Induced Gravity, the Higgs VEV, and Leptons from Spectral Geometry of Vacuum Entanglement on the Cube and Prism

The vacuum is taken as a network of entanglement bonds on a MERA over Q₃, each Planck-area bond a harmonic oscillator with coordinate D_KL, the relative entropy, bounded by the Bekenstein–Hawking value S_χ = S_BH ℓ_P²/A = 1/4. The partition function of Q₃ gives the vacuum entanglement S* = 2/9 and the residual capacity ξ* = 1/36. The 1+2 split makes z isometric and x, y disentanglers. The 5-star, the isometric edge and the four edges that share its endpoints, carries 5ξ*, delivered along the face diagonals of length √2 as c_D = 5ξ*/√2, and the disentangler row 1+1+2 = 4 sets the x, y edge weight ω = 1 + 4c_D = 1 + 5√2/18 of the vertex operator of Q₄ = Q₃ × I. Its sixteen modes are the products of (−1)^x, (−1)^y, (−1)^z, (−1)^τ, and its quadratic form, a vertex term m² and an edge term ω_ij, gives the nine cell factors R_ab = 1/(m² + λ(a,b)). The trace of its inverse gives Sakharov's induced gravitational constant G_N = 0.999057 ℓ_P² (−0.094%). The descent suppression e^(−1/ξ*) = e^(−36) is set by the residual capacity, and the Higgs mode carried down by e^(−36) gives v = 246.897 GeV (+0.28%). The inverse determinant and cells of the vertex operator give the neutrino Yukawa couplings in normal ordering, with Δm²₂₁ = 7.44659 × 10⁻⁵ eV² (−0.58%), Δm²₃₁ = 2.51451 × 10⁻³ eV² (+0.060%), Δm²₃₂ = 2.44004 × 10⁻³ eV² (+0.080%) and Σm_ν = 0.0596 eV (DESI bound < 0.0642 eV), and the charged-lepton Yukawa couplings y_τ = 9.929656 × 10⁻³ (−0.082%), y_μ = 5.850827 × 10⁻⁴ (+0.007%) and y_e = 2.759932 × 10⁻⁶ (−0.619%). Anthropic's Claude (Opus versions 4.6–5.5) was used for computational combinatorics/algebraic derivations, Python scripting, and manuscript editing carried out interactively, and PocketCAS for iOS and NetworkX with NumPy in Google Colab to independently verify the matrix calculations. Theoretical direction was provided by the author. Harmonic Entanglement Theory is a proposed theory and is being worked on at this time. No claim is made for the validity of these works. The results set out here are the output of the construction described in the paper, and the verification script checks the internal consistency of that construction, that the geometry built from NetworkX with the paper's stated rules reproduces the matrices in the text and that those reproduce every derived number. It does not establish that the underlying physics is correct. The agreement with measured values reported on this site is presented as it stands, without a claim that the framework is established.