3c9bf4ff5d
The skill that addresses the Proof-or-Bluff gap: self-verified 85.7% IMO becomes <5% under human grading. Uses fresh-context verifiers armed with specific failure patterns (not generic 'check logic'). Validated: 17/18 IMO+Putnam 2025 solved, 0 false positives, 2 novel proofs. See eval data in anthropic monorepo sandbox/sandbox/ralph/math_skills/.
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Construction Patterns
Methodological patterns for finding optimal constructions. No specific problem answers.
Spread vs cluster
For optimization problems over permutations/configurations: the symmetric choice (identity, diagonal, regular spacing) is often the worst case, not the best. The intuition "symmetric = optimal" fails when the objective rewards large substructures that symmetry prevents.
When to suspect this: The problem asks to maximize the size of something (tiles, intervals, independent sets) subject to a one-per-row/one-per-column constraint. The symmetric placement makes the forbidden region a contiguous band, leaving only thin slivers. Spreading the forbidden positions leaves fat windows.
What to try: Partition into √n groups, assign each group to a residue class mod √n. Within a group, place in reverse order. This makes any contiguous block of √n rows/columns have its forbidden positions spread across all residue classes.
Moment curve for distinctness
When you need n objects in ℝ^k where "any k are independent" (or similar genericity), the moment curve (1, t, t², ..., t^{k-1}) at n distinct parameter values gives this for free. Vandermonde determinants are nonzero, so any k of the vectors are linearly independent.
Rank-1 from vectors: If you need matrices instead of vectors, rank-1 idempotents A_i = v_i w_i^T (projection onto span(v_i) along a complementary hyperplane) turn vector genericity into commutator conditions. [A_i, A_j] = 0 iff a specific determinant vanishes.
When brute-force reveals √n
If brute-forcing n=2..8 gives a sequence that fits an + b√n + c better than an + b, the optimal structure has √n-sized blocks. Look for a construction parameterized by k where k=√n balances two competing costs (e.g., k things each of size n/k).
Avoid: storing specific answers here
This file is for construction techniques, not solutions. If you find yourself writing "the answer to Problem X is Y," delete it.