Face saturation and the electroweak spectrum : three boson masses from the pentachoron

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 mₖ/mₑ = u^{nₖ²}.

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 (P1 Theorem II.1), edge excitations—including trapped gauge bosons—have vanishing component sums and are therefore 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 mass ratio is

mH/mW = u^{rank(L⁽⁰⁾) + dim ker(L⁽⁰⁾)} = u^{4+1} = u⁵,

verified at +0.73% against PDG 2024 (within the W resonance width ΓW/mW = 2.6%).

Two additional observations are reported: sin²θW = α*(1 − α*) = (ρ* − 1)/ρ² (matching the MS-bar value at −0.27%, status T3) and αEM ≈ α/Σnₖ² (matching at +1.16%, status T3, not derived). Combined with one anchor mass, the framework predicts mZ and mH (or mW and mZ) at sub-percent precision with zero free parameters.

A variational framework is introduced: the pentachoric action S[ρ̃] = ½⟨ρ̃, L⁽⁰⁾ρ̃⟩ − α* Σρ̃ᵥ ln ρ̃ᵥ generates the local conformal factor e^{−αρ̃} of P1 via δS = 0, the causal flux F(ρ) = ρ e^{−αρ} as a mean-field reduction, and the face saturation threshold as a Hessian instability. The DOF conservation 10 → 10 and the 4+1 surplus become variational identities.

The companion verification script (128 tests, all PASS) provides a reproducible audit trail.