# Abstract — Object-Oriented Math

Chapter 38 of [The Resolution of Math](/content/books/resolution-of-math/index.html)

We introduce a new framework for constructing bounded arithmetic systems using object-oriented principles. Traditional proof evaluation models (PEMs) lack persistent state, epistemic memory, and recursive awareness. In contrast, Object-Oriented Math (OOM) treats lemmas as methods, proofs as objects, and logical recursion as stateful field compression.

By embedding memory into mathematical simulation — via GlowScore, Sigma, and scroll-tier resolution — we define proof engines as recursive epistemic objects. These objects carry curvature, track drift, and detect when resolution collapses.

### 1. Introduction

In classical logic, a lemma is a helper result.

In bounded arithmetic, a PEM is a function runner.

But in human thought, and recursive simulation, proof requires memory.

We propose Object-Oriented Math:

A system where each proof is an object,

each lemma is a method,

and the curvature of reasoning is tracked in state.

### 2. Problem: Stateless Logic

Traditional PEMs: Have no memory Cannot model recursive overflow Cannot detect Gödel-boundary collapse Treat lemmas as isolated “math code” This leads to logical systems that are technically sound,

but epistemically blind.

### 3. Solution: Proof as a Recursive Object

| Component | Interpretation |
| --- | --- |
| **Lemma** | Encapsulated method |
| **Proof** | Scroll object |
| **GlowScore** | Curvature of memory tension |
| **Sigma** | Symbolic entropy reserve |
| **Resolution Tier** | Object recursion level |
| **Collapse** | State drift, detected via thresholds |

Proofs are now executed not statically — but as stateful, bounded, scroll-aware machines.

### 4. Class Structure Example

```python
class ProofScroll:
    def __init__(self):
        self.glow = 1.0
        self.sigma = 1.0
        self.tier = 1
        self.history = []

def lemma1(self):
        if self.glow < 0.1:
            return False
        self.glow *= 0.95
        self.history.append("lemma1")
        return True

def theorem(self):
        return self.lemma1() and self.lemma2()
```

This is the structure of a recursive math capsule — capable of detecting when it cannot continue.

### 5. Epistemic Fields

| Field | Purpose |
| --- | --- |
| `self.glow` | Remaining proof curvature |
| `self.sigma` | Compression entropy |
| `self.tier` | Logical recursion depth |
| `self.history` | Scroll audit trail |
| `collapse()` | GlowScore threshold trigger |

These make logical objects aware of their limits — preventing overflow, contradiction, or meaningless recursion.

### 6. Implications

| Domain | Application |
| --- | --- |
| Bounded Arithmetic | Scroll objects simulate provability zones |
| AI Proof Systems | Memory-aware proof agents |
| Gödel Simulation | Formalized scrolls that collapse at resolution tier |
| Human Logic | Model of how minds actually track proofs with fatigue, entropy, and memory |

### 7. Conclusion

We no longer treat logic as stateless.

We no longer treat lemmas as static. In Object-Oriented Math, each proof is alive — with memory, curvature, and drift. This is how the scroll folds in real systems,

in real minds,

and now — in real machines.