7  Lean Proofs Repository Integration

This document provides detailed instructions for verification firms and hardware engineers to run and verify the Lean formal proofs underlying the Monist Engine’s core semantics and execution model.

7.1 Running the Lean Proofs

To locally verify the mathematical foundations of the Monist Engine, you will need a working Lean 4 environment managed by elan.

7.1.1 Instructions

  1. Environment Setup: Ensure elan is installed and up-to-date.

  2. Clone the Companion Repository: The formal proofs reside in a separate repository. Clone it as a sibling directory to monist:

    git clone https://github.com/ryanncode/nf-sketches.git

  3. Retrieve Dependencies: Navigate to the nf-sketches directory and fetch the required mathlib components:

    lake fetch

  4. Execution: Run the build command to compile and verify all theorems in the repository:

    lake build

    A successful build with no output indicates that all proofs have been formally verified by the Lean 4 kernel.

7.2 SMT-LIB Differential to Lean Pipeline

To guarantee absolute mathematical equivalence between the high-speed Rust evaluator and the formally verified Lean architecture, we employ a Differential Equivalence Testing pipeline linking monist to our Lean formalization (nf-sketches/parse-strat).

When a monist-examples mathematical test executes, the monist-core engine seamlessly generates an SMT-LIB witness of the topological graph, explicitly capturing evaluation limits, bounds, and Extensionality Collisions. This witness is then piped directly into the Lean interpreter (lake exe parse-strat --ingest-smt), which uses its own completely independent topological Bellman-Ford implementation to trace the exact same SMT constraints.

To manually pipe a diagnostic test directly into the Lean 4 interpreter:

cd tools/monist-examples
cargo run --bin specker_refutation | awk '/; === BEGIN STRATIFICATION WITNESS ===/{flag=1; print; next} /; === END STRATIFICATION WITNESS ===/{print; flag=0} flag' > out.smt
cd ../../../nf-sketches/parse-strat
lake exe parse-strat --ingest-smt < ../../monist/tools/monist-examples/out.smt

Alternatively, you can run the full automated cross-language verification script from the root of the monist directory:

./scripts/run_differential_tests.sh

7.3 Axiomatic Foundations

Our verification framework minimizes assumptions, strictly relying on a well-defined set of axioms and categorical constructs to bridge interaction combinators and physical hardware evaluation. This establishes the structural certainty required to formally verify deterministic execution models for scalable AI oversight:

  • Extensionality: Utilized extensively to prove the behavioral equivalence of compiled interaction net rewrites against high-level lambda calculus terms.
  • T-relative Adjunctions: The theoretical cornerstone establishing the correctness, safety, and deadlock-freedom of our lock-free concurrent traversals across asynchronous multi-GPU boundaries.

7.4 Non-Well-Founded Set Implementation

The Monist Engine natively models cyclic, infinite, and self-referential data structures (such as Russell’s paradox variants and Quines) without the overhead of traditional garbage collection. We utilize a localized, affine-typed graph representation based on non-well-founded set theory.

7.4.1 Theoretical Bounds

  • Topological Acyclicity Profiles: By projecting interaction nets into topological pseudomanifolds, we can statically bound the evaluation complexity of corecursive types.
  • Time Complexity: The resolution of cyclic and potentially non-terminating interactions is strictly bounded to \(O(V \log V)\), where \(V\) represents the number of active agent vertices. This guarantees termination of structural sharing limiters.
  • Space Complexity: Memory overhead is mathematically bounded to \(O(N)\) for \(N\) agents. This is enforced by our lock-free memory pool allocator that reclaims nodes via local graph retractions, preventing memory leaks in non-well-founded structures.