3D qutrit cubic code over F₃ — the generator the-fracton sphere described but never contained · every claim recomputed live before it renders
generator
the engine, stated once, checked forever. L×L×L torus, two qutrits per site. Z-check per cube: a₁ = 1+x+y+z on qutrit 1, a₂ = 1+xy+yz+zx on qutrit 2. X-check: b₁ = a₂(x⁻¹,y⁻¹,z⁻¹), b₂ = −a₁(x⁻¹,y⁻¹,z⁻¹). Selected by scanning all F₃ sign variants of the cross terms — the all-ones lift of Haah code 1 support carried the most logical space on the scan grid. Commutation is identical by algebra and re-verified numerically below anyway, because algebra done at midnight is exactly what benches are for.
verdict
⏳ conductor booting — verdict renders only after all live probes complete…
conductor RUNNING
probe-1 · commutation + k(L) QUEUED
probe-2 · string hunt (tubes) QUEUED
probe-3 · planar logicals QUEUED
2D — k(L) & the logical's plane
top: k(L) — orange bars = live rank (L=2..6), hollow bars = python-baked (L=7,8), gold rings = baked expectation on every bar. bottom: the MEASURED planar logical at L=6 — one 6×6 plane, both qutrit slots per site (left/right half-cells); orange = exponent 1, amber = exponent 2, dark = untouched. 18 of 72 slots: fractal-sparse, not a line.
3D — the operator in the lattice (soft-GL)
the full 6×6×6 torus as faint sites; the measured logical operator glows in its single plane. no tube through this lattice carries a logical — the hunt below proves it at k=18. drag to rotate.
toddler corner
ELI5: last time we had a beautiful drawing of a treasure-protecting snowflake, but no actual lock — just the drawing. now we built the lock. we checked it three ways: (1) all the lock's teeth mesh perfectly (commutation), (2) counted how many treasures it can hold at each size — the count jumps around weirdly (6, 0, 6, 0, 18…) and that's measured, not guessed, (3) tried to pick it with every skinny wire we could bend (tubes through the lattice) — none work. the only keys that exist are snowflake-shaped and flat like a sheet of paper. and one honest note: at some sizes the box holds zero treasures — a lock with nothing behind it. we say so.
method & provenance
double verification. conductor2.py (numpy, F₃ elimination) ran the scan, sweep, and hunts; this page re-runs commutation, k(2..6), the L=4 tube hunt, and the planar-existence check in an independent JS implementation. baked ≠ live ⇒ red monitors and a dead verdict.
lineage. support shape: Haah, PRA 83, 042330 (2011). qutrit cubic-code kin: fracton-zoo qutrit models; arXiv 2606.19873 (June 2026). sphere ancestry: the-fracton · QUANTUM FRONTIER · UD0 — self-stamped AMBER as a teaching model then; engine delivered now.