README.md

# From Reasoning Structure to the Ancient Problem of Primes

[![DOI](https://img.shields.io/badge/DOI-10.57967%2Fhf%2F7156-blue)](https://doi.org/10.57967/hf/7156)
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**Author:** Zixi Li (Oz Lee)
**Date:** 2025
**Publisher:** Hugging Face

## Citation

```bibtex
@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](https://doi.org/10.57967/hf/7110), 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.

---

## 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)

---

## 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 |

---

## 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)

---

## 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
```

---

## Building the Paper

### Requirements

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

### Compilation

```bash
# 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

```bash
python3 experiments.py
```

**Outputs:**
- CSV files in `outputs/` (classification results, mod structure, AST depth)
- PNG figures in `outputs/` (5 figures with seaborn visualizations)

---

## 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

---

## Relationship to Prior Work

This paper synthesizes insights from two companion works:

1. **[When Euler Meets Stack](https://doi.org/10.57967/hf/7110)** (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](https://doi.org/10.57967/hf/7067)**:
   - Phase transitions in NP-hard problems
   - Critical density `d_c(L)` marking solvable/unsolvable boundary

3. **[Reasoning Incompleteness](https://doi.org/10.57967/hf/7060)**:
   - 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.

---

## 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.

---

## 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**.

---

## 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?

---

## 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

---

## Contact

**Oz Lee (Zixi Li)**
Independent Researcher
Email: [email protected]
HuggingFace: [@OzTianlu](https://huggingface.co/OzTianlu)

---

## 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

---

**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.**