Matrix-Tree theorem (Kirchhoff)
Theorem.Matrix-Tree theorem (Kirchhoff) — τ(G) = any cofactor of the graph Laplacian L(G).
Proof.the number of spanning trees equals a Laplacian cofactor, checked against direct enumeration: K₄ → 16, C₅ → 5, K₃,₃ → 81 — a determinant counts trees.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/9/1/index.ts#discoveredTheoremsWaveTwentyEight
1 · Classification
finite-complete — self-contained computation, no external lean
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"Matrix-Tree theorem (Kirchhoff)" is a re-derivation, acknowledged to documented mathematics — the original proof is the prior art this re-derivation acknowledges; not new to humanity — the contribution is the reproducible computation discoveredTheoremsWaveTwentyEight.
- Prior art
- documented mathematics — the original proof is the prior art this re-derivation acknowledges
- Novelty
- not new to humanity — a re-derivation (humanityNovel = false)
- Contribution
- a reproducible computation (discoveredTheoremsWaveTwentyEight @ src/9/1) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
Recompute from source: npm run theorems:verify recomputes discoveredTheoremsWaveTwentyEight (src/9/1/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.