τ-Inhomogeneous Maxwell Equations

The τ-inhomogeneous Maxwell equations (`IV.T43`) are the τ-categorical theorem that the source equation d⋆F = ⋆J on the τ-EM gauge bundle is the Euler–Lagrange equation of the B-sector variational principle. Its chart-shadow projection is Gauss's law div E = ρ/ε₀ together with the Ampère–Maxwell law curl B − ∂_t E/c² = μ₀ J.

Categorical invariant. On the τ-EM gauge bundle: d⋆F = ⋆J as the B-sector Euler–Lagrange equation; chart shadow gives Gauss + Ampère–Maxwell.

Primary registry anchor: IV.T43

Supporting items: IV.T44, IV.T42

1. I.K0 — Universe Postulate
2. IV.T42 — Homogeneous Maxwell — dF = 0 (Bianchi half)
3. IV.T43 — Inhomogeneous Maxwell — d⋆F = ⋆J from B-sector variation
4. IV.T44 — Complete τ-Maxwell System — both halves assembled

Lean modules referenced: TauLib.BookIV.Electroweak.TauMaxwell

Calibration anchor: PG-P01-neutron

Calibration chain:

1. B-sector variational principle on the τ-EM gauge bundle
2. ε₀, μ₀ from c = L·H cascade and ℏ_τ → ℏ bridge
3. SI bridge via m_n anchor for charge / current units

Manuscript reference: manuscript-sources/book-04/part03/ch28-electroweak-synthesis.tex

Status: Formalized

Module: TauLib.BookIV.Electroweak.TauMaxwell

Lean kind: theorem

Lean symbol: Tau.BookIV.Electroweak.InhomogeneousMaxwellEquations

Related glossary entries

• PG-L03-tau-maxwell-system τ-Maxwell System (Complete)
• PG-L06-tau-bianchi-identity τ-Bianchi Identity