Infinite-Depth Reasoning Frameworks

• Infinite-depth reasoning frameworks are systems that remove fixed depth limits, enabling unbounded inferential chains in neural, symbolic, and formal settings.
• They utilize mechanisms like dynamic mixture-of-experts, looped architectures, and latent diffusion to adaptively scale computation for complex tasks.
• These frameworks enhance efficiency, interpretability, and adaptability across domains such as large language models, formal proofs, and mathematical logic.

Infinite-depth reasoning frameworks encompass neural, symbolic, and formal systems designed to support unbounded or arbitrarily deep sequences of inferential computation. These frameworks underlie a broad range of advances in LLMs, argumentation, symbolic logic, neural scaling theory, algebraic structures, and formal proof systems. By relaxing the conventional limitations imposed by fixed network depth, architectural layer count, context window, or recursion depth, infinite-depth frameworks permit reasoning chains of indefinite length, improved expressivity, and fundamentally different behaviors compared to finite-depth analogues.

1. Motivations and Theoretical Foundations

Infinite-depth reasoning frameworks arise from core observations about the limitations of fixed-depth computation in both artificial intelligence and mathematical logic:

• In traditional deep neural networks (DNNs), especially Transformers, reasoning depth is bounded by the number of layers. This creates inefficiencies—simple queries are over-computed, while complex chains are under-served (Roy et al., 24 Sep 2025). In symbolic or formal systems, fixed derivational depth limits the expressiveness for infinite or circular reasoning.
• Infinite-depth models allow iterative refinement (as many steps as necessary), supporting Turing-completeness, dynamic allocation of reasoning, and unbounded recursion (Zhu et al., 8 Jul 2025, Endrullis et al., 2015).
• The motivation is to decouple reasoning complexity from pre-defined architectural or resource constraints, enabling models to adaptively allocate computation according to problem requirements.

In infinite-depth regimes, critical phenomena—such as the non-commutativity of sequential limits (width-then-depth vs. depth-then-width in neural nets (Hayou, 2022)) and the emergence of non-Gaussian statistics (Bassetti et al., 22 Nov 2024, Li et al., 2021)—emerge, which are invisible in traditional, finite-depth analyses.

2. Neural Architectures and Mechanisms for Infinite Depth

A diverse set of neural strategies have been developed to instantiate infinite- or unbounded-depth reasoning:

a) Dynamic Mixture-of-Experts and Routing

The Dynamic Reasoning Chains through Depth-Specialized Mixture-of-Experts (DS-MoE) architecture leverages a pool of depth-specialized Transformer modules (experts) controlled by a routing network that dynamically assembles custom reasoning chains per input (Roy et al., 24 Sep 2025). Each expert is specialized for a specific cognitive function (shallow pattern recognition, compositional reasoning, logical inference, memory integration, or meta-cognitive supervision). The routing mechanism analyzes input complexity, using features such as parse tree depth and semantic density, and composes an optimal chain of experts. This enables, in principle, arbitrarily deep reasoning chains, with empirical evidence of substantial computational savings and improved accuracy on deep inference tasks.

b) Looped and Recurrent Models

Looped Transformer architectures repeatedly apply a shallow set of layers (with shared weights) to achieve effective infinite depth (Saunshi et al., 24 Feb 2025). A $k$-layer transformer looped $L$ times achieves the computational depth of a $kL$-layer model but with only $k$ times the parameters. Theoretical results show such looping is sufficient for simulating chain-of-thought (CoT) reasoning and algorithmic processes with potentially unbounded logical steps. Looped-based regularization can also inject this depth-inductive bias into standard (deep) transformer training. The observed dichotomy is that looping enables reasoning (open-book QA, math word problems) but not memorization (perplexity, closed-book QA).

c) Infinite-Depth Latent Reasoning and Diffusion Paradigms

Masked Diffusion Models (MDMs) and gradient-state recurrence architectures enable spatially or temporally infinite-depth latent reasoning by iterative global updates and optimization over latent states (Zhu et al., 8 Jul 2025). Diffusion models perform globally consistent, reversible reasoning where each denoising iteration updates all tokens given full context, supporting any number of refinement steps at inference time. Optimization-driven frameworks further enable dynamic, unbounded computation by continual refinement ("trading time for depth").

d) Mechanisms for Adaptive Compute Scaling

Test-time compute scaling, such as Adaptive Computation Time (ACT) (Rodkin et al., 22 Aug 2025), allows the model to control the number of computational steps per input, thus adaptively increasing reasoning depth as needed. Empirical findings demonstrate that while recurrence, memory, and ACT can extend effective depth, significant gains for deep multi-step tasks require explicit intermediate supervision or reinforcement learning to induce sufficiently deep computation.

3. Symbolic, Logical, and Algebraic Infinite-Depth Frameworks

Infinite-depth reasoning frameworks are also fundamental in formal reasoning, algebraic structures, and argumentation:

a) Coinductive and Infinitary Rewriting Frameworks

The coinductive framework for infinitary rewriting and equational reasoning (Endrullis et al., 2015) provides a fixed-point characterization of infinite-depth reasoning steps. Using greatest and least fixed-point operators ($\nu$, $\mu$), it captures rewrite sequences of arbitrary ordinal length, supporting proofs, reductions, and rewrites that are neither well-founded nor limited by ordinal notation. This approach (distinct from ordinal- or metric-based definitions) is particularly amenable to mechanized reasoning in theorem provers.

b) Logical Frameworks with Infinitary Terms

CoLF$^\omega$ (Chen, 2023) generalizes logical frameworks (LF, CoLF) to reason about all infinitary regular and non-regular structures, introducing productive Böh m trees as a syntactic basis. These trees support meta-encoding and hereditary substitution of infinite terms, allowing formal encoding and reasoning about structures with infinite observation depth, such as irrational reals and non-regular streams.

c) Argumentation Theory in Infinite Domains

SCC-recursiveness in infinite argumentation frameworks has motivated new semantics (cf1.5, stg1.5) that guarantee existence and modular directionality for unbounded or evolving argument graphs (Andrews et al., 9 Jul 2025). Transfinite recursion and SCC-prioritization allow infinite-depth evaluation of argument status, overcoming the breakdown of naive recursive semantics in ill-founded (non-terminating) AFs.

d) Infinite Limit-Depth in Valuation Theory

Algebraic constructions such as Maclane–Vaquié chains of valuations demonstrate the existence of objects whose definitional complexity—measured as "limit-depth"—is truly infinite; they are constructed via an infinite sequence of limit augmentations, never stabilizing at a finite stage (Alberich-Carramiñana et al., 2022). This has deep analogies to infinite recursion in logic and is related to pathologies like defect in valuation theory.

4. Scaling Laws, Mathematical Characterization, and Regime Transitions

Infinite-depth reasoning frameworks reveal fundamentally new mathematical phenomena:

• In the proportional infinite-depth and infinite-width neural limit, deep linear nets exhibit non-Gaussian output distributions governed by mixtures of Brownian-motion-based random matrices; predictive covariance depends explicitly on observed labels—a qualitative departure from Gaussian process (NNGP) limits (Bassetti et al., 22 Nov 2024).
• In the pure infinite-depth limit at fixed width, pre-activations converge to stochastic differential equations with distributions sensitive to activation function choice (log-normal, Ornstein–Uhlenbeck, or more exotic) (Hayou, 2022). The sequential limits (depth-then-width vs. width-then-depth) yield different variances, demonstrating non-commutativity.
• In deep ReLU ResNets, the infinite-depth-and-width regime is log-Gaussian, with variance determined by the ratio $d/n$. Hypoactivation and interlayer correlations appear as structural byproducts; Balanced ResNets correct for these, stabilizing variance and theoretical tractability (Li et al., 2021).

5. Applications and Empirical Achievements

Infinite-depth frameworks support a range of practical advancements:

• Efficiency and Accuracy: DS-MoE achieves up to 70% lower computation and double the inference speed of fixed-depth transformers, while boosting complex reasoning accuracy (Roy et al., 24 Sep 2025). InftyThink achieves unbounded depth reasoning with constant per-step compute, breaking quadratic context-scene scaling (Yan et al., 9 Mar 2025).
• Interpretable Chains: Looped and Mixture-of-Experts models yield explicitly inspectable, transparent reasoning chains, enhancing model interpretability (Saunshi et al., 24 Feb 2025, Roy et al., 24 Sep 2025).
• Generalization Across Domains: Symbolic and infinite argument frameworks guarantee existence and modularity in evolving domains, while latent reasoning architectures enable consistent, revisable global plans (mathematical reasoning, program synthesis, planning, etc.).
• Self-Correcting and Global Consistency: Diffusion-based infinite-depth architectures allow outputs to be revisited and revised indefinitely, supporting logical consistency enforcement and self-correction (Zhu et al., 8 Jul 2025).

6. Limitations, Challenges, and Open Directions

Key challenges and limitations for infinite-depth reasoning frameworks include:

• Training Stability: Deep iterative computation increases risk of vanishing gradients, memory constraints, and convergence instability, especially for massively parallel refinement steps (Zhu et al., 8 Jul 2025).
• Alignment and Interpretability: Infinite-depth latent computations break token-aligned reasoning chains, complicating alignment and mechanistic interpretability. Unified benchmarks are needed.
• Design of Scaling Laws: For block depth ≥ 2 (e.g., transformers), no depth parametrization currently guarantees all desired properties (stability, maximal feature diversity, universal hyperparameter transfer) in the infinite-depth limit (Yang et al., 2023).
• Empirical Saturation of Adaptive Depth: Mechanisms like ACT or memory-augmented networks show diminishing returns on longer/harder tasks unless coupled with explicit intermediate supervision (Rodkin et al., 22 Aug 2025).
• Theoretical Pathologies: Infinite chains in algebra (defect, non-Noetherianity) and logic (non-wellfounded derivations) reveal foundational boundaries—some infinite-depth frameworks model genuine mathematical pathologies requiring infinite construction steps (Alberich-Carramiñana et al., 2022).

7. Comparative Summary Table

Framework Type	Expressivity	Scalability	Key Mechanism
DS-MoE (Roy et al., 24 Sep 2025)	Arbitrary depth	High	Dynamic expert chains, input routing
Looped Transformers (Saunshi et al., 24 Feb 2025)	Arbitrary, induction-limited	High	Shared-weight layer looping
Diffusion/Latent (Zhu et al., 8 Jul 2025)	Unlimited (global self-refinement)	Moderate	Iterative plan refinement, diffusion
Formal/Coinductive (Endrullis et al., 2015, Chen, 2023)	Ordinal/beyond	Fully general	Greatest/least fixed-point induction
Argumentation (Andrews et al., 9 Jul 2025)	Ordinal	Varies	Transfinite and non-wellfounded rec.
Limit-depth Valuations (Alberich-Carramiñana et al., 2022)	True infinite (no stabiliz.)	Algebraically limited	Infinite MLV chains, transfinite

8. Outlook

Infinite-depth reasoning frameworks represent an overview between neural, symbolic, algebraic, and logical perspectives, all converging on the principle that constraints on reasoning depth are, in numerous settings, both artificial and limiting. Their development is not only a driver of new empirical capabilities in LLMs, but is also deepening theoretical understanding in areas as diverse as subfactor theory (Bisch et al., 29 Aug 2025), argumentation, and formal systems. Unresolved questions revolve around optimal parametrizations for highly modular neural architectures, formal guarantees of correctness and interpretability in deep latent chains, and the practical limits of resource allocation and data supervision in truly unbounded-depth reasoning systems.