compute-the-smallest-nontrivial-case

name

compute-the-smallest-nontrivial-case

description

When a general claim arrives too easily — a formula, an invariant, "this mechanism works", "the query logic is correct" — force it down to the smallest concrete instance that still exercises the core mechanism and compute it by hand before you generalize. The failure it prevents: leaping to a clean universal statement that was never made to survive contact with even one concrete case. Triggers on derivations, counting arguments, algorithm claims, edge behavior, and anywhere the words "obviously for all n" appear before one nontrivial n was worked.

compute-the-smallest-nontrivial-case

Feynman's concretize-down fused with von Neumann's invariant-reading: do not trust the general statement until the smallest live instance has been made to speak. Organon: P15 vary + P19 oracle + P11 gap-find. Phase: GENERATE -> JUDGE, with ORIENT reading the invariant off the small orbit. Respiratory: shrink the problem until the mechanism is exposed, then generalize only what survives.

Procedure

1. Find the smallest case that is still alive. Strip size, parameters, and decoration until one more simplification would remove the core mechanism. The smallest case is not the easiest case; it is the first case where the phenomenon actually occurs.

2. State what the general claim predicts on that case. Reduce the claimed formula, rule, or invariant to a concrete expected value or behavior for the chosen instance before you compute it.

3. Hand-compute the instance. Do the arithmetic, trace the execution, or enumerate the configurations directly. No appeal to the general argument is allowed inside the hand computation; otherwise the check is circular.

4. Compare prediction to computation. If they differ, reject the general claim. Do not patch the prose around the miss and keep the claim alive.

5. Read off the invariant only after the case agrees. Ask what stayed fixed through the small orbit, and only lift that structure back to the general statement.

The gate

Pass only if computed == predicted on the hand-worked case. If the smallest nontrivial case disagrees with the claim, the claim is rejected, not softened. A narrated "the small case seems consistent" does not pass; a fully worked case with explicit values does.

Example

Claim: "this dedup query returns one row per paper." Take the smallest live join where one paper has two citation rows and one author row. The general claim predicts one output row for that paper. Hand-compute the join: two rows appear. The small case already falsifies the universal statement, so the claim dies before a larger dataset can hide the fan-out.