Logical Operator Holonomy and Non-Orientable Algebra in a Klein Bottle Stabilizer Code

We demonstrate that the logical operator algebra of the Klein bottle stabilizer code is non-orientable, confirming that the code encodes non-orientable topology at the level of the logical Hilbert space, not merely at the level of the stabilizer construction. Using dynamic circuits with mid-circuit reset on IBM Fez (Heron r2, 156 qubits), we apply the b-cycle logical gate X_L2 to each of the four logical sectors and measure the resulting syndrome in a second round. The Klein code produces different sector transitions than the toric code under the identical gate, confirming that X_L2 anticommutes with Z_L1 in Klein but commutes in the toric code. No orientable stabilizer code can produce this result. Key results: Sector Job ID Klein R2 Z Toric R2 Z |10>_L (holonomy) d73rqh5koquc73e22ni0 10000010 at 11.79% 90σ 10001011 at 22.96% 272σ |00>_L d73u5ttkoquc73e251eg 10000001 at 13.9% 99σ 10001000 at 21.9% 312σ |10>_L d73u5ttkoquc73e251eg 10000010 at 12.7% 105σ 10001011 at 18.0% 230σ |01>_L d73u5ttkoquc73e251eg 10001000 at 11.9% 45σ 10000001 at 16.0% 70σ |11>_L d73u854vllmc73anvom0 dominant confirmed — dominant confirmed — All dominant patterns match theoretical predictions exactly. Cross-contamination below 1.2% in all cases. The transition table: Klein: |00> ↔ |11>, |10> ↔ |01> (X_L2 flips BOTH Z_L1 and Z_L2) Toric: |00> ↔ |01>, |10> ↔ |11> (X_L2 flips ONLY Z_L2) The two holonomy maps act on different sector pairs, the non-orientable twist permutes the logical sector labels relative to the orientable case. This is the third in a series of three papers establishing complete experimental characterisation of the Klein bottle stabilizer code: Paper 1 (doi:10.5281/zenodo.19284050): Code exists on hardware, GSD=4, kill test Z=691σ Paper 2 (doi:10.5281/zenodo.19286677): Boundary parameter δ is tunable, Z=316–606σ Paper 3 (this paper): Logical algebra is non-orientable, Z=45–312σ Live demo: https://kleincode.pythonanywhere.com Paper 1: https://doi.org/10.5281/zenodo.19284050 Paper 2: https://doi.org/10.5281/zenodo.19286677 All hardware experiments performed on IBM Fez (Heron r2, 156 qubits) via IBM Quantum Runtime. Code: MIT license. Paper: CC BY 4.0.