Every Big-O claim in this course ships with three independent receipts: an EMPIRICAL bench (criterion measures the curve), a STRUCTURAL guard (Rust types and proptest enforce it), and a FORMAL theorem (Lean 4 machine-checked, displayed as a [verified] stamp). Complexity is a CLAIM about how runtime or memory scales with input size. In this module you learn the c24 workflow: write the claim as a contract, bind it to your Rust pub item, then attach the three receipts. A CLAIM of O(N) survives only if the slope on a log-log plot stays linear from N=10 to N=1,000,000.

In this module you will master Empirical Wins — the speedups you can measure with cargo bench. You will translate three Python patterns to Rust idioms that drop runtime and memory: list comprehensions become fused iterator chains (filter+map in one pass, O(1) memory vs O(N)); x in dict becomes HashMap::contains_key with hasher choice as a c24 receipt (FxHash, AHash, or default SipHash, each measured); and sorted() becomes sort_unstable for cases where stability is not required. Every claim ships with criterion HTML reports as the EMPIRICAL receipt, a STRUCTURAL guard from Rust's type system, and a [verified] stamp from a FORMAL theorem.

Python Optional[T] becomes Rust Option<T> with proptest sweeping every combinator chain; Python try/except becomes Result<T, E> with @semantic: total promoting every fallible function — the structural receipt that makes silent failure impossible.

Systems-level translation makes the runtime trade-off visible. Python generators allocate per-call frame state and pay GIL overhead; Rust Iterator is monomorphized into the caller's stack — the structural receipt is the criterion bench showing a ~50x tight-loop gap. subprocess.run with sh -c re-introduces CWE-78 shell injection; Command::new().arg() invokes execve directly and removes the attack surface entirely. threading on CPython is serialized by the GIL; rayon::par_iter() distributes work across all cores via a Send + Sync work-stealing scheduler whose data-race bugs are compile errors. Each translation in this module is a runtime claim with a measurable receipt.

The capstone closes the loop with a five-step playbook — transpile, inspect, refine, bench, prove — that you can apply to any hot Python loop and ship a receipt-backed Rust translation. Every Pull Request lands four receipts: criterion HTML report, proptest run, contract pv check, and a Lean status: proved stamp. The second lesson is the inverse: when the receipt-driven math says keep Python. Breakeven analysis, BREAKEVEN.md, and a final graded quiz that synthesizes Modules 1–5.