Zero has no inverse
Theorem.Zero has no inverse — in any ring with 1≠0, 0·a=0 kills every candidate inverse (swept over ℤ/nℤ);.
Proof.in any ring with 1≠0, 0·a=0 kills every candidate inverse (swept over ℤ/nℤ); the zero ring (0=1) is the lone self-inverting exception; on the projective line [z:w]↦[w:z] makes inversion TOTAL with 0↔∞ — 1/0 is undefined, ∞, or NaN depending on which completion you bought.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/9/1/index.ts#divisionByZeroComputes
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
"Zero has no inverse" 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 divisionByZeroComputes.
- 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 (divisionByZeroComputes @ 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 divisionByZeroComputes (src/9/1/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.