Every quantitative sub-problem can be solved in at least three ways. A closed-form derivation produces an exact symbolic expression. A symbolic manipulation produces a transformed expression that exposes structure the original hid. A numerical evaluation produces a concrete value within a stated tolerance. Each method has a domain where it is the cleanest available choice — and using a stronger method when a weaker one suffices is just as wrong as using a weaker method when a stronger one is required.
Quantm's resolution stage selects the method explicitly and records the choice alongside the result. When closed form is available, the engine refuses to fall back to numerical approximation, because numerical approximation discards information the closed form would have preserved. When a symbolic identity collapses a complex composition to a trivial one, the engine applies the identity rather than computing through the complexity, because the simpler form propagates more cleanly through the rest of the derivation. When the structure resists symbolic treatment, the engine drops to numerical evaluation with explicit tolerance, and that tolerance flows into the audit trail.
The rule the engine refuses to break is the silent downgrade. If a closed form fails — because of a domain violation, a non-elementary integral, a transcendental obstruction — the failure is recorded and reported. The engine does not quietly substitute a numerical approximation and present it as the answer. That refusal is what makes the method-choice metadata trustworthy, and it is what makes the result downstream-auditable.