ALP-ttbar Toy MC v1.7: Post-registration Results for a Three-Dimensional Two-Hypothesis Vertex Fit (Adding the r–z Opening Angle) of Displaced Light ALP → e⁺e⁻

Post-registration results of version 1.7 of a small, numpy-only toy Monte Carlo study motivated by the displaced-dielectron extension (Alimena et al., arXiv:2610.38932) of the top-associated long-lived ALP strategy of Rygaard et al. (JHEP 10 (2023) 138). The preregistration was frozen and public before any v1.7 result (doi:10.5281/zenodo.23995668); an amendment adding the all-pixel variant and prediction P7 was committed before any computation. v1.7 adds the r–z (longitudinal) view to the two-hypothesis vertex fit of v1.5/v1.6 (conversion with a common polar angle vs. ALP with free polar angles), on the unchanged transverse fit of v1.6 config R (3.8 T, nominal material; identity 100.000%). Δχ²_3D = Δχ²_T + Δχ²_L. Three z geometries are compared: pixel z only (default), stereo strips, and z at every layer (all-pixel). Limit degradation from conversions (cτ = 1 mm, 17/50/100/200 MeV): transverse only 127/68/28/8.2×; 3D default 127/64/24/6.1×; 3D stereo 127/61/21/4.5×; 3D all-pixel 117/38.5/13.1/3.9×. The r–z information helps from 50 MeV upward; at 17 MeV it is limited by acceptance and by bremsstrahlung-reduced momenta, where extra multiple scattering fakes polar openings. Predictions P2–P6 confirmed, P1 at the edge of its range, P7 refuted (7.8% gain). One deviation is reported (sanity gate 3: both r–z hypotheses use the path length from the conversion-hypothesis transverse fit; the preregistered version is stored for the record). Future directions include a 3D Gaussian-sum fit, an all-pixel tracker with reduced material in the context of next-generation trackers (ALICE 3, FCC-ee), a joint 3D fit, and the dimuon channel. This is an independent personal project, not affiliated with the authors of the referenced papers or with any collaboration. Only qualitative trends and ratios are meaningful; no claim is made about the performance of a real detector. GitHub: https://github.com/wspaik5/ALP_ttbar_Toy_MC_v1.7_Postregistration_3D_Vertex_Fit