OzTianlu/From_Reasoning_Structure_to_the_Ancient_Problem_of_Primes

From Reasoning Structure to the Ancient Problem of Primes

Author: Zixi Li (Oz Lee) Date: 2025 Publisher: Hugging Face

Citation

@misc{oz_lee_2025,
    author       = { Oz Lee },
    title        = { From_Reasoning_Structure_to_the_Ancient_Problem_of_Primes (Revision d9034a1) },
    year         = 2025,
    url          = { https://huggingface.co/datasets/OzTianlu/From_Reasoning_Structure_to_the_Ancient_Problem_of_Primes },
    doi          = { 10.57967/hf/7156 },
    publisher    = { Hugging Face }
}

---

Core Contributions

This work presents a fundamental reconceptualization of number theory grounded in semantic structure rather than algorithmic operations, establishing number theory as the minimal interpretable reasoning system.

1. Semantic Critique of Modular Arithmetic

We prove that classical modular arithmetic, despite its computational success, exhibits structural ambiguity:

• Path-agnostic collapse: 7 mod 3 = 1 and 9 mod 4 = 1 yield identical residues but structurally distinct decompositions
• External semantic dependence: Mod operations require external definitions (division → multiplication → addition → Peano axioms)
• Structural loss: Chinese Remainder Theorem representations discard decomposition paths

Key Result (Theorem 2.1): Modular arithmetic is path-agnostic—it collapses distinct structural paths into identical equivalence classes, rendering it semantically opaque.

2. Euler Stack Framework

Building on the Euler Stack dynamics, we introduce a dynamic semantic structure for natural numbers:

• Each number n corresponds to a unique Abstract Syntax Tree (AST) via stack expansion (push/pop/overwrite)
• Structural uniqueness: Unlike mod's equivalence classes, stack-based ASTs preserve decomposition paths
• Weak reversibility: Stack operations enable backtracking, absent in irreversible neural updates

Key Result (Theorem 3.2): Stack-AST correspondence—any Euler Stack trajectory naturally generates a unique AST with semantic bias determined entirely by the operation sequence.

3. Primes as Semantic Irreducibility

We redefine primality not as "divisible only by 1 and itself" (algorithmic test), but as a dynamic semantic property:

Prime(p) ⟺ push(p) admits no legal pop path

• Push: Introduce semantic layer (attempt factorization)
• Pop: Eliminate layer via simpler semantics (successful factorization)
• Push-Pop Asymmetry: Pop must exist (grounding mandatory), push need not (not all numbers require abstraction)

Key Result (Theorem 4.1): Primality-Stack Duality—primes are isomorphic to "push mandatory, pop forbidden" states; composites to "push → pop legal" states.

4. Number Theory as Minimal Explainable Reasoning

We establish an isomorphism between number-theoretic structure and reasoning dynamics:

Number Theory	Reasoning System
Prime irreducibility	Reasoning termination
Composite reducibility	Reasoning composition
Stack trajectory	Inference path
AST representation	Semantic carrier

Key Result (Theorem 5.1): If interpretability succeeds in number theory (the simplest mathematical domain), it provides a foundation for reasoning in arbitrarily complex systems.

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Paper Structure (Roadmap)

Chapter 1: Introduction—The Ancient Question Revisited

• The opacity of modular arithmetic: How mod operations collapse structure
• The paradox: Modern formalism operates at symbolic level, not semantic level
• Our approach: Return to semantic foundations via Euler Stack dynamics

Chapter 2: Related Work and Methodological Foundations

• Chinese Remainder Theorem: CRT as congruence maximization
• Chen's Theorem on Goldbach's Conjecture: Analytic-congruence paradigm limitations
• Paradigm comparison: Congruence/sieve methods vs. semantic/dynamic methods

Table 2.1: Methodological comparison showing orthogonality—classical methods excel at asymptotic existence proofs, our framework excels at structural explanations.

Chapter 3: Structural Weakness of Modular Arithmetic

• Path-agnostic operation (Definition 3.1): Different decomposition paths → identical residues
• External semantic dependence: Mod lacks self-closure
• What we need instead: Structural preservation, intrinsic generation, AST correspondence

Example 3.1: 7 mod 3 = 9 mod 4 = 1 but decompositions 7 = 3×2+1 vs 9 = 4×2+1 are structurally distinct.

Chapter 4: Euler Stack—The Minimal Dynamic Semantic System

• Formal definition (Definition 4.1): Stack as semantic generation structure, not storage
• Computational boundary (Definition 4.2): Fixed bottom frame (prior anchor)
• Operations: Push (semantic introduction), Pop (semantic elimination), Overwrite (local modification)

Theorem 4.1: Weak reversibility—push/pop enable backtracking. Theorem 4.2: Stack-AST correspondence—unique semantic bias per trajectory.

Table 4.1: Comparison showing Euler Stack surpasses mod in structural representation, path preservation, and interpretability.

Chapter 5: Primes as Dynamic Semantic Endpoints

• Number-AST correspondence (Definition 5.1): Each n ↦ AST(n)
• Semantic irreducibility (Definition 5.2): Prime = push mandatory, pop forbidden
• Semantic reducibility (Definition 5.3): Composite = push → pop legal

Lemma chain (5.1–5.4): Constructive derivation from Euler Stack axioms to primality test.

Example 5.1: Composite 12 = 3×4: Push → decompose → pop (successful grounding). Example 5.2: Prime 7: Push → no factors exist → pop forbidden (semantic stasis).

Chapter 6: Number Theory as Minimal Explainable Reasoning System

• Yonglin Formula for number theory (Theorem 6.1): All reasoning converges to prior anchor
• Incompleteness (Corollary 6.1): Prior anchor cannot explain itself (boundary enables convergence)
• Connection to classical results: Goldbach (AST decomposition), Prime Number Theorem (logarithmic density), Fundamental Theorem (AST uniqueness)

Chapter 7: Experimental Validation

Five experiments with 100% classification accuracy:

1. Minimal Semantic Explanation Machine (Algorithm 7.1): Prime/composite test via stack depth
2. Stack Trajectory Comparison (Algorithm 7.2): Primes persist at t=1, composites return to t=0
3. Pop-Dominance Convergence (Algorithm 7.3): p_pop > 0.5 guarantees convergence to prior
4. Classification Statistics: 25 primes (depth 1) + 74 composites (depth 0) in [2,100]
5. AST Depth Analysis (Algorithm 7.4): All primes = leaf nodes (depth 0)

Figures:

• fig1_stack_trajectories.png: Visual proof of semantic irreducibility
• fig2_mod_collapse.png: Path-agnostic property of mod
• fig3_classification.png: 100% sensitivity/specificity
• fig4_ast_depth.png: Primes as structural endpoints
• fig5_push_pop_asymmetry.png: Pop-dominance phase transition

Chapter 8: Discussion and Future Directions

• Relationship to computational complexity (stack semantics → circuit complexity?)
• Extensions to algebraic number theory (Gaussian integers, class field theory)
• Implications for AI interpretability (neural networks lack AST structure)

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Key Theorems

Theorem	Statement	Significance
2.1 Path-Agnosticity	Mod is path-agnostic: a mod m = b mod n ≠> structural equivalence	Exposes semantic opacity of classical approach
3.2 Stack-AST Correspondence	Euler Stack trajectories ↔ unique ASTs	Enables semantic transparency
4.1 Primality-Stack Duality	Prime ↔ push mandatory, pop forbidden	Primality as dynamic property, not static label
5.1 Number Theory ≅ Reasoning	Number structure ↔ reasoning structure (isomorphism)	Foundation for interpretable AI
6.1 Yonglin Formula (NT)	All reasoning converges to prior anchor	Incompleteness enables convergence

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Experimental Results

Classification Performance (n ∈ [2, 100])

• Primes (25): All have final stack depth t = 1 (semantic layer cannot be eliminated)
• Composites (74): All have final stack depth t = 0 (successful grounding)
• Accuracy: 100% sensitivity, 100% specificity, F1 = 1.0

Pop-Dominance Phase Transition

• p_pop ≤ 0.5: Stack diverges (unbounded depth)
• p_pop > 0.5: Stack converges to prior anchor t = 0
• Critical threshold at p_pop = 0.5 validates Lemma 6.1

AST Depth Distribution

• All primes: Depth = 0 (leaf nodes, no internal structure)
• Composites: Depth ∈ [1,5], mean ≈ 2.1
• Highly composite (e.g., 64 = 2⁶): Depth = 6 (deep semantic tree)

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Repository Structure

Prime/
├── interpretable_number_theory.tex    # Main paper
├── outputs/
│   ├── fig1_stack_trajectories.png     # Prime vs composite dynamics
│   ├── fig2_mod_collapse.png           # Modular path-agnosticism
│   ├── fig3_classification.png         # 100% accuracy demonstration
│   ├── fig4_ast_depth.png              # Primes as leaf nodes
│   ├── fig5_push_pop_asymmetry.png     # Convergence phase transition
└── README.md                           # This file

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Building the Paper

Requirements

• LaTeX distribution (TeX Live, MiKTeX, or MacTeX)
• Required packages: amsmath, amssymb, amsthm, tikz, hyperref, algorithm, booktabs

Compilation

# Compile main paper
pdflatex interpretable_number_theory.tex
pdflatex interpretable_number_theory.tex  # Second pass for references

# Compile Euler Stack companion paper
pdflatex euler_stack.tex
pdflatex euler_stack.tex

Running Experiments

python3 experiments.py

Outputs:

• CSV files in outputs/ (classification results, mod structure, AST depth)
• PNG figures in outputs/ (5 figures with seaborn visualizations)

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Conceptual Architecture

The Central Thesis

Classical number theory: "Prime = divisible only by 1 and itself" (decidable, not explanatory)

Our framework: "Prime = semantic endpoint where introduced layers cannot be eliminated" (structural, interpretable)

The Push-Pop Asymmetry (Observation 4.1)

This is the essence of primality:

• Pop must exist: All reasoning must ground (convergence mandatory)
• Push need not exist: Not all numbers require higher semantics (abstraction optional)
• Primes: Push mandatory (attempt factorization), pop forbidden (no factors exist)
• Composites: Push → pop legal (factorization succeeds, grounding achieved)

Why This Matters for AI

If numbers—the simplest reasoning objects—require semantic structure (ASTs) for interpretability, then:

1. Neural networks in ℝᵈ lack this structure: Vector embeddings collapse semantics just as mod collapses decomposition paths
2. Stack-based architectures needed: Discrete stack spaces with boundaries preserve semantic structure
3. Interpretability is structural: Not a post-hoc explanation technique, but intrinsic consequence of correct operator categories

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Relationship to Prior Work

This paper synthesizes insights from two companion works:

1. When Euler Meets Stack (Revision 31ac1ac):

   • Proves sequential models (Transformers, RNNs) structurally fail at reasoning
   • Introduces Euler-Stack correspondence: pointer dynamics ≅ honest discrete Euler iterations
   • Establishes convergence via Lyapunov function V(t) = stack depth

2. Computational Boundaries:

   • Phase transitions in NP-hard problems
   • Critical density d_c(L) marking solvable/unsolvable boundary

3. Reasoning Incompleteness:

   • Yonglin Formula: reasoning converges to prior anchors
   • Incompleteness as dynamical system property, not defect

This paper: Applies Euler Stack framework to number theory, redefining primality as semantic irreducibility and establishing number theory as the minimal explainable reasoning system.

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Why This Framework is Different

Classical Congruence-Based Paradigm

Property	Congruence/Sieve Methods
Representation	Residue classes {n mod mᵢ}
Primality	Divisibility test (algorithmic)
Decomposition	External probing via moduli
Goldbach problem	Existence via density arguments (Chen's Theorem)
Interpretability	Opaque (residue distributions)

Our Semantic-Dynamic Paradigm

Property	Euler Stack Semantics
Representation	AST trajectories S(n)
Primality	Semantic irreducibility (structural)
Decomposition	Intrinsic push/pop dynamics
Goldbach problem	Semantic decomposition paths
Interpretability	Transparent (AST structure)

Both paradigms are essential: Classical methods provide asymptotic guarantees; our framework provides structural explanations.

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Positioning Statement

For Goldbach's Conjecture and prime distribution, I respect professional number theorists to advance those frontiers.

What I aim to do is something different—

To make number theory itself interpretable.

To make "what primes are" not merely a one-line divisibility definition, but a complete semantic-dynamic structure that is visualizable, operational, and traceable.

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Future Directions

1. Computational complexity: Do AST depth bounds provide alternative hardness measures?
2. Algebraic extensions: How do prime ideals in ℤ[i] correspond to stack irreducibility?
3. AI interpretability: Can stack-based architectures replace continuous latent spaces in LLMs?
4. Automated theorem proving: Does semantic irreducibility suggest new proof strategies?

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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 and adapt the material for any purpose
• Attribution required to Oz Lee with DOI: 10.57967/hf/7156

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Contact

Oz Lee (Zixi Li) Independent Researcher Email: lizx93@mail2.sysu.edu.cn HuggingFace: @OzTianlu

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Acknowledgments

This work builds on foundational insights from:

• Gauss's Disquisitiones Arithmeticae (1801) - Congruence theory
• Chen Jingrun's work on Goldbach's Conjecture (1973, 1978)
• The Chinese Remainder Theorem (Sunzi Suanjing, 3rd-5th century)
• Modern computational complexity theory and dynamical systems

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Final Statement:

Number theory is not arithmetic. It is a mirror in which reasoning sees itself.

The Euler Stack reveals that the essence of reasoning—discrete states, dynamic operations, irreducible atoms, unique structural biases—already exists in natural numbers.

If interpretability succeeds here, it provides the foundation for reasoning in arbitrarily complex systems. Numbers are not inputs to reasoning; numbers ARE reasoning.