Abstract This paper shows that the discrete Lorentzian causal diamond D — the two-complex built on the twelve lightlike nearest-neighbour vectors of the ternary Minkowski lattice -1, 0, +1⁴ — generates not two but four CSS quantum error-correcting codes via a geometric duality, and that this duality makes the distance asymmetry dZ = 2 of the original construction algebraically inevitable rather than merely observed. The central mechanism is that the causal diamond supports two natural orientations of its boundary: the Lorentzian orientation, in which past links are incoming, produces the temporal charge n⁰ₑff = 12 and motivates an all-ones X-check; the Euclidean orientation, in which all links are outgoing and the boundary sum vanishes, yields a dual code family in which the 21 plaquettes serve as X-type checks and the three spatial-axis groups serve as Z-type checks. These two orientations are related by Wick rotation t → ix, so the duality between the primal and dual code families is a quantum error-correction realisation of Wick rotation. The four codes derived from this geometry are: Code Parameters Highlights Code I [12, 4, (4, 2) ] Rate 1/3; corrects X-errors, detects Z-errors Code II [12, 1, (4, 3) ] Balanced; circuit-level threshold pc ≈ 3. 5% Dual A [12, 2, (2, 6) ] New; corrects all weight-1 and weight-2 Z-errors Dual II [12, 1, (3, 4) ] New; dZ = 4, preferred for dephasing-dominated hardware Additional contributions include: a two-stage combined protocol that measures the 21 plaquettes alternately in Z- and X-basis to correct both error types simultaneously (Pₗog = 0. 006 at p = 0. 01 with kₑff = 2), a Pigeonhole No-Go theorem proving that dZ ≥ 3 with k ≥ 2 is impossible in the primal CSS family, and an X-Decoration Equivalence theorem extending this bound to weight-≤6 non-CSS codes. Repository Contents Code: Numerical verification script Paper: Full PDF pre-print and original LaTeX source files. License Information Please note the dual-licensing structure of this repository: Software/Source Code: Licensed under the Apache License 2. 0. PDF Document & LaTeX Source: Licensed under the Creative Commons Attribution 4. 0 International (CC BY 4. 0).
Yannick Schmitt (Tue,) studied this question.