Three-Plane Scaffold: Numerical Field Data — Borromean Junction and Dirac Zero Modes

This dataset provides the numerical fields underlying the three-planescaffold framework for scalar-frame locking, Borromean colour holonomy,and the conditional strong-sector spectral gap. It contains the relaxedthree-wall junction scalar field, the inputs from which it was built, andthe sixteen Dirac eigenmodes nearest zero computed in that background,together with the three protected zero-mode profiles. The configuration is a triple domain-wall junction on an 80x80x80 lattice(spacing dx = 0.2, coordinate range [-8.0, +7.8]). Three scalar kink wallslie in the coordinate planes x=0, y=0, z=0 and meet at a common triplejunction at the origin (grid index 40,40,40). The phi^4 / Allen-Cahn widthparameter is delta = 1.0; the relaxed wall broadens to w ~ 1.41. The fieldis a converged relaxed solution (max|dphi/dt| = 4.93e-6); the relaxationcorrection relative to the separable product ansatz isepsilon_0 = ||delta_phi|| / ||phi|| = 0.1374. Contents (NumPy .npy arrays): - phi_exact.npy Relaxed three-wall junction field (80^3, float64) - phi_approx.npy Separable product ansatz tanh(x)tanh(y)tanh(z) - delta_phi.npy Relaxation correction; phi_exact = phi_approx + delta_phi - grid_x.npy 1-D coordinate array (shared by all three axes) - dirac_evals.npy 16 Dirac eigenvalues nearest zero - dirac_evals_squared.npy Their squares - dirac_evecs.npy 16 eigenvectors (512000 sites x 2 spinor components) - psi_x/y/z_zero_mode.npy Three named protected zero-mode profiles - README.md Full file specification and reproducibility checks Symmetry and index structure. The junction field is exactly invariant underthe S_3 permutation and Z_3 cyclic symmetries of the three axes (to machineprecision), while the relaxed solution carries an approximately 16% reflectionasymmetry across the walls arising from the asymmetric grid range and therelaxation. The Callias / Jackiw-Rebbi index of the junction is N = 3; thethree named zero-mode profiles are numerically identical up to roundoff andtheir overlap matrix is rank one, a direct consequence of the exactpermutation symmetry. Scope and limitations. This dataset is published with its limitations statedexplicitly. (i) The junction couplings (lambda_J, eta_J) are not included, sothe junction-core equation of motion is not independently verifiable fromthese files; far from the junction the field solves the one-dimensional kinkequation at the relaxed width to a residual of order 1e-2. (ii) The datasetcontains only the sixteen Dirac modes nearest zero, not the full chargedspectrum; it is therefore not sufficient to evaluate the induced U(1) vacuumpolarisation or any heat-kernel trace, which additionally require chargeassignments and a finite subtraction scheme. (iii) A meaningful strong-CPdeterminant phase is not computable from these data: it requires independent,complex, full-rank up/down flavour sectors and an explicit mass functionM(Phi, G, theta), none of which are contained here. The README includes a short, self-checking Python snippet that verifies thefield decomposition, the junction location, and the exact permutation symmetry.