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 $\text{kg, m, s}$ units (without auxiliary electrical dimensions $\text{C, F}$), achieving nine-significant-figure numerical parity across the eight single-particle and electroweak benchmarks and $0.004\%\text{--}0.34\%$ 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 ($\psi$) $\Longleftrightarrow$ Cartan moving frame (tetrad $\hat{\mathbf{e}}_i$) tracking a cavitated Kelvin vortex core ($4\pi$ untangling stator); Scalar Bilinear ($\bar{\psi}\psi$) $\Longleftrightarrow$ Inverse Stokes added-mass displacement volume ($1/V_{\text{added}} = 2/V_\tau$, with $C_m = 1/2$); Linear Kinetic Operator ($i\hbar c\,\gamma^\mu\partial_\mu\psi$) $\Longleftrightarrow$ Rotational Clebsch vorticity ($\nabla \times (\lambda\nabla\beta_\Omega)$); Rest-Mass Term ($-mc^2\bar{\psi}\psi$) $\Longleftrightarrow$ Cauchy core yield stress ($-P_c = -\frac{1}{2}\rho_\tau c^2$) and coordinate bag stress ($-P_{\text{bag}}$); Gauge Covariant Derivative ($D_\mu = \partial_\mu + \frac{iq}{\hbar}A_\mu$) $\Longleftrightarrow$ Eulerian velocity superposition ($\mathbf{v}_{\text{tot}} = \mathbf{v}_\circlearrowleft + \mathbf{v}_\perp$); QED Interaction Vertex ($-qc\bar{\psi}\gamma^\mu\psi A_\mu$) $\Longleftrightarrow$ Acoustic Magnus cross-pressure ($\rho_\tau\,\mathbf{v}_\circlearrowleft \cdot \mathbf{v}_\perp$); Maxwell Field Invariant ($-\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu}$) $\Longleftrightarrow$ Antisymmetric shear strain invariant ($-\frac{1}{4}P_c A_{\mu\nu}A^{\mu\nu}$); Fine-Structure Constant ($\alpha$) $\Longleftrightarrow$ Viscoelastic shear-to-bulk modulus ratio ($\Pi_{\text{shear}} / P_c = \mu_s / K = v_s^2 / c^2$). Summary of the Ten Side-by-Side Benchmark Evaluations Free Electron Rest State ($E_0$): Stationary Dirac kinetic integral $\hbar\omega_c$ vs. Cauchy core yield $P_c V_\tau = P_{\text{bag}} V_{\text{eff}} = 8.18710578 \times 10^{-14}\text{ J}$ ($0.51099895\text{ MeV}$). $U(1)$ Magnetic Coupling & $g = 2$ Zeeman Split ($B_0 = 1.0\text{ T}$): Gordon spin decomposition $\mu_B B_0$ vs. equatorial Kelvin-ring Magnus coupling $\rho_\tau V_{\text{added}} c\,v_\perp(r_c) = 9.27401007 \times 10^{-24}\text{ J}$. Schwinger Anomalous Moment ($a_e$) & Force at the Bohr Radius ($a_0$): 1-loop vertex $F_2(0) = \alpha/(2\pi)$ vs. toroidal shear-to-bulk loop $\Pi_{\text{shear}}/(2\pi P_c) = 1.16140973 \times 10^{-3}$; Coulomb force vs. spherical Bernoulli traction $\vert{}\mathbf{F}_{\text{restore}}(a_0)\vert{} = 8.23872350 \times 10^{-8}\text{ N}$. Hydrogen Lyman-$\alpha$ Photo-Excitation ($n=1 \to 2$): Eigenstate subtraction $E_2 - E_1$ vs. isochoric strain transmutation $P_c(\Delta V_{\text{pot}} + \Delta V_{\text{kin}}) = P_c \Delta V_\gamma = 1.63490427 \times 10^{-18}\text{ J}$ ($10.204270\text{ eV}$). Compton Scattering ($\vartheta = 90^\circ$): 2-vertex virtual fermion propagator $h/(m_e c)$ vs. Total Internal Reflection equatorial core circumference $2\pi r_c = 2.42631024 \times 10^{-12}\text{ m}$. Pair Annihilation ($e^-e^+ \leftrightarrow 2\gamma$) & Schwinger Cavitation Limit: 1-loop Euler–Heisenberg action vs. core boundary cavitation force $F_{\text{cav}}(r_c) = P_c V_\tau / r_c = 0.21201371\text{ N}$ ($E_{\text{crit}} = 1.32328547 \times 10^{18}\text{ V/m}$). Electroweak Vacuum Expectation Value ($v$): Mexican-hat Higgs minimum $(\sqrt{2}G_F^0)^{-1/2}$ vs. $45^\circ$ Tresca shear dilution across the Fermi fracture area $v = E_{\text{yield}}(A_F)/\sqrt{2\pi} = 246.219651\text{ GeV}$ ($3.94487371 \times 10^{-8}\text{ J}$). Relativistic Heavy-Ion Compliance ($\text{U}^{91+}, Z = 92$) & $Z_{\max} = 1/\alpha$: Point-nucleus Dirac $1s$ spinor eigenvalue vs. metric compliance $P_c V_\tau(1 - \alpha_s) = 132.279934\text{ keV}$. Two-Electron Helium Ground State ($1s^2, Z = 2$): Evaluates both (i) a 3-mode Rayleigh–Ritz variational rational polynomial in Hylleraas perimetric coordinates $(s, t, u)$ yielding the strict upper bound $-2.902432\text{ Ha} = -78.9792\text{ eV}$ ($99.97\%$ of NIST) and (ii) a constitutive monopole–dipole susceptibility closure ($I_0 = 5/16, \, I_1 = 3/16 \implies \frac{3717}{1280}\mu_M\text{ Ha} = -79.0085\text{ eV}$, within $0.004\%$ of NIST experiment $-79.0051\text{ eV}$, with first ionization potential $I_1 = 24.5932\text{ eV}$). Solid-State Hadronic Flux-Tube Confinement & Pion Fracture ($De > 1$): Isochoric necking of a 3-fault baryon core yields the Hookean stiffness $\kappa = \alpha m_{pr}^3 c^4 / \hbar^2 = 0.154803\text{ GeV/fm}^2$, dual-tether Ultimate Tensile Strength $F_{\text{ext}} = 2\kappa r_e = 0.872450\text{ GeV/fm}$ ($1.3978 \times 10^5\text{ N}$, matching Lattice QCD string tension $\sigma_{\text{QCD}}$), and $De = 1$ brittle fracture energy $2E_{\text{crit}} = 140.0505\text{ MeV}$ ($0.34\%$ from $m_{\pi^\pm}$) at $L_{\text{snap}} = 0.9512\text{ fm}$. 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 ($\rho_\tau, P_c, \epsilon_\tau, e_\tau, \Phi_{\perp(e)}$) and ten benchmark observables resolve on the integer dimensional lattice $\langle K, \Omega, E \rangle \in \mathbb{Z}^3$ in pure $\text{kg, m, s}$ units. Operator Transitions & Closed-Form Integral Evaluators (transition1–transition5): Proves the $4\times 4$ Dirac–Clifford matrix identities and spatial spin commutator $[\gamma^1, i\gamma^2] = 2\Sigma_3$, the full 5-term squared Pauli–Dirac pressure ledger, the Stokes exterior dipole antiderivative identities ($F_{\text{rad}}'(u) = u^{-4}$, $F_{\text{ang}}'(x) = 3x^2+1 \implies C_m = 1/2$), the Coulomb-elimination bridge $\sqrt{k_z} = e_\tau/e$, the Laplace–Gamma shell evaluators ($I_0 = 5/16, \, I_1 = 3/16$), and all 24 integer coefficients of the 3-mode Hylleraas perimetric rational polynomial from the master integral $\mathcal{M}(a,b,c)$. 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).