# ISO: An RLVR-Native Optimization Stack

arXiv: https://arxiv.org/abs/2607.19331 · HTML: https://mlsanity.com/paper/2607.19331/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 31 of its 41 references that are in the map.
- First posted on arXiv 21 Jul 2026.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)

## Builds on (most cited first, 30 of 31)

- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [LiveCodeBench: Holistic and Contamination Free Evaluation of Large Language Models for Code](https://mlsanity.com/paper/2403.07974/) (2024)
- [Editing Models with Task Arithmetic](https://mlsanity.com/paper/2212.04089/) (2022)
- [TIES-Merging: Resolving Interference When Merging Models](https://mlsanity.com/paper/2306.01708/) (2023)
- [GLM-5: from Vibe Coding to Agentic Engineering](https://mlsanity.com/paper/2602.15763/) (2026)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [ToolRL: Reward is All Tool Learning Needs](https://mlsanity.com/paper/2504.13958/) (2025)
- [MemAgent: Reshaping Long-Context LLM with Multi-Conv RL-based Memory Agent](https://mlsanity.com/paper/2507.02259/) (2025)
- [REINFORCE++: Stabilizing Critic-Free Policy Optimization with Global Advantage Normalization](https://mlsanity.com/paper/2501.03262/) (2025)
- [ProRL: Prolonged Reinforcement Learning Expands Reasoning Boundaries in Large Language Models](https://mlsanity.com/paper/2505.24864/) (2025)
- [RL's Razor: Why Online Reinforcement Learning Forgets Less](https://mlsanity.com/paper/2509.04259/) (2025)
- [Task Singular Vectors: Reducing Task Interference in Model Merging](https://mlsanity.com/paper/2412.00081/) (2024)
- [Fantastic Pretraining Optimizers and Where to Find Them](https://mlsanity.com/paper/2509.02046/) (2025)
- [The Invisible Leash: Why RLVR May or May Not Escape Its Origin](https://mlsanity.com/paper/2507.14843/) (2025)
- [Dion: Distributed Orthonormalized Updates](https://mlsanity.com/paper/2504.05295/) (2025)
- [JustRL: Scaling a 1.5B LLM with a Simple RL Recipe](https://mlsanity.com/paper/2512.16649/) (2025)
- [Co-Evolving LLM Coder and Unit Tester via Reinforcement Learning](https://mlsanity.com/paper/2506.03136/) (2025)
- [The Path Not Taken: RLVR Provably Learns Off the Principals](https://mlsanity.com/paper/2511.08567/) (2025)
- [APOLLO: SGD-like Memory, AdamW-level Performance](https://mlsanity.com/paper/2412.05270/) (2024)
- [Reinforcement Learning Finetunes Small Subnetworks in Large Language Models](https://mlsanity.com/paper/2505.11711/) (2025)
- [Stabilizing Knowledge, Promoting Reasoning: Dual-Token Constraints for RLVR](https://mlsanity.com/paper/2507.15778/) (2025)
- [Embodied-R1: Reinforced Embodied Reasoning for General Robotic Manipulation](https://mlsanity.com/paper/2508.13998/) (2025)
- [Maximum Likelihood Reinforcement Learning](https://mlsanity.com/paper/2602.02710/) (2026)
- [BroRL: Scaling Reinforcement Learning via Broadened Exploration](https://mlsanity.com/paper/2510.01180/) (2025)
- [Spectral Adapter: Fine-Tuning in Spectral Space](https://mlsanity.com/paper/2405.13952/) (2024)
- [StelLA: Subspace Learning in Low-rank Adaptation using Stiefel Manifold](https://mlsanity.com/paper/2510.01938/) (2025)
- [Reparameterized LLM Training via Orthogonal Equivalence Transformation](https://mlsanity.com/paper/2506.08001/) (2025)
- [Pion: A Spectrum-Preserving Optimizer via Orthogonal Equivalence Transformation](https://mlsanity.com/paper/2605.12492/) (2026)
- [POET-X: Memory-efficient LLM Training by Scaling Orthogonal Transformation](https://mlsanity.com/paper/2603.05500/) (2026)

## Built on by (most cited first, 2 of 2)

- [ORCA: The Annealed Spectral Conditioning Optimizer for Faster, Better LLM Training](https://mlsanity.com/paper/2610.06116/) (2026)
- [When Does Momentum Help in Reinforcement Learning with Verifiable Rewards?](https://mlsanity.com/paper/2610.20413/) (2026)

## Abstract (arXiv)

Reinforcement learning with verifiable rewards (RLVR) is rapidly advancing the reasoning capabilities of language models, yet the optimization layer that converts reward feedback into weight-space updates remains poorly understood. Building on our prior analysis (Zhu et al., 2025), we study this missing layer through the singular structure of model weights and identify spectral inheritance: RLVR can reuse the base model's weight spectra while acquiring new behavior through changes in the associated input and output singular frames. We operationalize spectral inheritance as Isospectral Optimization (ISO), an RLVR-native, fixed-spectrum optimization framework with complementary offline and online instantiations. Offline, ISO-Merger combines the frame changes of shared-base specialists into a single fixed-spectrum model, requiring no post-merge data, rollouts, gradient updates, or on-policy distillation (OPD). It recovers complementary specialist capabilities and achieves the strongest aggregate performance among the compared data-free merging methods. Online, ISO-Optimizer applies a chosen base optimizer, including AdamW and Muon, to the frame variables while keeping the base spectra fixed. Across reasoning and coding tasks ranging from 1.5B to 8B parameters, ISO-Optimizer improves accuracy in the reported runs and reaches matched scores with substantially fewer training steps. On Qwen3-8B-Base, AdamW reaches an aggregate accuracy of 0.495 after 270 training steps. ISO-AdamW reaches the same accuracy after only 100 training steps and improves further to 0.509 after 210 training steps. Together, ISO offers a concrete answer to RLVR's missing optimization layer: rather than inheriting pre-training optimization wholesale, design post-training around the structure of reward-driven adaptation: inherit the spectrum, optimize the frames.