Classical Stability and Poincaré Orbit of a Localized Charged Cell in the Axis Model: Energy-Space Hamiltonian Stability and an Exact Lorentz Mass Hyperboloid
Andrew Morton · Public technical note · Classical linear stability and relativistic orbit
Proves linear Hamiltonian stability and the exact classical relativistic energy–momentum orbit of the localized charged-cell solution.
Abstract
A companion constructive parent theory supplies an exact finite-energy localized charged cell with conserved signed source charge and energy below the free charged-field threshold. This supplement proves two classical properties of that branch. After fixing cell charge and total momentum and quotienting global phase and translations, the quadratic Hamiltonian descends to a strictly positive closed form. The reduced Hamiltonian generator is maximal skew-adjoint in the associated energy inner product, generates a strongly continuous unitary group, and has spectrum on the imaginary axis. Linear perturbations therefore remain bounded in energy norm modulo phase and translation modulation. Lorentz covariance then generates an exact classical Poincaré orbit with invariant mass shell P_μ P^μ = M_cl² and E(𝐏) = √(M_cl² + |𝐏|²). The certified rest-energy interval is M_cl ∈ [5.098542010836098, 7.451725569951958], below the free charged-field threshold 10.000999950004999 by at least 2.549274380053041. The result establishes a linearly stable classical charged soliton together with its exact relativistic energy–momentum orbit. A quantum continuation remains open and requires a controlled renormalized continuum charged sector with an isolated mass shell or pole and nonzero particle spectral weight.
Scope
This technical note establishes linear stability and an exact classical relativistic orbit. Nonlinear and asymptotic stability and a controlled continuum quantum particle remain unestablished.
Paper and supporting files
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Abstract checked 28 August 2026 · v2 · public release 9 August 2026.