Phase-Controlled Coherence Switching in Off-Diagonal Aubry-André-Harper Systems: Z3 Symmetry, Brannen Orbit Identification, and IBM Q Experimental Confirmation

In the off-diagonal Aubry-Andre-Harper (AAH) model, the phase parameter phi in the hopping amplitude t_n(phi) = t + lambda cos(2 pi alpha n + phi) governs the coherence character of quantum states. We demonstrate analytically and experimentally that as phi varies over [0, 2 pi), the inverse participation ratio (IPR) undergoes three-fold (Z3) periodic oscillations with three minima at phi = 0, 2 pi/3, 4 pi/3. These delocalization maxima identify precisely with the Brannen orbit positions for the three charged lepton generations: electron (k=0), muon (k=1), tau (k=2). The Z3 periodicity follows from the Z3 equivariance of the Brannen orbit proved in Paper 5 (Theorem 2, Brannen Uniqueness) combined with the ergodic properties of the golden-ratio orbit (Paper 3, Gauss-Kuzmin). IBM Q hardware on ibm_fez (Heron r2, 156 qubits, June 2026) confirms: Z3 contrast C_Z3 = 1.247 and 1.229 across two independent six-point phi runs (consistent to 1.5%, p < 0.002); C_Z3 = 1.317x at twelve-point resolution; phi = pi chain-breaking gives IPR = 0.347 (highest of all 12 measured values, 4.7-sigma above the Z3 lepton average). The Z3 Fourier amplitude A3/mean(IPR) = 0.147 ± 0.031 (4.7-sigma significance, chi² = 28.4 for dof = 11, p = 0.0028). These results provide the experimental Z3 foundation for Papers 3–10 and offer the first physical picture of why the Standard Model requires exactly three charged lepton generations. The phase structure reported here is specific to the incommensurate off-diagonal AAH model; it does not arise in the commensurate or diagonal realizations. What's new in Version 6: Version 6 is a substantial revision superseding the v2 domain-wall/structural-resonance framing. The paper is refocused on the Z3 phase-structure identification connecting the phase parameter's periodicity to the Brannen orbit, with IBM Q (ibm_fez) hardware confirmation. Hardware contrast values are harmonized across the series (C_Z3 = 1.247 / 1.229 across two independent 6-phase runs; 1.317 at 12-phase resolution). Table 8.1 corrected. A version note in the PDF details the changes. Code: github.com/quantumcoherence1/pi-anomaly-simulation Version 2.0 — April 2026. Corrected Mechanism A switching ratio range to 7x-12x across N=100-1000. Removed overclaimed 22.7x figure. Added disorder robustness cross-reference to companion Paper 1 Appendix B. Lab-scale predictions confirmed: 3x-6x at N=6-11, 7x-12x for Mechanism A at N≥100. Patent number: USPTO Provisional 64/019,824. Related paper: Irrational Modulation Parameter Dependence in Aubry-André Systems, Zenodo DOI 10.5281/zenodo.19654792. Patent protected under USPTO Provisional Applications 64/008,843, 64/009,939, and 64/019,824 (filed March 2026). For correspondence: braddowney95@gmail.com