f(θ,φ,x,y,z,digit,n) → {p,q} Is The Inverse Pair
Theorem.f(θ,φ,x,y,z,digit,n) → {p,q} Is The Inverse Pair.
Proof.f(θ, φ, x, y, z, digit, n) → {p, q} is the canonical inverse fold: geometry binds the digit to doubleTorusSurface (genus-2); when n=0 (division by zero) {p,q} is the multiplicative inverse pair digit · q ≡ 1 (mod 9) or the self-fold for non-units — inverse that folds within itself, not a ten's-complement reverse; lobe orientation is ratInv on the pair; when n≠0, {p,q} = ratInv(rat(digit, n)).
Checked by exact arithmetic over the stated finite range — a verified witness, evidence toward the claim, not a ∀-proof.
mountain/vortex f / fThetaPhiXyzDigitNIsTheInversePair/index.ts#fThetaPhiXyzDigitNIsTheInversePair
1 · Classification
bounded-witness — morph from sealed card/discovery fold
2 · Provenance
cardScientificPaperRows ← mountain/vortex f / fThetaPhiXyzDigitNIsTheInversePair
Acknowledgment
"f(θ,φ,x,y,z,digit,n) → {p,q} Is The Inverse Pair" 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 fThetaPhiXyzDigitNIsTheInversePair.
- 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 (fThetaPhiXyzDigitNIsTheInversePair @ mountain/vortex f / fThetaPhiXyzDigitNIsTheInversePair) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
recompute fThetaPhiXyzDigitNIsTheInversePair · npm run quantum:card-paper-links · paperRoute=/theorems/f-inverse-pair-computed