Generative Recursive Reasoning

Junyeob Baek; Mengye Ren; Mingyu Jo; Minsu Kim; Sungjin Ahn; Yoshua Bengio

arxiv: 2605.19376 · v2 · pith:SVHPMX4Dnew · submitted 2026-05-19 · 💻 cs.AI

Generative Recursive Reasoning

Junyeob Baek , Mingyu Jo , Minsu Kim , Mengye Ren , Yoshua Bengio , Sungjin Ahn This is my paper

Pith reviewed 2026-05-20 05:53 UTC · model grok-4.3

classification 💻 cs.AI

keywords generative recursive reasoningstochastic latent trajectoriesvariational inferencemulti-hypothesis reasoningunconditional generationneural reasoningconstraint satisfaction

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The pith

GRAM turns recursive reasoning probabilistic so models can follow many trajectories instead of one.

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

The paper introduces Generative Recursive reAsoning Models to replace the single fixed path of existing recursive reasoning systems with a probabilistic process. Reasoning is represented as a sequence of random latent-state updates that share the same transition rule, so the model can draw multiple distinct trajectories for the same input. This change produces a generative model capable of both conditional inference and unconditional generation when no input is supplied. A reader would care because it adds the ability to explore alternatives and scale computation at test time without changing the core recursive structure. The training uses amortized variational inference to learn these stochastic paths from data.

Core claim

GRAM models reasoning as a stochastic latent trajectory inside an iterative refinement loop that reuses the same transition function, producing a latent-variable generative model that supports p(y|x) for conditional reasoning and p(x) for unconditional generation when inputs are absent or fixed, and is trained by amortized variational inference to improve over deterministic recursive baselines on structured tasks.

What carries the argument

Stochastic latent trajectory formed by repeated application of a shared probabilistic transition function in latent space, which permits parallel sampling of distinct reasoning paths.

If this is right

The model outperforms deterministic recurrent and recursive baselines on structured reasoning and multi-solution constraint tasks.
Inference-time scaling is possible by increasing recursive depth or drawing more parallel trajectories.
Unconditional generation of inputs becomes possible by sampling from the marginal p(x).
Multiple hypotheses and alternative solution strategies arise naturally from different latent trajectories.

Where Pith is reading between the lines

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

The same mechanism could be tested on planning problems where several valid action sequences exist for one goal.
Parallel trajectory sampling offers a direct way to trade extra compute for higher solution coverage without retraining.

Load-bearing premise

That amortized variational inference applied to stochastic trajectories will yield coherent and diverse reasoning paths instead of mode collapse or incoherent outputs.

What would settle it

On a multi-solution constraint satisfaction task, if increasing the number of sampled trajectories produces no gain in solution diversity or validity compared with a single trajectory, the benefit of the generative component is falsified.

Figures

Figures reproduced from arXiv: 2605.19376 by Junyeob Baek, Mengye Ren, Mingyu Jo, Minsu Kim, Sungjin Ahn, Yoshua Bengio.

**Figure 2.** Figure 2: GRAM Architecture. A single stochastic latent transition in the hierarchical instantiation z = (h, l). After K low-level refinements via fL, the highlevel update fH produces a deterministic proposal ut, to which stochastic guidance ϵt is added: ht = ut + ϵt. Overview. GRAM models the conditional distribution pθ(y | x) by marginalizing over stochastic latent reasoning trajectories. Given an input x, GRAM… view at source ↗

**Figure 3.** Figure 3: Performance on puzzle benchmarks. On both Sudoku-Extreme and ARC-AGI, GRAM consistently outperforms all deterministic recursive baselines (Looped TF, HRM, TRM), demonstrating that stochastic latent transitions yield substantial gains within the recursive-reasoning paradigm. Looped TF results on ARC-AGI are omitted due to prohibitive training cost (see Section C.1.1) Note that large reasoning model scores a… view at source ↗

**Figure 4.** Figure 4: (Left) Inference-time scaling on Sudoku-Extreme. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: Unconditional Sudoku generation. Validity (%) of generated Sudoku puzzles. GRAM achieves higher validity than D3PM with substantially fewer parameters and steps. Setup. To investigate GRAM’s unconditional generative capability beyond conditional reasoning, we evaluate generation in two domains: structured constraint generation on Sudoku (from empty boards, evaluated by the fraction of generated boards … view at source ↗

**Figure 6.** Figure 6: Visualization of the generation process and samples. [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 7.** Figure 7: Qualitative examples of unconditional Sudoku generation by GRAM. [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

**Figure 8.** Figure 8: Full ELBO LELBO and surrogate objective LGRAM throughout training (plotted as −ELBO, smaller is better). On both Sudoku-Extreme (left) and N-Queens 8 × 8 (right), both quantities decrease monotonically over training, indicating that gradient updates of LGRAM consistently improve the full variational bound. The two curves do not coincide because LELBO sums KL contributions across all TTotal transitions whil… view at source ↗

**Figure 9.** Figure 9: Example of an 8 × 8 N-Queens puzzle instance. In this example, 5 queens are removed from the full board, leaving 3 queens. The model must find the positions of the remaining queens. This configuration admits exactly 3 valid solutions. Data Generation Details. The N-Queens problem requires placing N queens on an N × N chessboard such that no two queens attack each other—meaning no queens share the same row,… view at source ↗

**Figure 10.** Figure 10: Distribution of the number of valid solutions for generated N-Queens instances. [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗

**Figure 11.** Figure 11: Graph Coloring Example 3 6 9 12 15 18 Number of solutions 0 500 1000 1500 Counts Vertex 8 Graph Coloring 0 20 40 60 80 Number of solutions 0 500 1000 1500 2000 Counts Vertex 10 Graph Coloring [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗

**Figure 12.** Figure 12: Distribution of the number of valid solutions for generated graph coloring instances. [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗

**Figure 13.** Figure 13: Effect of sampling on ARC-AGI-1 without data augmentation. [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗

**Figure 14.** Figure 14: Effect of augmentation on sampling efficiency. [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗

**Figure 15.** Figure 15: Solution coverage analysis on N-Queens ( [PITH_FULL_IMAGE:figures/full_fig_p023_15.png] view at source ↗

**Figure 16.** Figure 16: Additional generated samples from GRAM. We provide 8 additional samples generated unconditionally on binarized MNIST using GRAM. Each row represents a single generated sample, visualized across its recursive refinement process. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_16.png] view at source ↗

**Figure 17.** Figure 17: illustrates the unconditional Sudoku generation setup. Starting from an empty board, the task is to generate complete boards, and validity is determined by whether the generated board satisfies all Sudoku constraints [PITH_FULL_IMAGE:figures/full_fig_p024_17.png] view at source ↗

**Figure 18.** Figure 18: Latent reasoning trajectory of TRM. The red dot indicates the initial state h0 and the green dot indicates the final state hT . Background color represents the loss landscape: bright yellow corresponds to high loss regions, while dark blue indicates low loss (optimal) regions. TRM follows a single deterministic path with no ability to escape suboptimal trajectories. 25 [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗

**Figure 19.** Figure 19: Latent reasoning trajectories of GRAM (50 samples). [PITH_FULL_IMAGE:figures/full_fig_p026_19.png] view at source ↗

read the original abstract

How should future neural reasoning systems implement extended computation? Recursive Reasoning Models (RRMs) offer a promising alternative to autoregressive sequence extension by performing iterative latent-state refinement with shared transition functions. Yet existing RRMs are largely deterministic, following a single latent trajectory and converging to a single prediction. We introduce Generative Recursive reAsoning Models (GRAM), a framework that turns recursive latent reasoning into probabilistic multi-trajectory computation. GRAM models reasoning as a stochastic latent trajectory, enabling multiple hypotheses, alternative solution strategies, and inference-time scaling through both recursive depth and parallel trajectory sampling. This yields a latent-variable generative model supporting conditional reasoning via $p_\theta(y \mid x)$ and, with fixed or absent inputs, unconditional generation via $p_\theta(x)$. Trained with amortized variational inference, GRAM improves over deterministic recurrent and recursive baselines on structured reasoning and multi-solution constraint satisfaction tasks, while demonstrating an unconditional generation capability. https://ahn-ml.github.io/gram-website

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.

Desk Editor's Note private letter to a colleague

GRAM turns deterministic recursive reasoning into a stochastic multi-trajectory model, which is a clean extension but still needs the full experiments to judge the gains.

read the letter

This paper's main contribution is GRAM, which reframes recursive reasoning models as generative latent-variable systems that run multiple stochastic trajectories instead of locking into one deterministic path. The setup supports both conditional reasoning and unconditional generation, with inference scaling via deeper recursion or parallel sampling of trajectories. It is trained end-to-end with amortized variational inference. That is the concrete step beyond prior deterministic RRMs. The framework itself is straightforward and directly addresses the limitation of single-hypothesis convergence that the authors flag in earlier work. The potential to allocate extra compute at test time by sampling more paths is a practical angle worth exploring. The central empirical claim is that this yields measurable improvements on structured reasoning and multi-solution constraint tasks. From the description, the argument structure holds together without circularity or hidden fitting assumptions. The stress-test note is right that no load-bearing internal contradiction shows up in the abstract or high-level setup. The weakest point remains the risk that variational training collapses to low-diversity paths or produces incoherent samples, which is a standard concern for latent-variable models and not unique to this one. Without the actual numbers, ablations, or diversity metrics, it is impossible to tell how well the method avoids that problem or how large the reported gains really are. This is aimed at researchers working on neural reasoning architectures that want to combine recursion with generative modeling. A reader already familiar with RRMs or amortized inference will get the most out of the details. The work shows clear engagement with the literature and a coherent modeling choice, so it deserves a serious referee who can examine the full experimental controls and loss design.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces Generative Recursive reAsoning Models (GRAM), a probabilistic framework that extends deterministic Recursive Reasoning Models by modeling reasoning as stochastic latent trajectories. Trained via amortized variational inference, GRAM supports conditional reasoning p_θ(y|x) and unconditional generation p_θ(x), with inference-time scaling via recursive depth and parallel trajectory sampling. The central empirical claim is that GRAM outperforms deterministic recurrent and recursive baselines on structured reasoning and multi-solution constraint satisfaction tasks while enabling unconditional generation.

Significance. If the reported gains hold under rigorous controls, the work would be significant for neural reasoning systems by addressing the single-trajectory limitation of prior RRMs and providing a generative model for multi-hypothesis reasoning. The combination of recursive structure with latent stochasticity and unconditional generation capability is a notable modeling advance.

major comments (2)

[§4] §4 (Experiments): The abstract and framework description assert measurable improvements and diversity benefits from stochastic trajectories, yet the manuscript must include explicit quantitative results (e.g., accuracy deltas, error bars, dataset sizes, and ablation controls on the number of trajectories) to substantiate the central empirical claim; without these, the degree of support for outperformance over deterministic baselines cannot be assessed.
[§3.2] §3.2 (Training and Inference): The claim that amortized variational inference on stochastic latent trajectories yields diverse, useful reasoning paths (rather than mode collapse or incoherent samples) is load-bearing for the multi-trajectory advantage; the manuscript should report specific diversity metrics (e.g., sample entropy or distinct solution counts) and controls for posterior collapse to verify this assumption holds in the reported tasks.

minor comments (2)

[Abstract] Notation for the generative model p_θ(x) in the unconditional case should be clarified with respect to the input x being absent or fixed, to avoid ambiguity in the conditional vs. unconditional distinction.
[Abstract] The website link is provided but the manuscript should include a brief summary of any additional results or visualizations hosted there to make the paper self-contained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments. We address the two major comments point by point below. We have made revisions to the manuscript to provide the requested quantitative details and metrics, which we believe strengthen the empirical section.

read point-by-point responses

Referee: [§4] §4 (Experiments): The abstract and framework description assert measurable improvements and diversity benefits from stochastic trajectories, yet the manuscript must include explicit quantitative results (e.g., accuracy deltas, error bars, dataset sizes, and ablation controls on the number of trajectories) to substantiate the central empirical claim; without these, the degree of support for outperformance over deterministic baselines cannot be assessed.

Authors: We agree that the manuscript would benefit from more explicit quantitative results to allow readers to assess the improvements. In the revised manuscript, we have included additional tables in Section 4 that provide accuracy values with error bars (standard deviations over 5 independent runs), the exact dataset sizes for each experiment, and ablation studies showing performance for different numbers of trajectories (specifically 1, 5, and 10). These results demonstrate the advantage of GRAM's stochastic trajectories. revision: yes
Referee: [§3.2] §3.2 (Training and Inference): The claim that amortized variational inference on stochastic latent trajectories yields diverse, useful reasoning paths (rather than mode collapse or incoherent samples) is load-bearing for the multi-trajectory advantage; the manuscript should report specific diversity metrics (e.g., sample entropy or distinct solution counts) and controls for posterior collapse to verify this assumption holds in the reported tasks.

Authors: We thank the referee for this suggestion. To verify that the stochastic trajectories provide diverse reasoning paths without mode collapse, we have added in the revised manuscript specific metrics in Section 3.2, including the average entropy of the trajectory distributions and the number of distinct valid solutions obtained from sampling multiple trajectories. We also report the evolution of the KL divergence to confirm the absence of posterior collapse. These additions support our claim that the generative model produces useful diversity. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents GRAM as a modeling framework that extends deterministic recursive reasoning models into a stochastic latent-variable generative model trained via amortized variational inference. No derivation chain, equations, or first-principles results are shown in the abstract or description that reduce any claimed prediction or result to fitted inputs or self-citations by construction. The central claims concern empirical improvements on reasoning tasks and unconditional generation capability, which are presented as outcomes of the proposed architecture rather than identities or renamings of prior results. No load-bearing self-citation, ansatz smuggling, or uniqueness theorem from overlapping authors is visible. The framework is self-contained as an architectural proposal with independent empirical content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review is limited to the abstract; the ledger therefore records only the modeling assumptions stated or implied there.

axioms (1)

domain assumption Amortized variational inference can train a latent-variable model of stochastic reasoning trajectories without mode collapse or loss of diversity.
Standard VAE assumption invoked to justify training the generative recursive model.

invented entities (1)

Stochastic latent trajectory no independent evidence
purpose: To represent multiple alternative reasoning paths inside the recursive model.
New modeling object introduced to turn deterministic refinement into probabilistic multi-trajectory computation.

pith-pipeline@v0.9.0 · 5719 in / 1238 out tokens · 32738 ms · 2026-05-20T05:53:36.638888+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

GRAM models reasoning as a stochastic latent trajectory... z_t = u_t + ε_t with ε_t ~ N(μ_θ(u_t), σ²_θ(u_t)I)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

hierarchical instantiation... low-level K updates then high-level stochastic transition

What do these tags mean?

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.