Testing slow geometric motion with every centered deformation

The radial geometry has a nonzero wave source. Allowing it to move is a possible alternative to static force balance, but the proposed motion must respect the approximation used to calculate it. Including the actual moving spatial basis gives a precise test: the simplest first-order slow action predicts motion far outside its own range of validity.

This result applies to the stated Berry-only action and the tested symmetric profile. It does not rule out faster same-field dynamics, other shapes or a complete sustaining kinetic action. No formal velocity reported here is asserted to be a physical trajectory.

The candidate law and its required condition

Use the complete curvature from the moving normal basis, the fixed bounded hopping $h=z/\sqrt{1+r^2}$, and every centered coordinate $\alpha_k=x_k+i y_k$, $k=2,\ldots,d$. The proposed slow action for a unit canonical wave is

$$L_{\rm slow}=A_a(\alpha)\dot\alpha_a-\lambda(\alpha). \tag{1}$$

At a real radial reference state, the imaginary forces vanish and $F_{xx}=F_{yy}=0$. Put $M_{kh}=F^{\rm wave}_{x_k y_h}$, $f_k=\partial_{x_k}\lambda$. The complete first-order equations give

$$\boxed{M\dot y=f,\qquad \dot x=0.} \tag{2}$$

For the occupied wave to follow one instantaneous eigenbranch, every other signed eigenmode must satisfy the necessary condition

$$\boxed{ \eta_r= \frac{\left|\sum_h\dot y_h\langle r,H_{y_h}n\rangle\right|} {|\lambda_r-\lambda_n|^2}\ll1.} \tag{3}$$

It is enough to find one violation of (3); changing the normal gap cannot remove a violation within the retained carrier itself. A common canonical wave norm multiplies both terms in (1), so it cancels from (2). A common change of time unit also leaves (3) invariant. Neither convention can repair this test.

The small signed partner supplies the strongest constraint

The carrier has chiral form $H_P=\left(\begin{smallmatrix}0&T^\dagger\\T&0\end{smallmatrix}\right)$. At the real reference state, a real source has zero matrix element between the two opposite small signed partners. An imaginary source generally has a nonzero matrix element there. Thus that partner's contribution to the real–imaginary curvature is zero, even though it imposes the strongest condition (3). Its two roles cannot be inferred from curvature alone.

The calculation retains all centered coordinates, both surface signs and all maintained particle controls. At $\gamma=10000$, examples for the geometric carrier on the positive surface sign are:

Even the smallest value among the positive-sign geometric controls is about $1.63\times10^{15}$. The original carrier also violates (3). Independent radial-grid and outer-domain refinements change the selected ratios by less than $5\times10^{-7}$ relatively. The violation is not a consequence of resolving the finite curvature matrix inaccurately.

The asymptotic mismatch

For the geometric carrier define

$$B_d=\frac12\left[ (F^{\rm normal}_{x_d y_d})_{00} +(F^{\rm normal}_{x_d y_d})_{d-1,d-1}\right]. \tag{4}$$

At every tested depth $B_d>0$. The large-hopping normal density has equal endpoint weights. Its full centered limiting curvature is diagonal and nonsingular; cross-order entries vanish in that limit. Let $\kappa_d$ be the endpoint hopping derivative and let $\lambda_{\rm lead}\propto\gamma^{-(d-1)}$ be the leading small frequency. The dominant source and opposite-partner vertex both scale as $\gamma\kappa_d$. Equations (2) and (3) therefore give

$$\boxed{\eta_{\rm pair}\sim \frac{\gamma^2\kappa_d^2}{4\lambda_{\rm lead}^2|B_d|} \propto\gamma^{2d}.} \tag{5}$$

The full-coordinate calculations at hopping strengths 1000 and 10000 verify this leading expression. Improving the simplex approximation by increasing hopping worsens this particular slow-motion condition.

The result identifies what the candidate action lacks: a self-consistent sustaining response capable of keeping the occupied motion in the regime being modeled. The complete particle's positive inertia is still a joint energy-and-momentum question. The signed modes in (3) have no assigned particle–antiparticle interpretation.

Verification

normal_motion_checks.py uses the complete spatial curvature and a high-precision signed singular system. Its 293-check report checks all centered equations, the zero real partner vertex, the strongest adiabatic condition, asymptotic scaling and domain refinements. This is a necessary consistency test of (1), not a numerical integration outside the approximation's validity.