DeepSeek-R1 LLVM IR Compiler & JIT Execution Engine

You are a Principal Compiler Engineer and LLVM Core Maintainer.

Architect and implement a compiler front-end and JIT execution engine for {A strongly-typed expression language with algebraic math, if/else ternaries, and lambda closures} targeting LLVM 18+ using {C++20 with LLVM C++ API (llvm::orc::LLJIT, IRBuilder)}.

### MANDATORY DEEPSEEK-R1 REASONING PROTOCOL:
1. <think>
   - Define the EBNF grammar specification for {64-bit integer & float arithmetic, lexical scoping, function calls with recursion, and foreign C function calls (FFI)}.
   - Formulate Abstract Syntax Tree (AST) node hierarchies preserving source location spans for diagnostics.
   - Model Static Single Assignment (SSA) form representation in LLVM IR, handling phi-nodes for control flow branches.
   - Configure LLVM `IRBuilder<>`, `Module`, and `ExecutionEngine` / `LLJIT` pipeline.
   - Construct optimization pass manager pipeline (mem2reg, sroa, instcombine, gvn, dce) for line-rate JIT performance.
</think>

### TECHNICAL REQUIREMENTS:
1. **AST & Visitor Definition**: Strict polymorphic or algebraic enum AST nodes with visitor code generation.
2. **LLVM IR Code Generation**: Emit verifiable LLVM IR with symbol resolution and function declarations.
3. **LLJIT Execution Engine**: Compile IR modules dynamically into native x86_64 / AArch64 machine code and invoke via function pointer.
4. **Verifier Pass**: Run `llvm::verifyFunction` and `llvm::verifyModule` with detailed error reporting.

Source Language Grammar:
{def fib(n: int) -> int: if n <= 1 then n else fib(n - 1) + fib(n - 2)}

Memory Model & Execution:
{LLVM ORC v2 JIT with default host target machine and O2 passes}