Continuum Translation of the QED Lagrangians

This record contains the manuscript "Continuum Translation of the QED Lagrangians" alongside its standalone, machine-checked Lean 4 / Mathlib formal verification module (QEDTranslation.lean). This work demonstrates across ten benchmark observables that the evaluations of Quantum Electrodynamics (QED) are reproduced by classical continuum mechanics in pure kg, m, s units (without auxiliary electrical dimensions C, F), achieving nine-significant-figure numerical parity across the eight single-particle and electroweak benchmarks and 0.004%--0.34% agreement across the multi-body atomic and hadronic confinement regimes. Every QED and Standard Model operator admits a direct continuum equivalent built from classical machinery established a generation or more prior: Dirac Bi-Spinor (ψ) Cartan moving frame (tetrad êᵢ) tracking a cavitated Kelvin vortex core (4π untangling stator); Scalar Bilinear (ψ̄ψ) Inverse Stokes added-mass displacement volume (1/Vadded = 2/V_τ, with Cₘ = 1/2); Linear Kinetic Operator (i c\,γ^μ∂_μψ) Rotational Clebsch vorticity (∇ × (λ∇β_Ω)); Rest-Mass Term (-mc²ψ̄ψ) Cauchy core yield stress (-Pc = -1/2ρ_τ c²) and coordinate bag stress (-Pbag); Gauge Covariant Derivative (D_μ = ∂_μ + iq/_μ) Eulerian velocity superposition (vₜₒₜ = v_ + v_⊥); QED Interaction Vertex (-qcψ̄γ^μψ A_μ) Acoustic Magnus cross-pressure (ρ_τ\,v_ · v_⊥); Maxwell Field Invariant (-1/4μ₀FμνFμν) Antisymmetric shear strain invariant (-1/4Pc AμνAμν); Fine-Structure Constant (α) Viscoelastic shear-to-bulk modulus ratio (Πₛₕₑₐᵣ / Pc = μₛ / K = vₛ² / c²). Summary of the Ten Side-by-Side Benchmark Evaluations Free Electron Rest State (E₀): Stationary Dirac kinetic integral ωc vs. Cauchy core yield Pc V_τ = Pbag Veff = 8.18710578 × 10⁻¹⁴ J (0.51099895 MeV). $U(1)$ Magnetic Coupling & $g = 2$ Zeeman Split (B₀ = 1.0 T): Gordon spin decomposition μB B₀ vs. equatorial Kelvin-ring Magnus coupling ρ_τ Vadded c\,v_⊥(rc) = 9.27401007 × 10⁻²⁴ J. Schwinger Anomalous Moment (aₑ) & Force at the Bohr Radius (a₀): 1-loop vertex F₂(0) = α/(2π) vs. toroidal shear-to-bulk loop Πₛₕₑₐᵣ/(2π Pc) = 1.16140973 × 10⁻³; Coulomb force vs. spherical Bernoulli traction Fᵣₑₛₜₒᵣₑ(a₀) = 8.23872350 × 10⁻⁸ N. Hydrogen Lyman-α Photo-Excitation (n=1 → 2): Eigenstate subtraction E₂ - E₁ vs. isochoric strain transmutation Pc(Δ Vₚₒₜ + Δ Vₖᵢₙ) = Pc Δ V_γ = 1.63490427 × 10⁻¹⁸ J (10.204270 eV). Compton Scattering (ϑ = 90^∘): 2-vertex virtual fermion propagator h/(mₑ c) vs. Total Internal Reflection equatorial core circumference 2π rc = 2.42631024 × 10⁻¹² m. Pair Annihilation (e⁻e⁺ ↔ 2γ) & Schwinger Cavitation Limit: 1-loop Euler–Heisenberg action vs. core boundary cavitation force Fcav(rc) = Pc V_τ / rc = 0.21201371 N (Ecrit = 1.32328547 × 10¹⁸ V/m). Electroweak Vacuum Expectation Value (v): Mexican-hat Higgs minimum (√2GF⁰)-1/2 vs. 45^∘ Tresca shear dilution across the Fermi fracture area v = Eyield(AF)/√2π = 246.219651 GeV (3.94487371 × 10⁻⁸ J). Relativistic Heavy-Ion Compliance (U⁹¹⁺, Z = 92) & Zₘₐₓ = 1/α: Point-nucleus Dirac $1s$ spinor eigenvalue vs. metric compliance Pc V_τ(1 - αₛ) = 132.279934 keV. Two-Electron Helium Ground State (1s², Z = 2): Evaluates both (i) a 3-mode Rayleigh–Ritz variational rational polynomial in Hylleraas perimetric coordinates $(s, t, u)$ yielding the strict upper bound -2.902432 Ha = -78.9792 eV ($99.97%$ of NIST) and (ii) a constitutive monopole–dipole susceptibility closure (I₀ = 5/16, \, I₁ = 3/16 3717/1280μM Ha = -79.0085 eV, within $0.004%$ of NIST experiment -79.0051 eV, with first ionization potential I₁ = 24.5932 eV). Solid-State Hadronic Flux-Tube Confinement & Pion Fracture ($De > 1$): Isochoric necking of a 3-fault baryon core yields the Hookean stiffness κ = α mₚᵣ³ c⁴ / ² = 0.154803 GeV/fm², dual-tether Ultimate Tensile Strength Fₑₓₜ = 2κ rₑ = 0.872450 GeV/fm (1.3978 × 10⁵ N, matching Lattice QCD string tension σQCD), and $De = 1$ brittle fracture energy 2Ecrit = 140.0505 MeV ($0.34%$ from mπ^±) at Lₛₙₐₚ = 0.9512 fm. Included Companion Lean 4 Formalization (QEDTranslation.lean) The accompanying file QEDTranslation.lean formally verifies the algebraic identities, closed-form integral evaluators, and rational numerical bounds in Lean 4 (Mathlib) with zero kernel axioms (0 axiom) and zero incomplete proofs (0 sorry): Dimensional Closure (unit_audit_primitives_and_benchmarks): Verifies that all five native continuum primitives (ρ_τ, Pc, ε_τ, e_τ, Φ⊥(e)) and ten benchmark observables resolve on the integer dimensional lattice K, Ω, E ∈ Z³ in pure kg, m, s units. Operator Transitions & Closed-Form Integral Evaluators (transition1–transition5): Proves the 4× 4 Dirac–Clifford matrix identities and spatial spin commutator [γ¹, iγ²] = 2Σ₃, the full 5-term squared Pauli–Dirac pressure ledger, the Stokes exterior dipole antiderivative identities (Frad'(u) = u⁻⁴, Fang'(x) = 3x²+1 Cₘ = 1/2), the Coulomb-elimination bridge √kz = e_τ/e, the Laplace–Gamma shell evaluators (I₀ = 5/16, \, I₁ = 3/16), and all 24 integer coefficients of the 3-mode Hylleraas perimetric rational polynomial from the master integral M(a,b,c). Symbolic & Numerical Output Parity (benchmark1–benchmark7, codata2022_and_supercomputer_qed_qcd_audit): Proves symbolic equivalence between the QED and continuum expressions across all ten benchmarks, verifies 6-axis local minimality and strict Pekeris upper-bound ordering for the 3-mode Helium polynomial, and machine-checks all nine-significant-digit CODATA 2022 and multi-body/hadronic bounds via exact rational arithmetic (norm_num).