about comparing the constraints that guide a complex network's behavior across layers.

Lu: I wonder if we are seeing a new way to measure the "knowledge" or information content of a representation, based on its structure rather than just how many features it has.

Meng: That ties back to my concern about efficiency; if the required topology is more stable, we might be able to design more robust hardware for these systems.

Lalam: The goal here is to use this quantitative lens to understand the "compression-reconstruction tradeoff," which is a universal problem in data science.

Tom: This leads us into the core concepts of the methodology, specifically looking at how they set up their models on Page two.

Page 2 of the paper: Jane: Page two introduces "Predictive Coding Networks" and Figure one giving us a visual breakdown of how these networks actually work. It’s not one single forward pass; it’s an iterative process that looks both top-down (making predictions) and bottom-up (calculating prediction errors).

Tom: The paper highlights that this structure allows for "approximate inversion," which is the ability to reverse the process and reconstruct the original input from a known output.

Lu: I find this "inversion" capability so powerful; it means we can're not just getting a label, we are actually seeing how to recover the original data.

Meng: From an engineering perspective, this iterative, bidirectional flow seems like a huge difference in terms processing time and resource allocation compared to a standard feedforward pass.

Lalam: The culture shift here is that we move away from just "guessing" the answer to actively trying to reconstruct the truth, which is a major conceptual leap.

Tom: So, Jane explained how the iterative process works, Tom clarified its ability to invert itself, and Lu found it exciting; Meng brought up the engineering challenge of resource allocation for this complex structure.

Jane: Exactly; we are looking at "Topological Simplification in Predictive Coding Networks" within a framework that allows us to see the entire process both forward and backward.

Lu: I'm curious about how they manage those local interactions between neighboring layers, as they mentioned that's how PCNs approximate backpropagation without needing full global signals.

Meng: That locality constraint is interesting for implementation; it suggests we can build these systems with a more localized hardware footprint than if we needed to move massive weight matrices across the entire network.

Lalam: The focus on this structure allows us to understand the constraints that drive information preservation in AI, which is a major contribution.

Tom: This detailed look at the architecture sets up our next segment, which is all about how they measure this simplification using "Topological Simplification in Predictive Coding Networks" methodology.

Page 3 of the paper: Jane: Page three dives into the methodology, explaining how they use persistent homology to track topological complexity across different layers. It’s a way to quantify structure that goes beyond just simple Betti numbers.

Tom: They are systematically varying two factors—the hidden-layer width and the activation function—to see what drives this simplification process in PCNs.

Lu: I think the choice of using persistent homology is very clever because it allows us to capture features that are "robust," meaning they aren't just noise or random fluctuations in the data.

Meng: From an engineering viewpoint, understanding these specific variables helps us decide where to allocate resources; if we know that a wider layer delays simplification, we can make informed decisions about network design.

Lalam: The cultural shift here is moving from thinking of AI models as black boxes to seeing them as systems whose internal structure can be analyzed and understood deeply.

Tom: So, Jane explained the measurement tool, Tom clarified the variables they are testing—width and activation function—and Lu saw the robustness in persistent homology; Meng brought up design decisions based on results.

Jane: We are using "Topological Simplification in Predictive Coding Networks" to see how these choices influence things like Betti number decay.

Lu: It's interesting that we are comparing this behavior to standard feedforward networks, which adds a layer of comparison and context to the our findings.

Meng: That comparison is vital for me because it tells us if this phenomenon is unique to recurrent architectures or if it’ applies more broadly across many different types of AI design.

Lalam: The study is looking at "Topological Simplification in Predictive Coding Networks" as a way to quantify how structure evolves, which is a major step toward understanding the inner workings of AI.

Tom: This leads us into the specifics of how they define and measure this simplification on Page four.

Page 4 of the paper: Jane: On Page four they introduce two specific metrics to measure what they're looking for in "Topological Simplification in Predictive Coding Networks." First, there's the Center of Mass of Topological Simplification (COM).

Tom: COM is essentially a way to define the *timing* of simplification—the expected depth at which the network starts merging connected components. A lower COM means it simplifies early.

Lu: I find that concept fascinating; it gives us a concrete number for when the "irreversible geometric mergers" occur in representation space, rather than just saying they happen eventually.

Meng: The Center of Mass is a practical way to measure performance degradation related to structural collapse, which is very valuable for monitoring system health and reliability.

Lalam: It also tells us about the relationship between our architecture's capacity and its stability, influencing how we perceive the intelligence of the AI model itself.

Tom: So, Jane explained COM as a timing metric, Tom clarified what it means in terms of component merging, Lu found the concept fascinating, and Meng saw its practical value for monitoring system health.

Jane: We are using "Topological Simplification in Predictive Coding Networks" to quantify exactly when this structural change happens.

Lu: I wonder how different types of complexity—like loops versus flat regions—contribute to this Center of Mass measurement?

Meng: That’s an important detail for me; if we can measure the contribution of specific features, we might be able to design specialized hardware for certain topological requirements.

Lalam: The goal is to tie this measurable timing directly into the overall impact and behavior of the AI system, which is a major step forward.

Tom: This brings us right into the heart of their results on Page five where they start showing how different architectures perform.

Page 5 of the paper: Jane: On Page five they present their findings for synthetic data models, showing a consistent trend that is central to "Topological Simplification in Predictive Coding Networks." They found that smaller models simplify their topology earlier than larger ones.

Tom: This is a clear trade-off; smaller models collapse those connected components sooner, which is quantified by the COM metric we discussed.

Lu: The capacity of the model seems to act as a buffer against this simplification, allowing me to see how hardware choices directly influence the structural integrity of my AI's thought process.

Meng: This confirms that increasing capacity delays simplification, and it’ also gives us a clear direction for optimizing our own AI designs—we can choose size based on required structural stability.

Lalam: The cultural implication is that we are no longer just building bigger models to get better results; we' are also considering the internal structure and complexity of how they process information.

Tom: So, Jane outlined the main finding—smaller models simplify earlier—and Tom defined what that means in terms structural collapse, while Lu saw capacity as a buffer, and Meng pointed out optimization opportunities.

Jane: The paper is demonstrating "Topological Simplification in Predictive Coding Networks" isn't just about size; it’ also relates to the specific activation function used.

Lu: I noticed the effect was strongest with ReLU activations, which suggests that different mathematical functions have a different impact on how information is organized structurally.

Meng: That means our choice of activation function has a quantifiable impact on the hardware efficiency and robustness of our AI systems too.

Lalam: The focus on these results will help us understand how to design more resilient and explainable AI systems, which is something that matters to the wider world.

Tom: This leads us into a critical finding that links structure directly to performance on Page six.

Page 6 of the paper: Jane: On Page six we see a strong negative correlation between when this topological simplification happens and how well the model can reconstruct its original input data.

Tom: This is what they call the "simplification-reconstruction tradeoff," and it’s very intuitive; if you lose too much structure too early, you can't reconstruct the data accurately.

Lu: I find that connection deeply satisfying, because it shows a direct relationship between internal structural collapse and external output quality.

Meng: From an engineering perspective, this is a clear warning that we must prevent premature structural simplification if we want to maintain high fidelity in our systems.

Lalam: The impact here is that we are learning how to balance efficiency—getting a smaller, faster model—with the critical requirement of preserving information for reconstruction.

Tom: So, Jane explained the negative correlation between simplification and reconstruction, Tom defined this as a tradeoff, Lu found it satisfying to see the direct link between structure and output quality.

Jane: We are using "Topological Simplification in Predictive Coding Networks" to quantify this relationship, showing how structural integrity is tied to performance.

Lu: I wonder if there are specific types of structures that are more vital for reconstruction than others? And how does the model prioritize them?

Meng: That would be key for me; identifying which components of the manifold need protection during training is a huge practical advantage.

Lalam: The focus on this tradeoff will help us move toward designing AI that is not only fast but also highly accurate and explainable in its generative capabilities.

Tom: This leads into the real-world application of these findings, moving to Page seven.

Page 7 of the paper: Jane: Moving to Page seven we see how these findings hold up when the models are trained on a real-world dataset, MNIST. It’s not as controlled as the synthetic data.

Tom: The challenge is that with real data, we don't know the true topology beforehand, so they use different ways to measure things and compare results across different architectures.

Lu: I think it’s really important to see this consistency in real-world data because it suggests that this isn't just a quirk of our synthetic test cases; the principles are universal.

Meng: From an engineering standpoint, seeing how these trends hold up tells us that we can apply these principles to massive, messy datasets without losing the insights from controlled experiments.

Lalam: The cultural shift is recognizing that complex real-world data structures behave similarly to abstract mathematical shapes, which is a very profound idea for AI.

Tom: So, Jane outlined the challenge of using real data versus Tom’s point about consistency across Lu's view on universality and Meng’s practical application.

Jane: We are using "Topological Simplification in Predictive Coding Networks" to see if these patterns survive in complex, real-world scenarios like image recognition.

Lu: The consistency of the trends is a huge validation of the research, proving that our mathematical tools are robust enough to handle messy data.

Meng: I’m glad it works; it means we can trust these insights when scaling up to massive AI deployments and can rely on the structural predictions we’ve made.

Lalam: The focus on consistency in real-world examples will help us understand the limits of our current AI capabilities and where the next breakthroughs might be.

Tom: This leads us to the head-to-head comparison against Page eight when contrasting PCNs with traditional MLPs.

Page 8 of the paper: Jane: On Page eight they compare these PCNs to "matched" Multilayer Perceptrons—MLPs—which are standard feedforward networks. This is a crucial comparison for "Topological Simplification in Predictive Coding Networks."

Tom: The result was very consistent: the PCNs always have a larger COM than the MLPs. In simple terms, they simplify their topology much later than the standard models of the same size and activation function.

Lu: I find that difference so interesting because it confirms that's not just a matter of how big we build the network, but fundamentally how we design its connections and its objective.

Meng: The average difference was three point six layers, which provides a concrete number for me—it tells us exactly how many more steps an engineer needs to account for in terms of computational depth when choosing between these two architectures.

Lalam: This highlights that the structural constraints of PCNs are fundamentally different from MLPs, and that is a major lesson for rethinking AI architecture design.

Tom: So, Jane explained the comparison with MLPs, Tom provided the concrete number (three point six layers), Lu saw it as a fundamental difference in design constraints, and Meng gave us a practical measure of depth.

Jane: We are using "Topological Simplification in Predictive Coding Networks" to show that the inherent nature of PCNs resists simplification compared to standard neural networks.

Lu: It suggests that the ability to reconstruct the input might be forcing a level of structural preservation we otherwise wouldn't see.

Meng: That’s very helpful for implementation; if we need a more complex, stable representation, we just choose the architecture that delays this collapse.

Lalam: The comparison will help us guide future AI development toward architectures that are more robust and structurally coherent.

Tom: This final technical comparison sets up our concluding discussion on Page nine.

Conclusion: Jane: We've reached the conclusion of "Topological Simplification in Predictive Coding Networks," summarizing the key findings we've discussed today. It’s a lot to take in!

Tom: In short, we found that smaller models simplify earlier, and that this structural decay is directly correlated with poorer reconstruction performance.

Lu: The core idea is that capacity influences the timing of topological collapse, and this relationship dictates the trade-off between efficiency and fidelity in AI systems.

Meng: My takeaway from "Topological Simplification in Predictive Coding Networks" is that we can now use structural metrics like COM to make informed engineering decisions about how to build robust and efficient AI.

Lalam: The ultimate impact is a much deeper understanding of the relationship between structure and performance, leading us to a more nuanced view of what it means for an AI system to be "intelligent."

Tom: So, Jane summarized the findings, Tom gave us the core takeaway regarding size vs. collapse, Lu framed it in terms of structural integrity and fidelity.

Jane: And Meng provided the engineering perspective on how we can use this data for practical design choices.

Lu: I just hope future work will validate these trends across broader datasets, proving that our mathematical insights hold true everywhere.

Meng: We're ready to apply these results to see if they hold up in large-scale production environments, as the next step in testing this "Topological Simplification in Predictive Coding Networks."

Lalam: The study has opened a new perspective on how we design and understand AI, giving us a powerful tool for the future of intelligent systems.

Tom: It’s been a fantastic deep dive into this paper! We’re ready to wrap up our discussion on "Topological Simplification in Predictive Coding Networks."

Adam Shaw, Jiayu Li, Michael Sperling, Michael Kim, Alvin Jin

University of Southern California, University of Southern California, University of Southern California, University of Southern California, University of Southern California

cs.LG, math.AT

Submitted: 2026-08-03

Updated: 2026-08-20

Code: https://github.com/jiayuliusc/Topological-Simplification-in-Predictive-Coding-Networks

Importance score: 84/100

The gist: Summary of "Topological Simplification in Predictive Coding Networks" This study investigates the evolution of representation topology within predictive coding networks (PCNs), a bidirectional,

Key concepts

Predictive Coding Networks (PCNs): A type of interconnected AI system where information processing is iterative, moving both top-down (making predictions) and bottom-up (calculating prediction errors). This structure allows the model to reconstruct the original input data.
Topological Simplification: The process where a network's internal representation simplifies its topological features as it processes data. It is measured by tracking how complexity changes across layers, contrasting PCNs with traditional feedforward networks.
Center of Mass (COM): A specific metric used to define the timing of simplification—the expected depth at which the network starts merging connected components in its representation space. A lower COM indicates an earlier simplification.

Terminology

Predictive Coding — 44× in this paper · explained in 2 episodes
Persistent Homology — 30× in this paper · explained in 3 episodes
Activation Function — 23× in this paper · explained in 1 episode
Accuracy — 11× in this paper · explained in 1 episode
Fidelity — 9× in this paper · explained in 2 episodes
Consistency — 7× in this paper · explained in 3 episodes
Mean Reconstruction Distance — 4× in this paper
Point Clouds — 4× in this paper · explained in 2 episodes
Structural Integrity — 4× in this paper · explained in 3 episodes
Center of Mass of Topological Simplification — 3× in this paper
Core — 3× in this paper · explained in 2 episodes
Generative Model — 3× in this paper · explained in 2 episodes
Reconstruction Error — 3× in this paper · explained in 2 episodes
Adam — 2× in this paper · explained in 2 episodes
Generative Models — 2× in this paper · explained in 2 episodes
Robustness — 2× in this paper · explained in 5 episodes
Benchmarking — 1× in this paper · explained in 3 episodes
Computational overhead — 1× in this paper · explained in 2 episodes
Generative Modeling — 1× in this paper · explained in 3 episodes
Latent Space — 1× in this paper · explained in 7 episodes
Representation Learning — 1× in this paper · explained in 2 episodes
Structural Constraints — 1× in this paper · explained in 2 episodes
alpha — 1× in this paper · explained in 1 episode

Summary

Summary of Topological Simplification in Predictive Coding Networks

This study investigates the evolution of representation topology within predictive coding networks (PCNs), a bidirectional, hierarchical generative architecture, using persistent homology. The research aims to determine if topological simplification is a general property of deep representations and how it relates to the ability of PCNs to support reconstruction.

Methodology and Framework:

The researchers utilize persistent homology as a quantitative tool for tracking topological structure in finite point clouds by summarizing homological features across scales via persistence barcodes (Edelsbrunner et al., 2002). The study employs two novel metrics:

Center of Mass of Topological Simplification (COM): This quantifies when simplification occurs. It is defined based on the layerwise Betti drop and a distribution p = /D, where D is the total simplification across all layers. The COM is calculated as the expected depth of topological simplification:

COM = sum=1 L+1 p(

A lower COM indicates earlier topological collapse.

Mean Reconstruction Distance (MRD): This measures reconstruction fidelity. For a set of N reconstructions i from M model seeds, the MRD is the mean Euclidean distance between the reconstruction and its nearest same-class neighbor x i:

MRD(A) = 1 over MN sum m=1 M sum i=1 N i - x i squared

This metric captures whether inversions lie near the correct data manifold.

The experiments were conducted on two datasets: a synthetic, topologically complex two-dimensional manifold (Ma embedded in Mb) and the real-world MNIST dataset. Models were trained to achieve high accuracy (≥ 99.9% for synthetic data; ≥ 95% for MNIST).

Key Findings from Synthetic Data Analysis:

The results indicate a systematic variation in topological simplification based on architectural capacity and activation function:

Model Size vs. Simplification Timing: Our results indicate that smaller models exhibit topological simplification earlier, on average, than larger models. This relationship was quantified by correlating COM with model size (the sum of hidden-layer widths). The correlation was strongly positive across different activations: rho = 0.76 for ReLU, 0.79 for Leaky ReLU, and 0.72 for tanh.
Capacity Sensitivity: A linear model was fitted to determine the effect of capacity (P) on COM: COM a(P) = alpha a + gamma a P + epsilon. The slopes were positive for all activations (e.g, 100 = 1.79 for ReLU). The effect was noted to be strongest for ReLU, suggesting that ReLU PCNs are more capacity-sensitive.
The Simplification-Reconstruction Tradeoff: A strong negative correlation (rho = -0.58, p < 10-3) was observed between COM and MRD. This means architectures that delay topological simplification preserve invertibility, while early simplification leads to poorer reconstruction quality.

Findings from Real-World Data (MNIST):

The analysis was extended to the MNIST dataset, tracking beta 0 and beta 1 across layers for the digit-0 class. Although correlation was limited by the small number of representative architectures, trends consistent with synthetic experiments were observed: PCNs exhibit progressive topological simplification across layers.

Comparison with Feedforward Networks (MLPs):

The PCNs were compared against matched Multilayer Perceptrons (MLPs). The results showed a consistent positive COM gap: PCNs have larger COMs than matched MLPs for all architectures and activations tested. The bootstrap analysis found an average COM difference (COM) of 3.6 layers, meaning PCNs simplified topology on average 3.6 layers later than an MLP of the same architecture and activation.

Conclusion:

The study concludes that topological simplification is not uniform across architectures: smaller models simplify earlier, while larger models (trained to the same accuracy) preserve topological structure deeper into the network. Furthermore, the findings demonstrate a capacity–topology–reconstruction tradeoff in PCNs: increasing capacity can delay topological simplification, improving invertibility, whereas smaller models with earlier simplification produce more compressed but less reconstructible representations.

Improvements for AI systems

As a diligent researcher, I have thoroughly analyzed this paper on Topological Simplification in Predictive Coding Networks (PCNs). The findings provide a powerful quantitative framework—using Persistent Homology metrics like COM (Center of Mass of Topological Simplification) and MRD (Mean Reconstruction Distance)—to address fundamental trade-offs in deep learning architectures.

The key insight is that the recurrent, bidirectional dynamics inherent in PCNs constrain topological collapse compared to standard feedforward networks (MLPs), and that this structural preservation directly influences the model's ability to perform accurate inversion/reconstruction.

Based on these results, I propose several highly specific improvements and corresponding applications for various AI systems.

Improvement: Implement dynamic architectural constraints based on target complexity and required invertibility, rather than relying solely on fixed layer counts or width.

Implementation: When designing a generative model (e.g, for image synthesis or complex data manifold reconstruction), the architecture should be selected such that its expected COM is maximized relative to the task requirements.

Mechanism: Utilize models with larger hidden-layer widths (P) and/or more layers, as these architectures exhibit a stronger positive correlation (gamma about 0.79 for Leaky ReLU) between capacity and delayed topological simplification.
What the System Does: The improved system can be designed to preserve critical structural information (e.g, holes in a manifold or specific connectivity) deep within its layers, ensuring that the generated output is structurally faithful to the input data distribution, thereby preventing premature collapse of complex features.

Improvement: Replace traditional loss-based early stopping criteria with topological degradation monitoring.

Implementation: During training, continuously calculate a running estimate of (the running minimum Betti number) for the target manifold M a. Define a Topological Degradation Threshold (tau) as the point where the rate of Betti number decay exceeds a predefined limit.

Mechanism: If (the change in) indicates rapid, irreversible simplification—suggestive of an early topological collapse—the training is halted or regularization is increased, even if the standard loss function suggests further convergence.
What the System Does: The improved system maintains its internal representational fidelity throughout the learning process. It prevents catastrophic forgetting or structural degradation in latent space, ensuring that the learned representation retains the necessary complexity to support high-fidelity reconstruction (minimizing MRD).

Improvement: Use COM as a diagnostic tool to predict when a model is likely operating outside of its optimal reconstructive regime.

Implementation: For a deployed PCN, monitor the calculated COM during inference or use it to estimate the expected structural integrity. If the measured operational COM deviates significantly from the designed capacity-driven COM, flag a potential failure mode.

Mechanism: Since early topological simplification correlates strongly with weak reconstruction (rho = -0.58), this metric serves as a proxy for structural instability.
What the System Does: The improved system can preemptively identify instances where its learned representation is insufficient to support inversion or accurate reconstruction, allowing for real-time error handling or switching to a higher-capacity backup model before data quality degrades.

Improvement: Refine the inversion process by prioritizing architectures that maximize MRD fidelity over simple gradient descent speed.

Implementation: When performing top-down reconstruction (inversion), use a cost function that penalizes not only the reconstruction error but also the deviation from a high-fidelity topological profile (i.e., choosing an architecture/initialization that yields a lower MRD).

Mechanism: Instead of relying on standard gradient descent starting from random initialization, we leverage the fact that PCNs naturally support approximate inversion. The optimization is guided by selecting the path through the energy landscape F that maximizes the preservation of connected components (i.e., minimizes early topological collapse).
What the System Does: The improved system generates reconstructions that are not just statistically plausible, but structurally consistent with complex input manifolds, leading to superior semantic fidelity compared to models that collapse structure prematurely.