Research-Stack/6-Documentation/papers/OTOM/12_PIST_Architecture.md

PIST: Perfectly Imperfect Square Theory

Authors: Research Stack Team
Date: April 2026
Domain: TTM Layer F (Control)
OTOM Version: 2.2

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Abstract

PIST (Perfectly Imperfect Square Theory) provides the control framework for parallel execution in OTOM. The "imperfect square" captures the essence of parallel non-orthogonal state exploration, while "perfectly" acknowledges the deliberate design of invariant-safe traversal. PIST ensures safety invariants are maintained during parallel state exploration across the manifold.

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1. PIST Core

1.1 Traversal Function

\text{pist}(S_0, \text{goal}, \text{invariant}) = \{s \in \mathcal{S} \mid \text{reachable}(s, S_0) \land \text{invariant}(s) \land \text{goal}(s)\}

1.2 Parallel Expansion

\text{expand}_{\parallel}(S) = \bigcup_{s \in S} \text{successors}(s)

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2. Safety Invariants

2.1 Prohibited State Avoidance

\forall s \in \text{pist}(S_0), s \notin \mathcal{P}

2.2 Conservation Laws

\forall s_1 \to s_2, \text{invariant}(s_1) \implies \text{invariant}(s_2)

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3. Shell Model

3.1 Shell Structure

\text{Shell}_n = \{s \in \mathcal{S} \mid \text{depth}(s) = n\}

3.2 Shell Counting

N_{\text{shell}}(n) = |\text{Shell}_n|

3.3 Gap Conservation

\Delta_{\text{gap}} = N_{\text{shell}}(n+1) - N_{\text{shell}}(n) = \text{constant}

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4. Bridge Operations

4.1 Domain Crossing

\text{bridge}(s, D_{\text{src}}, D_{\text{dst}}) = s' \in D_{\text{dst}} \mid \text{equivalent}(s, s')

4.2 PistBridge

\text{PistBridge}(P_1, P_2) = \{(s_1, s_2) \mid s_1 \in P_1 \land s_2 \in P_2\}

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5. Implementation

Lean 4 Modules:

• PIST.lean — Core traversal
• ShellModel.lean — Shell counting
• PistBridge.lean — Bridge operations
• PistSimulation.lean — Simulation harness

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6. Theorems

6.1 Safety Preservation

\text{safe}(S_0) \land \text{pist}(S_0, \cdot, \cdot) \implies \forall s \in \text{result}, \text{safe}(s)

6.2 Completeness

\exists s \in \mathcal{S}, \text{goal}(s) \implies s \in \text{pist}(S_0, \text{goal}, \cdot)

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7. References

• Korf, R.E. (1990). Real-time heuristic search.
• Research Stack, OTOM Ontology v2.2.