Abstract — Object-Oriented Math

Chapter 38 of The Resolution of Math

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

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

4. Class Structure Example

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

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

6. Implications

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.