# Coupled Scaling: A Representational Accessibility Framework for Neural Scaling Laws

arXiv: https://arxiv.org/abs/2609.03533 · HTML: https://mlsanity.com/paper/2609.03533/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 33 of its 39 references that are in the map.
- First posted on arXiv 3 Sep 2026.

## Builds on (most cited first, 30 of 33)

- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Deep Learning Scaling is Predictable, Empirically](https://mlsanity.com/paper/1712.00409/) (2017)
- [Are Emergent Abilities of Large Language Models a Mirage?](https://mlsanity.com/paper/2304.15004/) (2023)
- [Explaining Neural Scaling Laws](https://mlsanity.com/paper/2102.06701/) (2021)
- [Repeat After Me: Transformers are Better than State Space Models at Copying](https://mlsanity.com/paper/2402.01032/) (2024)
- [Zoology: Measuring and Improving Recall in Efficient Language Models](https://mlsanity.com/paper/2312.04927/) (2023)
- [Spectrum Dependent Learning Curves in Kernel Regression and Wide Neural Networks](https://mlsanity.com/paper/2002.02561/) (2020)
- [Spectral Bias and Task-Model Alignment Explain Generalization in Kernel Regression and Infinitely Wide Neural Networks](https://mlsanity.com/paper/2006.13198/) (2020)
- [The Quantization Model of Neural Scaling](https://mlsanity.com/paper/2303.13506/) (2023)
- [A Solvable Model of Neural Scaling Laws](https://mlsanity.com/paper/2210.16859/) (2022)
- [A Dynamical Model of Neural Scaling Laws](https://mlsanity.com/paper/2402.01092/) (2024)
- [Broken Neural Scaling Laws](https://mlsanity.com/paper/2210.14891/) (2022)
- [Scaling Laws vs Model Architectures: How does Inductive Bias Influence Scaling?](https://mlsanity.com/paper/2207.10551/) (2022)
- [How Feature Learning Can Improve Neural Scaling Laws](https://mlsanity.com/paper/2409.17858/) (2024)
- [Data Scaling Laws in NMT: The Effect of Noise and Architecture](https://mlsanity.com/paper/2202.01994/) (2022)
- [Superposition Yields Robust Neural Scaling](https://mlsanity.com/paper/2505.10465/) (2025)
- [Scaling Laws and Spectra of Shallow Neural Networks in the Feature Learning Regime](https://mlsanity.com/paper/2509.24882/) (2025)
- [SMELT: Scaling Laws for Compute-Matched MoE Looped Transformers](https://mlsanity.com/paper/2609.01343/) (2026)
- [What do Language Models Learn and When? The Implicit Curriculum Hypothesis](https://mlsanity.com/paper/2604.08510/) (2026)
- [Why Larger Models Learn More: Effects of Capacity, Interference, and Rare-Task Retention](https://mlsanity.com/paper/2605.29548/) (2026)
- [Optimal scaling laws in learning hierarchical multi-index models](https://mlsanity.com/paper/2602.05846/) (2026)
- [Towards Robust Scaling Laws for Optimizers](https://mlsanity.com/paper/2602.07712/) (2026)
- [Spectral Lens: Activation and Gradient Spectra as Diagnostics of LLM Optimization](https://mlsanity.com/paper/2605.05683/) (2026)
- [Neural Scaling Universality: If Exponents Are Fixed, Time to Understand Coefficients](https://mlsanity.com/paper/2606.25008/) (2026)
- [Scaling Laws and Symmetry, Evidence from Neural Force Fields](https://mlsanity.com/paper/2510.09768/) (2025)
- [A Qualitative Test-Risk Mechanism for Scaling Behavior in Normalized Residual Networks](https://mlsanity.com/paper/2605.08297/) (2026)
- [Same Architecture, Different Capacity: Optimizer-Induced Spectral Scaling Laws](https://mlsanity.com/paper/2605.21803/) (2026)
- [Scaling Domain Data Repetition in LLM Pretraining](https://mlsanity.com/paper/2608.14071/) (2026)
- [Data Scaling as Progressive Coverage of a Predictive Contribution Spectrum](https://mlsanity.com/paper/2605.20196/) (2026)

## Abstract (arXiv)

We ask when two learning systems trained on the same task under a common resource protocol should share a scaling rate and when their rates should differ. Coupled Scaling answers this through representational accessibility: the task-relevant geometry that a specified architecture-optimization system can reach and how that geometry is acquired as resources grow. In an orthogonal model, unsupported target energy sets the asymptotic floor, while unacquired supported energy sets the finite-budget residual. When acquisition can skip high-value directions, the largest fully acquired prefix no longer determines a unique exponent. For target powers aj asymp j-b, b>1, and prefix log-growth rate 0<ρ≤1, the sharp attainable interval is [ρ(b-1),minb-1,ρ b], and every rate in this interval is realized by a fixed acquisition order on a common support. A rank-window condition identifies when the product law α=ρ(b-1) applies, while a fixed-kernel specialization expresses the same task-system dependence through the task-weighted spectral tail. We then audit released capability trajectories to diagnose measurement and comparison effects and propose a staged test in which independently measured geometry predicts held-out loss, scaling rates, and cross-task reversals.