Face saturation and the electroweak spectrum:three boson masses from the pentachoron The companion papers P1 and P2 established the pentachoron (K₅) as the unique elementary cell of discrete gravity in d = 4, with a single coupling α* = 1/(4 ln 2), and derived nine fermion masses (mean 2.4%, max 5.3%) from a zero-parameter mass formula m_k/m_e = u^{n_k²}. This paper extends the programme to the electroweak sector. The Laplacian L⁽⁰⁾ of K₅ has five eigenmodes: four internal modes (rank 4, component sums vanishing) and one network mode ψ₀ (kernel, component sum √5). By the fermion conservation theorem, edge excitations have vanishing component sums and are mute on ψ₀. Face excitations have non-vanishing component sums and activate ψ₀. The Higgs boson, identified with the face excitation that creates and maintains the saturation, pays a surplus of five mode activations at unit cost u over the W boson. The predicted ratio is m_H/m_W = u^{rank(L⁽⁰⁾) + dim ker(L⁽⁰⁾)} = u^{4+1} = u⁵, verified at +0.73% against PDG 2024. Two additional observations are reported: sin²θ_W = α*(1−α*) (−0.27%, T3) and α_EM ≈ α*/50 (+1.16%, T3). The divisor 50 = Σn_k² = 2n_d² is now identified as a Pythagorean topological invariant (P15), the same integer that determines the Planck gap ln(m_Pl/m_e) = 50 + α*u¹⁶. The integer 16 = (|V|−1)² governing the Higgs surplus (4+1, where 4 = deg = |V|−1) also appears as the Schur renormalisation coefficient (P15), the up-quark mass exponent (P2), and the Planck gap correction exponent — all tracing to deg(K₅) = 4. A variational framework is introduced: the pentachoric action S[ρ̃] = ½⟨ρ̃, L⁽⁰⁾ρ̃⟩ − α*Σρ̃_v ln ρ̃_v generates the conformal Ansatz, the causal flux F(ρ), and the face saturation threshold as variational identities. Companion verification script: 154 tests, all PASS.