Topologically Robust Syndrome-Based Readout of a Classical Parameter in a Family of Non-Orientable Stabilizer Codes

This upload contains the manuscript, LaTeX source, and figures for the second paper in the Klein Bottle QEC series. Research Overview We introduce a one-parameter family of Klein bottle stabilizer codes C(δ) defined by shifting the orientation-reversing boundary identification of an Lx × Ly lattice. Although all members share identical topology (Klein bottle, GSD=4), each produces a distinct and theoretically predictable syndrome fingerprint P(δ) when a topological excitation is prepared. Key Results We verify all four fingerprints of the Lx=4, Ly=2 family on the 156-qubit IBM Fez processor (Heron r2), with statistical significance Z = 316–606σ (job d71582469uic73cl1q5g, 8192 shots each). Topological Protection The encoding map δ → P(δ) is injective and robust under noise. Because δ is embedded in the circuit topology rather than the quantum state, local errors cannot corrupt the parameter directly. Misidentifying δ requires ≥ d simultaneous errors, providing a misidentification probability scaling as pᵈ (where d=4) versus p¹ for standard gate-encoded parameters. Contents paper.pdf – Full manuscript paper.tex – LaTeX source (self-contained) fig_delta_lattice.pdf – Lattice diagrams and fingerprint encoding maps fig_delta_fingerprints.pdf – Hardware results: pattern frequencies and Z-scores fig_delta_depth.pdf – Signal strength vs. circuit depth analysis README.md – Detailed job IDs, API instructions, and reproduction notes Related Resources Paper 1 (Discovery): doi:10.5281/zenodo.19202945 Live API: https://kleincode.pythonanywhere.com GitHub: https://github.com/theoricline/klein-bottle-qec License: Creative Commons Attribution 4.0 International (CC BY 4.0)