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 XL2 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 XL2 anticommutes with ZL1 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> (XL2 flips BOTH ZL1 and ZL2) Toric: |00> ↔ |01>, |10> ↔ |11> (XL2 flips ONLY ZL2) 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.