Holomorphic binding and its responsive-frame test

There is a useful exact alternative to the polarization-current binding law. Its complete fixed-frame linearization is a complex-linear self-adjoint Dirac operator, so it preserves the spatial norm of every linear disturbance. Allowing the defining frame to respond changes that conclusion: the simplest positive frame kinetic extension has genuine resolved growing radial controls.

This comparison identifies which part of the construction supplied the apparently stable response. It does not derive the simplex mass weights or a stable physical mass operator. Mass remains the selected frequency, and the physical directions remain ordinary and uncompactified.

An analytic map with the same localized branch

Write the full field as $\Psi=(\phi,\chi)^T$, where each block is an $s\times s$ complex matrix. In a specified component frame, define

$$f_0=\frac{\operatorname{Tr}\phi}{s},\qquad a=\frac{\operatorname{Tr}\phi^2}{s},\qquad b=2f_0^2-a,\qquad t_h=\frac{a}{b}. \tag{1}$$

The map

$$F(\phi)=m_c\beta\phi\beta-\kappa t_h(\phi)\phi \tag{2}$$

is holomorphic and homogeneous of degree one wherever $b\ne0$. On the radial field $\phi=fI_s+i h\alpha_{\hat x}$, with real $f,h$,

$$a=f^2-h^2,\qquad b=f^2+h^2,\qquad t_h=\frac{f^2-h^2}{f^2+h^2}.$$

It therefore reproduces the lower-block binding source of the original polarization profile exactly. Use the real potential

$$U_h=\omega(\|\phi\|_F^2-\|\chi\|_F^2) +2\operatorname{Re}\operatorname{Tr}(\chi^\dagger F(\phi)). \tag{3}$$

The first term is a signed linear scalar current. It agrees with the $q=0$ current root on the positive-trace core branch, but is an explicit linear continuation outside that branch. It is not the nonsmooth absolute value $\omega|\operatorname{Tr}C|$.

The complete source is

$$S_\phi=\omega\phi+DF(\phi)^\dagger\chi,\qquad S_\chi=-\omega\chi+F(\phi). \tag{4}$$

Here the adjoint acts on the complete matrix space, not just scalar and vector coefficients. The stationary field $\Psi=e^{-i\omega t}(\phi_0,0)^T$ has the same radial equations and Bessel-ratio solution as the original core. Its shape is independent of $\omega$ and of a common normalization multiplier. The assigned $\omega_{N,d}=\omega_{0,d}\binom{N+d}{d}$ values still enter through prescribed inverse weights.

Full spatial fixed-frame linearization

At $\chi_0=0$, the rotating-frame Hessian is

$$\mathcal H= \begin{pmatrix}0&L^\dagger\\L&-2\omega\end{pmatrix}, \qquad L=-i\alpha_a\partial_a+DF(\phi_0(x)). \tag{5}$$

No conjugate disturbance is suppressed in obtaining (5); its absence is an actual consequence of holomorphicity at this branch. The derivative coefficient is uniformly bounded along the full core. For example, writing $t=(f^2-h^2)/(f^2+h^2)$ and $s_p=2fh/(f^2+h^2)$, one sufficient estimate is

$$\|DF\|\le m_c+\kappa(t+2s_p)\le m_c+\sqrt5\kappa.$$

Thus (5) is a self-adjoint bounded perturbation of the free Dirac operator on its $H^1$ domain. Its entire spatial linear evolution is unitary in canonical $L^2$, including every angular and partner component. A useful factorization is

$$(\mathcal H+\omega)^2= \operatorname{diag}(L^\dagger L+\omega^2, LL^\dagger+\omega^2). \tag{6}$$

This is an analytic fixed-frame result, stronger than a finite local spectrum scan. It is not yet a statement about the complete moving particle.

Removing off-branch poles

The holomorphic quotient has poles away from the displayed core. Put $\rho=(\|\phi\|_F^2+\|\chi\|_F^2)/s$, and multiply only the binding ratio term in (3) by the common cutoff $\eta(|b|^2/\rho^2)$. Use the same twice continuously differentiable cutoff that is zero below $1/16$ and one above $1/4$, described in Regular polarization binding.

At the core its argument equals one. The profile, source, and entire linearization (5) are therefore unchanged. Away from the core the regularized potential is no longer holomorphic, but it is regular on the unit field sphere and homogeneous of degree two. Its source extends as a globally Lipschitz map at the total-field origin. All density and cutoff derivatives are retained in the source.

The fixed-frame nonlinear Dirac equation consequently has global canonical $L^2$ mild evolution, conserved norm and finite propagation. These facts do not imply nonlinear orbital stability of the core. In particular, pointwise linearization around an exponentially decaying profile does not by itself prove uniform Frechet differentiability of the flow in the unweighted spatial $L^2$ topology.

Covariance requires the frame to be included

The traces in (1) identify spinor and partner components in a specified frame. They are not by themselves a Lorentz-covariant interaction. A conditional completion evaluates them on $\Psi_u=B(u)^{-1}\Psi$, where $u$ is a retained future unit timelike orientation of the same proposed field. Under a Lorentz transformation,

$$B(u')^{-1}S_\Lambda B(u)=\operatorname{diag}(R,R), \qquad \Psi'=S_\Lambda\Psi R^\dagger.$$

The rest-frame matrices transform by unitary conjugation, under which (1)–(3) are invariant. The induced partner connection must also be included in the kinetic term. This orientation and its realization in the master field are inverse premises; no additional interacting particle is being introduced.

The first frame source vanishes on the radial core because

$$\operatorname{Re}\operatorname{Tr}(\phi_0^\dagger\alpha_aF(\phi_0))=0.$$

Uniformly boosted cores therefore remain conditional solutions, with $E_0=\omega Q$ and positive canonical inertia $\omega Q$. The mixed second variation, however, is not zero. In the component-frame convention,

$$DF^\dagger(\alpha_a\phi_0)+\alpha_aF(\phi_0) =-2\kappa t\alpha_a\phi_0. \tag{7}$$

The wave and frame must both respond to this same mixed term.

One explicit completion uses the Lorentz scalar frame action

$$\mathcal L_u=-\frac{I_f(u\cdot j)}2 \bigl[c_f^2\eta^{\mu\nu}+(1-c_f^2)u^\mu u^\nu\bigr] \partial_\mu u^\alpha\partial_\nu u_\alpha, \quad u^2=1,\quad I_f>0,\quad 0<c_f<1.$$

It has no alignment potential. At the unboosted core, its quadratic term is $I_f\rho_{\rm total}(|\dot v|^2-c_f^2|\nabla v|^2)/2$. Equation (7) adds the reciprocal quadratic energy $2\kappa t\,v^a\operatorname{Re}\operatorname{Tr} (\delta\phi^\dagger\alpha_a\phi_0)$. Its complete rotationally invariant channel consists of four complex radial wave coefficients and the radial frame displacement and velocity. With $m_c=1$, $\kappa=2$, $I_f=1$, $c_f=1/2$, it has:

The rates are in the common test units. Meshes of 160, 320 and 640 points, and outer radii 20 and 28, give convergent values. The eigenfunctions have bounded spatial derivative norms, finite central extent, and eigen-equation residuals below $8\times10^{-14}$. They are not alternating grid modes. These results exclude this particular responsive-frame extension as a stable particle model; they do not exclude every possible support action. The full branch inertia remains positive despite the growing shape mode.

Collective drift is a different requirement

A free localized particle must also admit its translation–boost pair. If $T=\partial_a\Phi$ is a translation zero mode and $b$ is the boost tangent, the time generator has $\mathcal A b=-T$, up to the chosen translation sign. A complex-linear self-adjoint $\mathcal H$ with $\mathcal A=-i\mathcal H$ could not have both this generalized mode and $\mathcal H T=0$: its range is orthogonal to its kernel, whereas the required $\mathcal H b$ is a nonzero multiple of $iT$.

Thus fixed-frame unitarity cannot simply be promoted to uniformly bounded motion of the complete free particle. Legitimate drift has to be treated through the responding support or the full real Hamiltonian structure, and stability assessed modulo collective motion. This observation does not explain away the radial growing modes above: those are actual shape/frame disturbances in a separate invariant spatial channel.

Reproducible calculations

holomorphic_core_binding.py and regular_holomorphic_binding.py contain the complete analytic and regularized sources. The 1,110-check report verifies stationary Cartesian equations, full variations, adjoints, regularization, and frame covariance across depths 3, 4, 5, 6 and 10. The 38-check spatial frame report retains the refined growing eigenfunctions and all reciprocal frame terms. The independent nonlinear Cartesian control uses holomorphic_spatial_evolution.py; its finite periodic boxes are numerical approximations, not compactified physical dimensions or a proof of nonlinear particle stability. The 25-check nonlinear refinement report compares the complete disturbance fields across grids and time steps. Relative differences are about 0.172% for the depth-three grid comparison, 0.120% for the depth-four grid comparison, and 0.0314% for the depth-four time-step comparison. Canonical norm drift remains below six parts in a billion in the tested trajectories. These controls use a common illustrative frequency of one; they are not separate fitted particle assignments.