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
In Object-Oriented Math:
| 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
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.