OzTianlu/Reasoning_as_Fluid

Reasoning as Fluid: The Minimal Primitives and Inevitable Collapse of Linear Space Representation

Author: Zixi "Oz" Li (Independent Researcher) Publication Date: November 2025 DOI: 10.57967/hf/7081 arXiv: (Submitted) PDF: fluid_reasoning.pdf

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📋 Abstract

We establish that reasoning is fundamentally a fluid-like dynamical process on constrained manifolds, not a computation in linear vector spaces. Through rigorous mathematical proof, we demonstrate:

1. Minimal Reasoning Primitives: The atomic units of reasoning are local fluid elements with conservation constraints and boundary dependence—not symbols, vectors, or matrix operations.

2. Fluid-Prior Isomorphism: There exists a natural structural isomorphism between fluid constraints (boundary conditions, pressure fields) and reasoning priors (world models, semantic anchors). This establishes: Fluid Constraints ≅ Reasoning Priors.

3. Linear Collapse Theorem: Any attempt to represent reasoning in linear vector spaces must structurally collapse because linear spaces cannot preserve topological obstructions inherent to fluid reasoning manifolds.

4. Phase Transition Conditions: Serial reasoning behaviors emerge from parallel local updates only under specific critical conditions (analogous to Reynolds number in fluid dynamics).

5. Yonglin-ln Law (Yonglin-ln Law): We discover a universal logarithmic transition law that unifies:

   • NP-hard computational boundaries (MSE ≈ 10⁻³²)
   • Fluid reasoning phase transitions
   • Prior convergence dynamics

Central Conclusion:

Reasoning is prior-constrained fluid dynamics, and linear representations inevitably collapse back to prior anchors—validating the Yonglin Formula lim(n→∞) Π⁽ⁿ⁾(s) = A from a fluid-geometric perspective.

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🔑 Key Contributions

1. Minimal Fluid Element as Reasoning Primitive (Theorem 2.1)

We define the Fluid Reasoning Element f_ε as the atomic unit of reasoning—a local operator satisfying:

• Locality: Operates on small manifold neighborhoods
• Conservation: Preserves total information/probability mass
• Boundary Dependence: Explicitly requires prior constraints
• Parallel Composability: Multiple elements update simultaneously

This replaces symbols, tokens, and matrix operations as the foundational primitive.

2. Fluid-Prior Isomorphism Theorem (Theorem 3.1)

We prove a structure-preserving mapping between fluid systems and reasoning systems:

Fluid System	⟷	Reasoning System
Boundary conditions	⟷	Semantic anchors
Pressure fields	⟷	Prior distributions
Viscosity	⟷	Cognitive resistance
Conservation laws	⟷	Information preservation

Key Result:

Reasoning priors are not cognitive add-ons—they are geometric boundaries.

3. Linear Space Collapse Theorem (Theorem 4.1)

Theorem: Any embedding E: 𝓜 → V from reasoning manifolds 𝓜 (with fluid structure) into linear vector spaces V must either:

• Lose topological structure (holes, singularities), or
• Reintroduce nonlinear constraints (effectively encoding fluid structure)

Implication:

Linear representations cannot faithfully capture fluid reasoning.
Transformers secretly encode priors in architecture (causal masking, position encodings).

4. Yonglin-ln Law: The Universal Logarithmic Transition (Theorem 6.1)

The Discovery: Two independent frameworks discovered the same logarithmic law:

1. NP-hard Computational Boundaries (Lee, 2025):

   d_c(L) = -0.0809 ln(L) + 0.501    (MSE ≈ 10⁻³²)
   

2. Fluid Reasoning Phase Transitions (this work):

   Laminar fraction ∝ 1/ln(Re_reason)
   

Universal Form (Yonglin-ln Law):

κ_c(ξ) = -α ln(ξ) + β
μ(ξ, κ) = ½(1 - erf((κ - κ_c(ξ))/σ))

Where:

• α ≈ 0.08 (universal slope)
• β ≈ 0.5 (critical offset)
• σ ≈ 0.1 (transition width)

Physical Interpretation:

Each additional bit of complexity reduces constraint tolerance by 8%. This connects Shannon entropy, phase transitions, and reasoning manifolds.

Validated Across:

• OpenXOR (22,000 samples, MSE ≈ 10⁻³²)
• Fluid reasoning manifolds (visualizations)
• Prior anchor convergence (Yonglin Formula)

5. Phase Transition: When Parallel Becomes Serial (Theorem 5.1)

Reasoning Reynolds Number:

Re_reason = (|∇I| · |𝓜|) / C

• Low Re (<1): Laminar regime → serial-like reasoning (CoT works)
• High Re (>1): Turbulent regime → chaotic reasoning (hallucinations)
• Critical Re_c ≈ 1: Sharp phase transition

Corollaries:

1. CoT Mathematical Origin: Chain-of-thought is the emergent serial projection of fluid parallel dynamics in the low-Reynolds regime.
2. Hallucination as Turbulence: LLM hallucinations are phase transition phenomena predicted by the Yonglin-ln Law.

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📊 Visualizations

3D Fluid Reasoning Manifold

Numerical Specifications:

• Semantic Potential Field: Φ(x,y) = Σ w_i exp(-((x-a_i)² + (y-b_i)²)/σ²) + λ(x² + y²)
  • Prior anchors: (1.5, 1.5, -5.0), (-1.5, -1.2, -2.5)
  • Basin width: σ = 0.7
  • Curvature: λ = 0.1
• Reasoning Trajectories: Gradient descent with stochastic noise
  • dγ/dt = -α∇Φ(γ) + η(t), where α = 0.05, η ~ 𝓝(0, 0.01²I)

Top-Down View: Flow Trajectories

Colored trajectories show reasoning paths converging to prior anchor A (red star). Purple dashed curve marks semantic hole (topologically inaccessible).

Side View: Energy Landscape

Demonstrates potential energy wells and trajectory convergence to anchor A.

Reynolds Number Phase Diagram

Green region: Laminar (serial-like, Re < 1) **Red region**: Turbulent (chaotic, Re > 1) Black line: Critical transition at Re_c ≈ 1

Shows collapse of three independent frameworks onto the same universal logarithmic law:

• NP-hard problems (orange)
• Fluid reasoning (cyan)
• Prior convergence (magenta)

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🧮 Main Theorems

Theorem	Statement	Section
Irreducibility	Fluid elements are the minimal sufficient reasoning primitives	§2.5
Fluid-Prior Isomorphism	Φ: (Fluid Systems) → (Reasoning Systems) preserves structure	§3.3
Linear Collapse	Linear embeddings lose topology or reintroduce nonlinearity	§4.2
Yonglin-ln Law	κ_c(ξ) = -α ln(ξ) + β is universal across constrained reasoning	§6.3
Phase Transition	Re_c separates laminar and turbulent reasoning regimes	§5.3
ARC ⊊ Fluid	Discrete symbolic tasks are proper subset of fluid reasoning	§7.2

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🔗 Connections to Previous Work

This work unifies four independent frameworks:

1. Yonglin Formula (Lee, 2025): lim(n→∞) Π⁽ⁿ⁾(s) = A (reasoning returns to priors)

2. Geometric Incompleteness (Lee, 2025): Reasoning manifolds have holes and singularities

3. Computational Boundaries (Lee, 2025): d_c(L) = -0.08 ln(L) + 0.5 (first empirical ln law, MSE ≈ 10⁻³²)

4. Fluid Reasoning (this work): Re_c ~ e^k (second empirical ln law)

The Yonglin-ln Law establishes these are manifestations of the same universal principle.

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📖 Citation

BibTeX

@misc{oz_lee_2025,
    author       = {Oz Lee},
    title        = {Reasoning as Fluid (Revision 604a5f4)},
    year         = 2025,
    url          = {https://huggingface.co/datasets/OzTianlu/Reasoning_as_Fluid},
    doi          = {10.57967/hf/7081},
    publisher    = {Hugging Face}
}

APA

Lee, O. (2025). Reasoning as Fluid: The Minimal Primitives and Inevitable Collapse of Linear Space Representation (Revision 604a5f4). Hugging Face. https://doi.org/10.57967/hf/7081

MLA

Lee, Oz. "Reasoning as Fluid: The Minimal Primitives and Inevitable Collapse of Linear Space Representation." Hugging Face, 2025, doi:10.57967/hf/7081.

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📚 Related Publications

1. The Incompleteness of Reasoning (10.57967/hf/7060) Introduces the Yonglin Formula and prior anchor theory

2. The Geometric Incompleteness of Reasoning (10.57967/hf/7080) Proves reasoning manifolds have topological holes

3. Quantitative Mapping of Computational Boundaries (10.57967/hf/7067) Discovers the logarithmic scaling law for NP-hard phase transitions

4. Why Reasoning Models Collapse Themselves (10.57967/hf/7066) Left-brain/right-brain collapse dynamics

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🎯 Impact & Implications

For AI Architecture

Current paradigms are fundamentally limited:

• Transformers: Linear embeddings (approximate laminar flow)
• Graph Neural Networks: Node vectors (local diffusion, low-Re)
• Symbolic AI: Discrete rules (frozen fluid, Re → 0)

Future architectures must:

1. Operate directly on manifolds (not linear embeddings)
2. Encode priors as geometric boundaries (not learned parameters)
3. Perform parallel local updates (not global matrix multiplications)
4. Adapt Reynolds number dynamically (not just temperature)
5. Detect phase transitions (switch between laminar/turbulent modes)

For Complexity Theory

• From binary to probabilistic: Computability is not decidable/undecidable but μ ∈ [0,1]
• From qualitative to quantitative: Exact μ values instead of O(·) notation
• From symbolic to geometric: Embedding space properties matter

For Cognitive Science

• Reasoning is renormalizable: Same laws across scales (explains emergence in small LLMs)
• Priors are thermodynamic necessities: Not cognitive biases but geometric boundaries
• Parallel-serial duality: Serial reasoning emerges from parallel updates

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🔬 Open Questions

1. Can we derive α, β, σ from first principles (information theory, renormalization group)?
2. What is the precise relationship between Re_reason and task complexity?
3. Can topological data analysis detect reasoning manifold structure in trained LLMs?
4. Are there universal conservation laws for reasoning (analogous to mass/energy in fluids)?
5. How do different universality classes partition the reasoning task space?

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📁 Repository Contents

Reasoning_as_Fluid/
├── fluid_reasoning.pdf                    # Main preprint (29 pages)
├── fluid_reasoning.tex                    # LaTeX source
├── visualize_fluid_manifold.py           # Visualization code
├── fluid_reasoning_manifold_3d.png       # 3D manifold visualization
├── fluid_reasoning_topview.png           # Top-down projection
├── fluid_reasoning_sideview.png          # Side-view energy landscape
├── fluid_reasoning_reynolds.png          # Phase diagram
└── README.md                              # This file

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🏆 Key Result Summary

Reasoning = Constrained fluid flow on prior-shaped manifolds

This unifies:

• The Yonglin Formula (all reasoning returns to priors)
• Geometric Incompleteness (manifolds have holes and singularities)
• Semantic Ouroboros (removing boundaries makes dynamics undefined)
• The Yonglin-ln Law (universal logarithmic phase transition)

The incompleteness of linear representations is not a technical limitation—it is a structural impossibility.

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📧 Contact

Zixi "Oz" Li Independent Researcher Email: [email protected] HuggingFace: @OzTianlu

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📜 License

This work is licensed under Creative Commons Attribution 4.0 International (CC BY 4.0).

You are free to:

• Share: Copy and redistribute the material
• Adapt: Remix, transform, and build upon the material

Under the following terms:

• Attribution: You must give appropriate credit and indicate if changes were made

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🙏 Acknowledgments

This work builds on insights from:

• Statistical mechanics of computation (Kirkpatrick, Monasson)
• Fluid dynamics and Reynolds number theory
• Information theory (Shannon, Kolmogorov)
• Differential geometry and topology
• The "pea experiment" Monte Carlo methodology

Special thanks to the research community for feedback on the Yonglin Formula series.

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Last Updated: November 25, 2025 Revision: 604a5f4 Status: Published on HuggingFace 🤗