How to Check an AI's Engineering Calculation

An AI answer that looks right and one that is right are indistinguishable until someone checks. The good news: most wrong engineering answers fail one of a handful of fast checks that any reviewer can run in minutes, without redoing the whole calculation.

1. Dimensions first

Before reading a single number, check that the final expression is dimensionally consistent. It is the cheapest check there is, and it catches a whole class of error — a dropped g₀, a square root that should be a square, a pressure added to a stress per unit length. If the units on the right-hand side do not reduce to the units of the answer, stop there.

2. Order of magnitude

Every discipline has numbers an experienced engineer carries around: low Earth orbit is about 7.7 km/s, sea-level air is about 1.2 kg/m³, a transport aircraft cruises near Mach 0.8, a chemical upper stage delivers a few km/s. An answer an order of magnitude away from these is not a surprising insight — it is almost always a unit or formula slip.

3. Limiting cases

Push one input to zero or infinity and ask whether the formula still behaves. A rocket with no propellant must gain no velocity. A beam with infinite stiffness must not deflect. A drag estimate at zero airspeed must be zero. A formula that fails its limiting case is wrong everywhere else too, even where the numbers look plausible.

4. Is it the right physics for this regime?

Language models are fluent in textbook formulas and less careful about when they apply. Check the regime explicitly: incompressible relations are a reasonable approximation only up to around Mach 0.3 (density has already changed by roughly 4.5% there); thin-wall pressure-vessel formulas assume a radius-to-thickness ratio well above ten; small-angle and linear-elastic assumptions have limits the answer should state. A correct formula applied outside its regime is still a wrong answer.

5. Where did every input come from?

Each number in the setup should trace to one of three places: the question, a named reference, or a clearly labelled assumption. The dangerous case is the fourth — a plausible value the model supplied silently, such as the wing area of an aircraft it half-remembers or the specific impulse of an engine it has confused with another. If you can't tell where an input came from, treat the result as unverified.

6. Constants and conventions

Check the constants against a standard source: standard gravity g₀ = 9.806 65 m/s², Earth's gravitational parameter μ⊕ = 398 600.4418 km³/s², the specific gas constant for air ≈ 287.05 J/(kg·K), γ = 1.4 for air. Check conventions too — natural versus base-10 logarithms, degrees versus radians, gauge versus absolute pressure, true versus equivalent airspeed.

7. Re-derive it a different way

The strongest check is an independent derivation: energy instead of momentum, a closed form instead of a numerical sum, a different reference frame. Two methods that share no steps are unlikely to share an error. If they disagree, one of them — or the problem framing — is wrong, and you have found it before it reached the design review.

A worked example

Question: a stage with a specific impulse of 300 s and a mass ratio m₀/m_f = 3 — what is its ideal Δv? The correct result is the Tsiolkovsky rocket equation:

Δv = I_sp · g₀ · ln(m₀/m_f) = 300 × 9.806 65 × ln 3 ≈ 3 232 m/s

Now three answers an AI could plausibly give, and the check that catches each one:

ANSWER	SLIP	CAUGHT BY
1 404 m/s	log₁₀ used instead of ln	Conventions; re-derivation
329.6 m/s	g₀ dropped	Dimensions (s × 1 ≠ m/s); magnitude
3 232 m/s	—	Passes all seven checks

None of the wrong answers is absurd on its face. Each is caught in under a minute by a check that doesn't require redoing the calculation.

Build the checks into the tool

A checklist works, but it relies on the reviewer running it every time. The more reliable fix is architectural. In AETHER, ARITHMOS never lets the language model produce the number: the model writes the calculation, a deterministic engine executes it, and on multi-step problems a second, independent derivation must agree before the result is marked verified. Assumptions and validity limits are printed as labelled lines, so checks 4 to 6 take seconds. The principle is set out in Verifiable AI: no hallucinated math.

Published · 20 July 2026 · All technical notes

How to check an AI's engineering calculation