τ-Maxwell System (Complete)
The complete τ-Maxwell system is a τ-categorical theorem (`IV.T44`) assembling all four Maxwell equations: the homogeneous pair dF = 0 (from the Bianchi identity, kinematic) and the inhomogeneous pair d⋆F = ⋆J (from B-sector variation, dynamic). No equation is postulated — they emerge from the U(1) connection on the EM gauge bundle.
τ-Definition
The complete τ-Maxwell system is a τ-categorical theorem (`IV.T44`) assembling all four Maxwell equations: the homogeneous pair dF = 0 (from the Bianchi identity, kinematic) and the inhomogeneous pair d⋆F = ⋆J (from B-sector variation, dynamic). No equation is postulated — they emerge from the U(1) connection on the EM gauge bundle.
Categorical invariant. On the τ-EM gauge bundle: dF = 0 (homogeneous, IV.T42) and d⋆F = ⋆J (inhomogeneous, IV.T43); together IV.T44.
Primary registry anchor:
IV.T44
τ-Derivation Chain
-
I.K0— Universe Postulate -
IV.T234— Bianchi identity ∂_[μ F_νρ] = 0 — automatic from d² = 0 on any U(1) connection -
IV.T42— Homogeneous Maxwell — dF = 0 yields div B = 0 and Faraday's law -
IV.T43— Inhomogeneous Maxwell — d⋆F = ⋆J yields Gauss's law and Ampère–Maxwell from B-sector variation -
IV.T44— Complete τ-Maxwell System — all four equations assembled
Lean modules referenced:
TauLib.BookIV.Electroweak.TauMaxwell
SI Translation
Calibration anchor: PG-P01-neutron
Calibration chain:
- U(1) connection one-form on the EM gauge bundle
- ε₀, μ₀ from c = LH cascade and ℏ_τ → ℏ bridge
- SI bridge via m_n anchor for charge / current units
Manuscript reference: manuscript-sources/book-04/part03/ch28-electroweak-synthesis.tex
Lean Coverage
Status: Formalized
Module: TauLib.BookIV.Electroweak.TauMaxwell
Lean kind: theorem
Lean symbol: Tau.BookIV.Electroweak.CompleteTaumaxwellSystem