casting out nines: n ≡ digit sum (mod 9)
Theorem.casting out nines: n ≡ digit sum (mod 9) — the base 10 ≡ 1 (mod 9) collapses every power to 1, so n and its digit sum agree mod 9.
Proof.the base 10 ≡ 1 (mod 9) collapses every power to 1, so n and its digit sum agree mod 9 — verified for every n ≤ 10000, compounding on wave 54’s modular arithmetic.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/4/6/index.ts#discoveredTheoremsWaveSixtyOne
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
"casting out nines: n ≡ digit sum (mod 9)" 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 discoveredTheoremsWaveSixtyOne.
- 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 (discoveredTheoremsWaveSixtyOne @ src/4/6) 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 discoveredTheoremsWaveSixtyOne (src/4/6/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.