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arxiv: 2604.12946 · v1 · submitted 2026-04-14 · 💻 cs.LG

Recognition: no theorem link

Parcae: Scaling Laws For Stable Looped Language Models

Hayden Prairie , Zachary Novack , Taylor Berg-Kirkpatrick , Daniel Y. Fu
Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:29 UTC · model grok-4.3

classification 💻 cs.LG
keywords looped language modelsscaling lawsspectral normstable trainingtransformer architectureparameter efficientdynamical system
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The pith

Parcae stabilizes looped language models by constraining spectral norms of injection parameters, enabling predictable scaling laws that improve quality with fixed parameters by increasing FLOPs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Looped architectures reuse layers to raise training FLOPs without adding parameters or data, but prior versions suffer instability such as residual explosion. The paper models the loop as a nonlinear time-variant dynamical system over the residual stream and shows via linear approximation that instability stems from large spectral norms in injection parameters. Parcae introduces a negative diagonal parameterization that is discretized to bound these norms while preserving capacity. With stability achieved, the authors derive power laws for scaling loops during training and saturating exponential decay for test-time looping. At 1.3B parameters under fixed budget, Parcae delivers measurable quality gains over transformer baselines and reaches 87.5 percent relative quality of a model twice its size.

Core claim

By recasting looping as a dynamical system and using a linear approximation to locate instability in large spectral norms, Parcae discretizes a negative diagonal parameterization to constrain those norms, producing stable training and explicit power-law scaling for increasing FLOPs via loops at fixed parameter count.

What carries the argument

Negative diagonal parameterization of injection parameters, discretized to enforce spectral norm bounds within the looped residual-stream dynamics.

If this is right

  • For a fixed FLOP budget, training quality improves when loops and data are scaled together rather than one alone.
  • Test-time looping produces quality gains that follow a predictable saturating exponential decay.
  • Parcae yields up to 6.3 percent lower validation perplexity than earlier large-scale looped models.
  • At 1.3B parameters Parcae raises CORE and Core-Extended scores by 2.99 and 1.18 points over strong transformer baselines under identical parameter and data limits.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the derived power laws continue to larger scales, looped models could reduce peak memory during training by trading repeated passes for wider layers.
  • The spectral-norm control may apply directly to other iterative or recurrent blocks in sequence models.
  • Combining Parcae loops with mixture-of-experts routing could multiply the effective FLOP scaling without proportional parameter growth.

Load-bearing premise

The linear approximation to the nonlinear time-variant dynamical system accurately identifies instability sources, and discretizing the negative diagonal parameterization constrains norms without loss of capacity.

What would settle it

Train a 1.3B Parcae model and check whether residual explosion or loss spikes appear, or whether quality fails to rise as predicted when the number of loops is increased.

Figures

Figures reproduced from arXiv: 2604.12946 by Daniel Y. Fu, Hayden Prairie, Taylor Berg-Kirkpatrick, Zachary Novack.

Figure 1
Figure 1. Figure 1: Parcae and the Scaling Laws of Looping. (Left) Parcae constrains the spectral norm of A and normalizes the input injection, stabilizing the residual stream ht across loops. (Right) We observe looping to be an orthogonal axis of scaling compute which follows a power law. can be difficult to understand analytically. As a result, training requires sensitive hyperparameter selection and residual normalization … view at source ↗
Figure 2
Figure 2. Figure 2: Training Instability of Looped Architectures. (left) Pre-Norm looped models diverge, while residual norm. and Parcae converge. (right) Instability stems from an exploding recurrent state norm ||hT ||2, the hidden embedding norm after T recurrences. through first-order differential equations h˙(t) = Ah(t) + Be(t), y(t) = Ch(t) that describe the evolution of a hidden state h(t) ∈ R dh given an input signal e… view at source ↗
Figure 3
Figure 3. Figure 3: Spectral Radius of Unconstrained A. For a Pre-Norm RDM, we plot the ρ(A) throughout training using different learning rates, observing divergent runs learn ρ(A) > 1. The state explosion, in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Looping Scales Training Compute Optimally. (Left) Parametric isoLoss contours over µrec and data. The efficient frontier (blue line) traces the lowest FLOP budget required to achieve each loss level, showing that optimal training requires increased looping. (Right) Parabolic isoFLOP fits for 140M and 370M models reveal a clear optimum µrec at each FLOP budget, indicating that looping is an orthogonal scali… view at source ↗
Figure 5
Figure 5. Figure 5: Optimal µrec and Tokens Follows Predictable Power Laws. We fit a parabola to each isoFLOP budget for both 140M and 370M Parcae models, using its minima to approximate the optimal µrec and token budget at each scale. We observe that optimal recurrence (left plots) and tokens (right plots) follow a predictable power law with similar coefficients at both scales. 10 18 10 19 10 20 10 21 2.75 3.00 3.25 Val. Los… view at source ↗
Figure 6
Figure 6. Figure 6: Pareto Frontier of Loop￾ing. We observe that looping has a stricter IsoFLOP optimal loss frontier over fixed-depth, non-looped models. Dots are empirical points. FLOPs Optimal µ ∗ rec Fixed-Depth (×1018) µ ∗ rec Core Core Ext. Core Core Ext. 140M 1 2 7.6 5.7 7.9 6.1 2 2 9.0 6.2 10.5 6.4 4 4 11.2 8.4 10.7 8.1 8 6 10.5 7.8 11.8 7.7 16 8 14.6 9.8 13.0 8.8 64 10 16.2 11.0 15.0 9.5 370M 32 4 15.2 10.1 16.8 11.2… view at source ↗
Figure 7
Figure 7. Figure 7: Test-Time Scaling of Parcae. When evaluating Parcae models from [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Scaling Test-Time Compute follows a Predictable Power Laws. We plot the validation loss with different µrec as a function of test-time recurrence T, and find the fitted exponential decay (solid curve for each µrec) tightly captures the test-time performance of looping. 5.3 Test-Time Scaling Laws of Parcae We study looping as a mechanism for scaling test-time compute. We find the test-time compute follows a… view at source ↗
Figure 9
Figure 9. Figure 9: Training instability of recurrent depth models across different learning rates. We show [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Training curves showing per-sequence sampling effectively eliminates loss spikes in [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of recurrent residual and state norm metrics (defined in Section A.1), which [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of our sampling method with [ [PITH_FULL_IMAGE:figures/full_fig_p027_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: A distributional mismatch can be observed from the recurrent sampling method of [ [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Training and validation curves of three 100 million parameter Parcae models pretrained on [PITH_FULL_IMAGE:figures/full_fig_p028_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Validation curves of six different recurrent depth models, pretrained on 10 billion tokens, [PITH_FULL_IMAGE:figures/full_fig_p029_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Validation and training curves of looped models, pretrained on 8.5 billion tokens. Each [PITH_FULL_IMAGE:figures/full_fig_p030_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Late Stage Instability of 1.3B Parcae models. We observe loss spikes and state explosion at the final stages of our large-scale run. 0 25k 50k 75k 100k 125k 150k 175k Optimizer Step (k) 0.5 0.6 0.7 0.8 0.9 m a x(e x p( A)) Spectral Norm of A (Discretized A) 0 25k 50k 75k 100k 125k 150k 175k Optimizer Step (k) 1 2 3 4 5 6 7 8 9 max( B) Spectral Norm of B (Discretized B) 0 25k 50k 75k 100k 125k 150k 175k Op… view at source ↗
Figure 18
Figure 18. Figure 18: Spectral Norms of A, B, C throughout training 1.3B Parcae. We find that the spectral norm of A and B remain stable throughout training, while the spectral norm of C grows. We begin by exploring the spectral norm of A, B, C to see if our dynamical systems block was creating instability, results of which can be found in [PITH_FULL_IMAGE:figures/full_fig_p031_18.png] view at source ↗
Figure 20
Figure 20. Figure 20: We found that on the first recurrence, the recurrent state norm jumped drastically, and [PITH_FULL_IMAGE:figures/full_fig_p031_20.png] view at source ↗
Figure 19
Figure 19. Figure 19: Comparison of C Amplification with Spectral Norm. We observe that the actual expansion ratio of C is small and decreasing slowly throughout training. 0 5 10 15 20 25 Recurrence Iteration (T) 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 State Norm State Norm Across Recurrence Iterations 25k 50k 75k 100k 125k 150k 175k Optimizer Step [PITH_FULL_IMAGE:figures/full_fig_p032_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Empirical Average of Recurrent State Norm over T iterations. For each checkpoint we have for our failed 1.3B Parcae model run, we evaluate the recurrent norm through T = 24 recurrences at test time, on a held out validation set of fineweb-edu [60]. We find that after an initial explosion on the first recurrence, the state remains relatively stable. initial spike, we perform a fine-grained analysis of the … view at source ↗
Figure 21
Figure 21. Figure 21: Recurrent State Norm Progression After Each Transformer Block for T = 1. For each checkpoint we have for our failed 1.3B Parcae model run, we evaluate the recurrent norm after injection and each non-linear transformer block for only T = 1. We find that the non-linear parts of Parcae have little effect on explosion, which instead mainly stems from the initial injection of prelude output e. emb L0 L1 L2 L3 … view at source ↗
Figure 22
Figure 22. Figure 22: State Norm Progression Throughout each Transformer Layer in the Prelude Block. For each checkpoint we have for our failed 1.3B Parcae model run, we evaluate residual norm after each transformer block in the prelude P. We find that a single layer creates an explosion of the residual norm and leads to divergence. preventing the recurrent norm from growing too large (see [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 23
Figure 23. Figure 23: Prelude Norm Stabilizes Recurrent Norm. We find that prelude norm helps stabilize recurrent state norm in Parcae models following the setup in Section 5.1 for Transformers. 0 5 10 15 20 Optimizer Step (k) 3.0 3.2 3.4 3.6 3.8 Validation Loss Parcae 140M 0 10 20 30 40 50 Optimizer Step (k) 2.8 3.0 3.2 3.4 3.6 Validation Loss Parcae 370M Without PN With PN [PITH_FULL_IMAGE:figures/full_fig_p034_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Prelude Norm Improves Quality. We find that in our 140M and 370M Parcae models trained in the same setup as Section 5.1 for Transformers, normalizing the prelude output leads to better convergence. 34 [PITH_FULL_IMAGE:figures/full_fig_p034_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Parametric Fit of Looping. Visualization of our parametric function Lbtrain(µrec, D), which displays the IsoLoss contours for both 140M Parcae (left) and 370M Parcae (right) models. Model E A a B b Huber (×10−4 ) Small (140M) 2.662 522733.307 0.771 25420.102 0.525 0.44 Medium (370M) 2.439 832134.346 0.775 6386.865 0.448 0.01 [PITH_FULL_IMAGE:figures/full_fig_p035_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Out-of-Distribution Prediction of Unified Parametric Fit. We visualize the prediction of our unified parametric fit (orange) and an oracle fit using the empirical loss at T = µrec for Lbtrain (blue) against empirical validation loss with increasing T for models trained in Section 5.1. M Extended Evaluation Details and Setup We include a complete list of benchmarks used for evaluation in [PITH_FULL_IMAGE:… view at source ↗
read the original abstract

Traditional fixed-depth architectures scale quality by increasing training FLOPs, typically through increased parameterization, at the expense of a higher memory footprint, or data. A potential alternative is looped architectures, which instead increase FLOPs by sending activations through a block of layers in a loop. While promising, existing recipes for training looped architectures can be unstable, suffering from residual explosion and loss spikes. We address these challenges by recasting looping as a nonlinear time-variant dynamical system over the residual stream. Via a linear approximation to this system, we find that instability occurs in existing looped architectures as a result of large spectral norms in their injection parameters. To address these instability issues, we propose Parcae, a novel stable, looped architecture that constrains the spectral norm of the injection parameters via discretization of a negative diagonal parameterization. As a result, Parcae achieves up to 6.3% lower validation perplexity over prior large-scale looped models. Using our stable looped architecture, we investigate the scaling properties of looping as a medium to improve quality by increasing FLOPs in training and test-time. For training, we derive predictable power laws to scale FLOPs while keeping parameter count fixed. Our initial scaling laws suggest that looping and data should be increased in tandem, given a fixed FLOP budget. At test-time, we find that Parcae can use looping to scale compute, following a predictable, saturating exponential decay. When scaled up to 1.3B parameters, we find that Parcae improves CORE and Core-Extended quality by 2.99 and 1.18 points when compared to strong Transformer baselines under a fixed parameter and data budget, achieving a relative quality of up to 87.5% a Transformer twice the size.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes Parcae, a looped language model that stabilizes training by modeling the loop as a nonlinear time-variant dynamical system over the residual stream. A linear approximation identifies large spectral norms in injection parameters as the instability source, addressed via discretization of a negative diagonal parameterization. The work derives training scaling laws (power laws suggesting tandem increases in looping and data at fixed FLOP budget) and test-time scaling (saturating exponential decay), and reports that at 1.3B parameters Parcae improves CORE and Core-Extended scores by 2.99 and 1.18 points over strong Transformer baselines under fixed parameter/data budgets, reaching up to 87.5% relative quality of a twice-larger Transformer.

Significance. If the linear approximation accurately diagnoses instability and the scaling laws prove robust, Parcae offers a memory-efficient alternative to parameter scaling for quality gains via increased FLOPs. The empirical results at 1.3B scale and the derivation of predictable laws (rather than purely empirical fits) are notable strengths that could influence efficient architecture design.

major comments (3)
  1. [§3] §3 (Stability via Dynamical System): The central stability claim rests on the linear approximation to the nonlinear time-variant residual dynamics correctly identifying large spectral norms of injection parameters as the root cause, motivating the negative-diagonal discretization. No experiments or analysis verify that nonlinear interactions do not dominate, so the parameterization may constrain norms without addressing the actual failure mode; this is load-bearing for both stability and downstream scaling claims.
  2. [§4.1] §4.1 (Training Scaling Laws): The manuscript states that 'predictable power laws' govern scaling FLOPs at fixed parameter count and recommends increasing looping and data in tandem. However, no goodness-of-fit statistics (R², residual norms, or cross-validation details) or explicit functional forms are provided for the fitted laws, weakening the 'predictable' assertion and the recommendation for joint scaling.
  3. [§5.3] §5.3 (Test-time Scaling and 1.3B Evaluation): The saturating exponential decay for test-time looping is presented as predictable, and the 1.3B results claim 2.99/1.18 point gains plus 87.5% relative quality. The baseline comparisons lack explicit confirmation that Transformer controls received matched hyperparameter search or optimization budgets, risking attribution of gains to the Parcae parameterization rather than other factors.
minor comments (2)
  1. [Abstract] Abstract: The reported 'up to 6.3% lower validation perplexity over prior large-scale looped models' does not name the specific prior models or the scale at which the comparison holds.
  2. [§3] Notation and §3: The discretization step mapping the continuous negative diagonal parameterization to a discrete spectral-norm constraint would benefit from an explicit equation or pseudocode to clarify capacity preservation.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback on our manuscript. We address each major comment point-by-point below, providing clarifications and committing to revisions where they strengthen the work without misrepresenting our contributions.

read point-by-point responses

  1. Referee: [§3] §3 (Stability via Dynamical System): The central stability claim rests on the linear approximation to the nonlinear time-variant residual dynamics correctly identifying large spectral norms of injection parameters as the root cause, motivating the negative-diagonal discretization. No experiments or analysis verify that nonlinear interactions do not dominate, so the parameterization may constrain norms without addressing the actual failure mode; this is load-bearing for both stability and downstream scaling claims.

    Authors: The linear approximation follows standard practice in dynamical systems analysis to identify dominant instability modes, as nonlinear systems are often diagnosed via their linearized behavior near equilibria. While nonlinear interactions are present, the parameterization's empirical success in stabilizing training (where prior looped models exhibit explosion and spikes) validates its practical utility. We will revise §3 to explicitly discuss the approximation's limitations as a diagnostic tool and note that full nonlinear verification remains an open direction. revision: partial

  2. Referee: [§4.1] §4.1 (Training Scaling Laws): The manuscript states that 'predictable power laws' govern scaling FLOPs at fixed parameter count and recommends increasing looping and data in tandem. However, no goodness-of-fit statistics (R², residual norms, or cross-validation details) or explicit functional forms are provided for the fitted laws, weakening the 'predictable' assertion and the recommendation for joint scaling.

    Authors: We agree that explicit functional forms and quantitative fit metrics are needed to support the predictability claim. In the revision, we will report the exact power-law equations (e.g., loss as function of FLOPs, loops, and data), R² values, residual diagnostics, and fitting methodology to substantiate the recommendation for tandem scaling of loops and data. revision: yes

  3. Referee: [§5.3] §5.3 (Test-time Scaling and 1.3B Evaluation): The saturating exponential decay for test-time looping is presented as predictable, and the 1.3B results claim 2.99/1.18 point gains plus 87.5% relative quality. The baseline comparisons lack explicit confirmation that Transformer controls received matched hyperparameter search or optimization budgets, risking attribution of gains to the Parcae parameterization rather than other factors.

    Authors: The Transformer baselines followed standard hyperparameter configurations from the literature, with additional tuning to match the fixed parameter and data budgets. We will revise §5.3 to detail the hyperparameter ranges explored, optimization procedures, and final settings for the baselines, clarifying that gains are attributable to the Parcae architecture under comparable training conditions. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external empirical validation rather than self-reduction

full rationale

The abstract describes recasting looped models as a nonlinear dynamical system, applying a linear approximation to diagnose instability from spectral norms, and introducing a negative-diagonal discretization for stability. Scaling laws are stated as 'predictable power laws' and 'saturating exponential decay' derived for fixed-parameter FLOP scaling, with quality gains validated at 1.3B parameters against baselines. No equations or self-citations are provided that reduce any claimed prediction or uniqueness result to a fitted input or prior author work by construction; the central claims rest on observed stability and quality metrics rather than tautological reparameterization.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides insufficient detail to enumerate free parameters, axioms, or invented entities beyond the high-level dynamical-systems framing.

pith-pipeline@v0.9.0 · 5619 in / 1139 out tokens · 56197 ms · 2026-05-10T15:29:19.183577+00:00 · methodology

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