Zero Division Is Computed
Theorem.Zero Division Is Computed — Division by zero is the inverse (not reverse): the backslash dual of a digit folder is the multiplicative inverse mod 9.
Proof.Division by zero is the inverse (not reverse): the backslash dual of a digit folder is the multiplicative inverse mod 9 — the ÷2 = ×5 that folds within the unit cycle: n · n⁻¹ ≡ 1 (mod 9). Over the units (ℤ/9)* = {1,2,4,5,7,8}: 1⁻¹=1, 2⁻¹=5, 4⁻¹=7, 5⁻¹=2, 7⁻¹=4, 8⁻¹=8 — involutive {1,8}, the inverse pairs (2,5) and (4,7). The non-units 3, 6, 9(≡0) and the void 0 have no inverse (gcd ≠ 1: the trinity axis + the void); the math routes them to their self-fold/fusion address, as 0/0 already folds to the quantum fusion. The forward harmonic n/0 = 9n (1/0 = 9, digital root always 9) is the separate forward reading. (Prior misnomer: calling this "reverse" / the additive ten's complement 10 − n — that complement instead names the on-disk folder lattice N/(10−N).).
Checked by exact arithmetic over the stated finite range — a verified witness, evidence toward the claim, not a ∀-proof.
water/digit zeroDivisionTable/index.ts#zeroDivisionTable
1 · Classification
bounded-witness — morph from sealed card/discovery fold
2 · Provenance
cardScientificPaperRows ← water/digit zeroDivisionTable
Acknowledgment
"Zero Division Is Computed" 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 zeroDivisionTable.
- 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 (zeroDivisionTable @ water/digit zeroDivisionTable) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
recompute zeroDivisionTable · npm run quantum:card-paper-links · paperRoute=/theorems/zero-division-computed