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

this is a placeholder - version 3 is on its way; the current data has issues and shouldn't be used. **Version 2 — May 2026. Supersedes v1 (March 2026).** Numerical field data for the three-plane scaffold programme. The scaffold is a topological soliton framework in which the particle content and coupling constants of the Standard Model emerge from the geometry of three mutually orthogonal scalar domain walls in R³ whose zero-sets form a Borromean link. This deposit contains the Allen-Cahn relaxed junction field and associated mode data used throughout the scaffold paper series (see Related Identifiers). --- **Grid specification** 80³ points, dx = 0.2 δ (kink width units), coordinate range [−8.0, 7.8] in units of δ. Physical conversion: 1 grid unit of energy = mW = 80.4 GeV. The meson mass (continuum threshold of the fluctuation spectrum) is m_s = √2 · mW; the W-boson mass corresponds to mW = 1/δ in grid units. --- **Contents (Version 2)** *Field files (carried over from v1, unchanged):*- `grid_x.npy` — 1D coordinate array (80 points)- `phi_approx.npy` — product-ansatz field tanh(x)·tanh(y)·tanh(z)- `phi_exact_inner.npy` — Allen-Cahn gradient-flow relaxation of phi_approx; primary field for all calculations- `delta_phi_inner.npy` — correction field phi_exact − phi_approx; field-correction parameter ε₀ = ‖δφ‖/‖φ‖ = 1.87% *Zero mode files (regenerated — replaces v1 psi_{x,y,z}_zero_mode.npy):*- `psi_zero_mode_approx.npy` — product-ansatz Jackiw–Rebbi zero mode sech(x)·sech(y)·sech(z)/√8, normalised in 3D; same for all three walls by Oh symmetry- `psi_x_zero_mode_exact.npy`, `psi_y_zero_mode_exact.npy`, `psi_z_zero_mode_exact.npy` — JR zero mode amplitudes on phi_exact, computed as the lowest eigenvector of the SUSY Hamiltonian H₀ = −∇² + φ² − ∂φ/∂xₐ for each axis a; normalised in 3D; Oh permutation symmetry confirmed (overlap ⟨ψ_y | P_xy ψ_x⟩ = 1.000) *New in Version 2:*- `breathing_mode_exact.npy` — A₁g (Oh-symmetric) eigenvector of the scalar fluctuation operator L = −∇² + V″(φ_exact), V″ = 3φ² − 1; reflected to full 80³ grid; eigenvalue ω² = 1.6722, ω = 1.2931 (in units mW = 1), corresponding to mH = 104.0 GeV from the eigenvalue alone- `scaffold_scalar_results.npy` — Python dict (load with allow_pickle=True) containing all key scalar quantities: ε₀, ω_breathing on exact and approx fields, Z₂, Z₄, Z₄/Z₂², continuum threshold, operator normalisation conventions, and first-order correction Δω/ω = −3.36%- `README.txt` — full documentation including operator normalisation conventions, relationship between the eigenvalue frequency and the topological crossing formula, and deprecation notes for v1 files --- **Key numerical results** | Quantity | Value ||---|---|| Field correction ε₀ | 1.87% || A₁g breathing mode ω (phi_exact) | 1.2931 mW || A₁g breathing mode ω (phi_approx) | 1.3293 mW || Approx → exact shift in ω | −2.72% || Z₂ = ∫ sech²(x)sech²(y)sech²(z) d³x | 8.000 (exact: 2Nc) || Z₄ = ∫ sech⁴(x)sech⁴(y)sech⁴(z) d³x | 64/27 (exact: 1/Nc³ · Z₂²) || Continuum threshold m_s | √2 · mW || 1D shape mode frequency ω_S | √(3/2) · mW | Note: the breathing mode eigenvalue ω = 1.293 mW gives mH = 104 GeV, not 125 GeV. The value fH = 1.558 quoted in the Higgs paper (doi:10.5281/zenodo.20148808) comes from the topological crossing formula mH = ywall/(3π²), not from this eigenvalue. These are two distinct quantities; both are documented in README.txt. --- **Deprecation notice for v1 files** The following files from v1 (March 2026) are superseded and should not be used:- `psi_{x,y,z}_zero_mode.npy` — the spatial structure of these files was inconsistent with the v1 README description (not a localised sech profile; large at vacuum corners rather than junction core)- `dirac_evals.npy` / `dirac_evecs.npy` — all eight eigenvalues were ~0.015, inconsistent with the README claim that the two smallest should be ~4×10⁻⁸; the subgrid computation was unreliable The v1 field files (phi_approx, phi_exact_inner, delta_phi_inner, grid_x) are unchanged and remain valid. --- **Software and reproducibility** All computations use Python 3 with NumPy and SciPy (sparse eigensolver ARPACK via `scipy.sparse.linalg.eigsh`). The eigenvalue computations used the positive-octant reduction with Neumann boundary conditions and explicit Oh symmetry verification (permutation overlaps). Full computational details are in README.txt. Papers: https://doi.org/10.5281/zenodo.19267506 https://doi.org/10.5281/zenodo.19202565 https://doi.org/10.5281/zenodo.19301348