Formal Verification of Memory-Safe, Symplectic Runtimes for AI Inference and Plasma Control: A Lean 4 and Logic Tensor Network Approach

We present a unified formal technical specification and mathematical validation of the RunuX-AI memory-safe runtime and its application to active feedback 3D toroidal plasma control. Our framework guarantees compile-time memory safety, numerical preservation of quantized weights, and exact energy conservation under active control loops. Using the Lean 4 proof assistant, we formally verify key system correctness properties.

Core Verified Invariants

1. Arena Memory Allocator Safety (Zero-Overlap Invariant)

The state of the allocator tracks the total memory capacity and the current active offset:

structure BumpAllocatorState where
  capacity : Nat
  offset   : Nat
  offset_le_capacity : offset <= capacity

We formally prove that consecutive allocations yield completely disjoint index spaces (no-aliasing invariant).

2. PolarQuant Norm Preservation (Zero-Distortion Guarantee)

We formally prove that block-wise pseudo-random orthogonal rotations preserve Euclidean norm perfectly under the Johnson-Lindenstrauss Lemma:

theorem polarquant_norm_preserving (U : E →L[ℝ] E) (hOrth : IsOrthogonal U) (x : E) :
  ‖U x‖ = ‖x‖

3. Speculative Rejection Sampling Correctness

Given a target distribution p and a draft distribution q, the acceptance probability and residual fallback distribution are modeled as:

def accept_prob {α : Type*} [DecidableEq α] [Fintype α]
  (p q : Distribution α) (x : α) : Real :=
  if q.prob x = 0 then 1.0 else Real.min 1.0 (p.prob x / q.prob x)

def residual_prob {α : Type*} [DecidableEq α] [Fintype α]
  (p q : Distribution α) (x : α) : Real :=
  let diff := p.prob x - q.prob x
  if diff > 0 then diff else 0.0

We formally prove that the expectation of the accepted step combined with the residual fallback step exactly reconstructs the target distribution p(x).

4. Symplectic Energy Conservation Guarantee

We formally verify that rescaling toroidal plasma states by $k = \sqrt{E_0/E_{\text{now}}}$ guarantees exact energy conservation ($E_0$) under FNO boundary active feedback damping coils.

5. Unified Co-Inference & Logic Tensor Network Boundaries

We extend our Lean 4 specification to formal boundaries governing the neuromorphic learning layers and quantum simulator stubs:

5.1 Soundness of Rust Memory Boundaries

We specify that the neural diff-optimizer validates bounds checks, guaranteeing memory-safe execution:

theorem SUPERSONIC_Rust_DiffOptimizer_memory_safety_sound
  (c : RustCode) (h : valid_bounds c) : safe_execution c

5.2 WARS-Quantum-LTN Unitary Preservation

theorem WARS_Quantum_LogicTensorNetwork_unitary_preservation
  (v : StateVector) (h : polarquant_contract v) : norm_equal v

5.3 Biomimetic Co-Inference DFA Error Boundedness

theorem biomimetic_dfa_weight_bounded (w : WeightMatrix) : biomimetic_dfa_weight_bounded_prop w
theorem biomimetic_dfa_error_bounded (e : ErrorVector) : biomimetic_dfa_error_bounded_prop e

Licensing and Academic Use: Mathematical specifications are dual-licensed under CC-BY-4.0 and the MIT license. Developed by Socrate AI Lab.