[Section 2.2 insertion]
An auxiliary mesh-resolution study for the baseline phi = 1.0 case is provided in Online Resource 2. Three geometrically similar structured meshes (256 x 24 x 24, 320 x 30 x 30, and 384 x 36 x 36) were compared under otherwise identical settings. Because the finest mesh develops earlier severe timestep restriction in the late stage, the comparison is restricted to the common stable interval t <= 0.60 ms. Within that interval, the same center-plane-dominant bifurcation topology, the same layered three-dimensional propagation pattern, and the same qualitative growth trend of Db and dpAhead are recovered (Fig. S12-S15 and Table S5), arguing against a directional-grid interpretation of these structures.

[Discussion insertion]
The supplementary mesh study is therefore used to assess robustness rather than to claim strict whole-window late-stage grid independence. Within the common stable interval t <= 0.60 ms, the coarse, baseline, and fine meshes consistently recover the same center-plane double-branch topology, the same layered propagation pattern, and the same qualitative structural-compressive growth trend. The finest mesh exhibits earlier severe timestep restriction in the late stage and is therefore not used to support a whole-window late-stage convergence claim.

[Supplementary information insertion]
Online Resource 2 (ESM_2.pdf) contains the supplementary mesh-resolution study for the baseline phi = 1.0 case, including center-plane temperature and streamwise-velocity comparisons on coarse, baseline, and fine structured meshes, metric evolution within the common stable interval, spanwise temperature comparisons at t = 0.60 ms, and the corresponding tabulated mesh-comparison values.

[Captions]
Fig. S12  Center-plane temperature slices (z = 0.010 m) for the coarse, baseline, and fine meshes at t = 0.40, 0.50, and 0.60 ms. The same T = 1200 K front definition and the same color scale are used throughout. The center-plane double-branch flame head is recovered on all three meshes within the common stable window.

Fig. S13  Center-plane streamwise-velocity slices (z = 0.010 m) for the coarse, baseline, and fine meshes at t = 0.40, 0.50, and 0.60 ms. The same color scale is used throughout. In all three meshes, the branch tips remain dynamically favored relative to the recessed center.

Fig. S14  Mesh-resolution comparison of the precursor metrics within the common stable window (t <= 0.60 ms): (a) Db(t), (b) dpAhead(t), and (c) trajectories in the (Db, dpAhead) plane.

Fig. S15  Temperature slices at t = 0.60 ms on z = 0.003, 0.010, and 0.017 m for the coarse, baseline, and fine meshes. In all three meshes, the center plane exhibits the clearest branch separation, while the off-center planes remain narrower and less fully bifurcated.

Table S5  Mesh-resolution comparison for the baseline phi = 1.0 case within the common stable window.
