The inverted circuit
Theorem.The inverted circuit — mirror(circuit) = /0\9/6/3\5\7\8/4/2/1/0\ — a total involution; x⁻¹ exists on exactly the six units and fails on {0,3,6,9}; x² = 1 has gcd(2,6) = 2 solutions {1, 8}.
Proof.inverting /0\1\2\4\8/7/5/3\6\9/0\ has two readings that part on the non-units: the MIRROR (reverse + swap slashes) is total and involutive and the derivative law survives it — reversal negates every delta, the swap negates every mark, two negations cancel — giving /0\9/6/3\5\7\8/4/2/1/0\, which meets the axis first (inversion swaps the cycles); the MULTIPLICATIVE inverse is partial, existing on exactly the six units (the x2 orbit) and failing on exactly {0,3,6,9}, the axis and the void; where both are defined they agree — inverting the orbit reverses it up to rotation (angle negation); and only 1 and 8 = ±1 are self-inverse, since x²=1 has gcd(2,6)=2 solutions in a cyclic group of order 6.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/1/9/index.ts#invertedCircuitComputes
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
"The inverted circuit" 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 invertedCircuitComputes.
- 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 (invertedCircuitComputes @ src/1/9) 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 invertedCircuitComputes (src/1/9/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.