Relativistic Localization and Stable Particle-Like States: Scaling No-Go Theorems, Broken-Vacuum Dirac–Scalar Bound States, Collective-Coordinate Mass, and Antiparticle-Inclusive Linear Stability

This paper develops the localization and relativistic-mass sector of the Canvas programme as a self-contained mechanism study. The analysis begins by replacing historical identification of recurrence eigenvalues with particle masses by the physical definition E_{\rm rest} = \int d^3x\,T^{00}, M = E_{\rm rest}/c^2. A sequence of no-go results then narrows the admissible localization mechanism. What the Paper Does The paper systematically tests whether a relativistic, finite-energy, linearly stable particle-like composite can emerge from a self-consistent field theory with a broken vacuum. The chain of reasoning proceeds through several stages: Physical definition of mass: For any Lorentz-covariant field theory with stress tensor T^{\mu\nu}, the rest energy of a static localized configuration is E_0 = \int d^3x\,T^{00}, and the corresponding rest mass is M = E_0/c^2. This is fundamentally different from identifying an internal recurrence eigenvalue, inverse threshold, or lattice band curvature with physical rest mass. Nonrelativistic band-curvature obstruction: For a generic lattice quasiparticle, the inertial tensor is defined by dispersion curvature, whereas gravitational charge is controlled by the total energy. No generic identity enforces M_i c^2 = E_{\min} for a nonrelativistic lattice cluster whose inertia is defined by band curvature. Therefore the old nonrelativistic voxel-cluster picture cannot by itself be a fundamental equivalence-principle matter theory. Collective-coordinate relativistic mass theorem: For a finite-energy localized solution in a Lorentz-invariant continuum, Lorentz covariance generates the boosted configuration. The leading worldline action is S_{\rm eff} = -\mu c \int ds, and matching gives \mu c^2 = E_0. Thus the relativistic inertial mass is exactly M_i = E_0/c^2. If gravity couples universally to the same T_{\mu\nu}, the weak-field argument yields M_g = M_i = E_0/c^2. Massless exterior tail obstruction: Consider a stationary Dirac field outside a compact defect. If the asymptotic Dirac mass vanishes, the exterior equation yields oscillatory e^{\pm i|\omega|r}/r tails rather than exponentially decaying e^{-\kappa r}/r. A compact scalar defect on an asymptotically massless Dirac vacuum does not generically support an isolated normalizable real-frequency massive bound state for \omega \neq 0. A nonzero asymptotic gap is therefore required. Contact nonlinear-Dirac dilation no-go: For a fixed-charge spinor normalization, under the charge-preserving dilation \psi_\lambda(\mathbf x) = \lambda^{3/2}\psi(\lambda\mathbf x), the Dirac kinetic term scales as \lambda K, while a local attractive quartic interaction scales as \lambda^3 V. The energy function is E(\lambda) = \lambda K - (G_\psi/2)\lambda^3 V. Stationarity requires K = (3/2)G_\psi V, but the second derivative is E''(1) = -3G_\psi V < 0. Therefore every stationary fixed-charge solution of the minimal massless attractive contact model has a negative second variation along the canonical dilation. This is a genuine instability direction, not merely a failure to find a numerical solution. Propagating mediator reverses the scaling sign: Introduce a real propagating scalar X with a positive mass term and Yukawa-type coupling. Under joint scaling \psi_\lambda = \lambda^{3/2}\psi(\lambda x), X_\lambda = \lambda X(\lambda x), the Dirac kinetic energy, scalar gradient energy, and Yukawa energy all scale as \lambda, while the mediator mass term scales as \lambda^{-1}. Hence E(\lambda) = A\lambda + B/\lambda, with B > 0. At a stationary point, E'(1) = A - B = 0, so A = B, and E''(1) = 2B = M_X^2\int X^2d^3x > 0. A massive propagating mediator removes the universal negative scale mode of the minimal contact completion along the canonical joint dilation. However, if X \to 0 asymptotically, the exterior fermion remains massless and the tail obstruction survives. Broken-vacuum Dirac–scalar completion: The constructive completion is a real scalar X with broken vacuum X \to v \neq 0 and Yukawa coupling yX\bar\psi\psi. The action is \mathcal L = \bar\psi(i\gamma^\mu\partial_\mu - yX)\psi + \frac12\partial_\mu X\partial^\mu X - (\lambda/4)(X^2 - v^2)^2. The vacuum is X_{\rm vac} = \pm v. The asymptotic fermion mass is m_\infty = yv, and the scalar fluctuation mass is M_X^2 = 2\lambda v^2. The same broken vacuum supplies both a nonzero fermion gap and a finite scalar interaction range. Spherical stationary ansatz and boundary conditions: For the lowest j = 1/2 channel with \kappa = -1, the radial Dirac system is derived. Finite energy and regularity require G(0) = F(0) = X'(0) = 0. At infinity, G, F \to 0, X \to v. The single-particle radial normalization is \int_0^\infty (G^2 + F^2)dr = 1. The total conserved carrier number is N for N occupied equivalent states. Asymptotic fermion localization: At large r, X(r) \to v, m(r) \to m_\infty. The radial system has solutions governed by \kappa_{\rm tail} = \sqrt{m_\infty^2 - \omega^2}. For |\omega| < m_\infty, G, F \propto e^{-\kappa_{\rm tail}r}, up to algebraic radial factors. Thus |\omega| < m_\infty is the ordinary bound-state gap condition. If |\omega| > m_\infty, the exterior is oscillatory. Representative benchmark: The audited mechanism benchmark uses v = 1, y = 2, \lambda = \frac12. Therefore m_\infty = 2, M_X = 1. For N = 10, the boundary-value calculation gives \omega \simeq 1.69013, X(0) \simeq 0.38423, R_{\rm rms} \simeq 1.637, E_{\rm tot} \simeq 19.6141. The asymptotic decay constant is \kappa_{\rm tail} \simeq 1.0693, so \kappa_{\rm tail}^{-1} \simeq 0.935. At the numerical outer boundary r = 30, the fermion amplitudes had fallen to approximately the 10^{-15} level. Since Nm_\infty = 20, E_{\rm tot}/(Nm_\infty) \simeq 0.98070 < 1, so the composite is energetically bound. Continuation in carrier number: The persisted coarse branch from N = 8 to N = 15 shows that energetic binding begins between N = 8 and N = 9. The central scalar field decreases monotonically with N and changes sign between N = 14 and 15. Since m_{\rm eff}(r) = yX(r), the high-charge branch develops a dynamically generated mass-sign-inverted core. Thermodynamic identity: For a family of stationary fixed-charge solutions, dE/dN = \omega. The numerical branch satisfies this identity at roughly the 10^{-3} level on the well-bound branch. The branch also has dN/d\omega < 0, favorable in Vakhitov–Kolokolov turning-point diagnostics. Linear stability analysis: The paper formulates the complete stability problem. The radial sector shows no physical growing eigenvalue on the tested N = 10 and N = 14 backgrounds. Translation invariance supplies the exact L = 1 zero mode. Static nonradial shape operators are positive through L = 6. An antiparticle-inclusive signed-energy RPA, using \Delta_n = E_n - \omega for both positive- and negative-energy Dirac states, exhibits no converged localized growing mode through L = 6. Small high-frequency complex quartets drift with box size and disappear under domain enlargement, identifying them as discretized-continuum spectral pollution. Stability statement: Within the representative broken-vacuum Dirac–scalar completion and the tested numerical domains, the reconstructed N = 10 and N = 14 localized backgrounds have no converged growing localized physical mode through angular multipoles L = 0, \ldots, 6, including explicit negative-energy Dirac perturbations. Why This Matters The localization programme has passed through a sequence of increasingly restrictive tests. A generic nonrelativistic cluster fails the relativistic mass/equivalence requirement. An asymptotically massless defect fails ordinary localization. The minimal attractive contact nonlinear Dirac model has an exact dilation instability. A massive propagating mediator removes that universal scale instability but still requires a gapped exterior. A broken scalar vacuum supplies precisely that missing ingredient: m_\infty = yv \neq 0. Within a representative dimensionless completion, the coupled Dirac–scalar equations then possess a self-consistent finite-energy branch whose well-bound portion satisfies energetic and thermodynamic consistency tests. The stability programme goes beyond those diagnostics: the tested N = 10 and N = 14 configurations have no converged localized growing mode through L = 6 after explicit inclusion of negative-energy Dirac perturbations and careful separation of true modes from finite-box continuum artifacts. The result is not a Standard Model mass prediction. It is a mechanism theorem-plus-computation: a broken relativistic field vacuum can dynamically support stable particle-like composites. Together with Lorentz invariance, their conserved rest energy yields the correct relativistic inertial mass; with universal metric coupling, the same energy yields gravitational mass. The remaining fundamental work is upstream parameter derivation and downstream identification with the chiral gauge/matter spectrum, not the existence of a viable relativistic localization mechanism itself. Keywords: relativistic localization, Dirac–scalar bound states, broken vacuum, collective-coordinate mass, linear stability, antiparticle-inclusive RPA, spectral pollution, Canvas programme, emergent particles, nonlinear field theory