Shape-Frequency Mobility for Classical Spatial Patterns

A localized field profile and translation symmetry do not by themselves establish particle inertia or a moving solution. For the Gaussian-window pattern action, a uniform phase gradient has a positive restricted kinetic response, yet the complete field equation excludes the corresponding differentiable Gaussian traveling family. Two explicit changed phase actions supply exact motion. The second follows a branch's actual positive stationary frequency and includes the full functional response of that frequency.

These are classical fields on ordinary shared $\mathbb R^{10}$. Density is squared field amplitude, not position probability; amplitude and phase are variables of one field, not additional physical dimensions or exchanged particles. An internal profile and its moving center remain distinct, including for an electron with a literal deterministic trajectory in six ordinary spatial dimensions. All ten center-motion directions are permitted independently of the pattern's active-depth classification. Neither observed-particle identification nor a local Lorentz-covariant field theory follows from the results below.

Canonical Signs and the Gaussian Branch

For $\psi=a e^{i\theta}$, $a>0$, $\rho=a^2$, the canonical integrand is

$$\frac i2(\psi^*\dot\psi-\dot\psi^*\psi)=-\rho\dot\theta.$$

Thus $S=\int dt[-\int\rho\dot\theta-H]$ gives $\dot\rho=\delta H/\delta\theta$, $\dot\theta=-\delta H/\delta\rho$. If the phase derivative is $-\nabla\cdot(\rho v)$, the continuity equation has sign $\dot\rho+\nabla\cdot(\rho v)=0$. A relative equilibrium $\psi=e^{-i\omega_vt}U(y-vt)$ must solve

$$G[U]=\omega_vU-i v\cdot\nabla U. \tag{1}$$

Write $c$ for the positive speed parameter. GC-R1 has

$$H_{\rm GC}=\mu N+C_\chi^{-1}\int n_x\eta_x(h_x-\mu)dx, \quad i\psi_t=G_{\rm GC}[\psi],\quad dH[v]=2\operatorname{Re}\langle v,G_{\rm GC}\rangle.$$

Use the windows, moments, centroid cap and full variation defined in the linked note. For any rank-$d$ spatial projector $P$, $d\in\{2,3,4,5,6,10\}$, its positive centered ground amplitude is

$$B'=b_1P+b_0(I-P),\quad B=B'-\alpha I>0,\quad B_{\rm in}=B(I-\sigma^2B)^{-1},\quad \sigma^2B<I,$$ $$a_0(y)=A e^{-y^TB_{\rm in}y/2},\qquad u_0(y)=A\det(I+\sigma^2B_{\rm in})^{-1/2}e^{-y^TBy/2},\qquad A>0. \tag{2}$$

Then $G_{\rm GC}[a_0]=\mu a_0$, and $N=A^2\pi^5/\sqrt{\det B_{\rm in}}$ remains uncalibrated. Define

$$Q=(I+\sigma^2B_{\rm in})^{-1}=I-\sigma^2B,\quad S=B(B')^{-1},\quad X_x=(B')^{-1}\alpha x,\quad c_x=-Sx, \quad e(x)=\theta(|Sx|^2/L_c).$$

For the following GC trial calculation, assume the other moment caps are flat one near the ground moments and the considered small momenta. The centroid cap is nonnegative, compact in anchor space, and positive near zero. No assumption that it is constant is made.

Original GC-R1: Complete Residual and Restricted Inertia

For $\psi_p=e^{ip\cdot y}a_0$, exact Gaussian convolution gives

$$J_\sigma\psi_p=e^{-\sigma^2p\cdot Qp/2}e^{i(Qp)\cdot y}u_0(y).$$

Local centroid and centered covariances are unchanged; the local momentum is $p_s=Qp$, and $h=\sqrt{\mu^2+c^2|p_s|^2}$. The complete source, including centroid variation, is

$$G_{\rm GC}[\psi_p]=\mu\psi_p+C_\chi^{-1}J_\sigma\int g_xdx,$$ $$g_x=\chi_x^2u_p\left\{ e(x)(h-\mu)+i e(x)\frac{c^2}{h}p_s\cdot B'(y-X_x) +(h-\mu)(\partial_{c_x}\eta)\cdot(y-X_x)\right\}. \tag{3}$$

The last term is essential at finite $p$. The normalized oscillator part annihilates the phase-removed Gaussian; the other moment-cap derivatives vanish only under the stated flat-cap assumption. A trial translated field $e^{ip\cdot y-i\Omega t}a_0(y-vt)$ requires source $\Omega\psi_p-i e^{ip\cdot y}v\cdot\nabla a_0$, which must be compared with all of (3).

The restricted energy is exactly

$$H_{\rm trial}(p)=N\mu+N\kappa_Be^{-\sigma^2p\cdot Qp} [\sqrt{\mu^2+c^2|Qp|^2}-\mu],\quad \kappa_B=(NC_\chi)^{-1}\int n_x(0)e(x)dx. \tag{4}$$

Strict positivity and the finite nontrivial centroid cap give $0<\kappa_B<\|u_0\|^2/N=\sqrt{\det Q}<1$. Hence

$$H_{\rm trial}=N\mu+\frac{N\kappa_Bc^2}{2\mu}p^TQ^2p+O(|p|^4), \quad \dot X=\frac{\kappa_Bc^2}{\mu}Q^2p+O(|p|^3),\quad \mathcal P=Np,$$ $$M_{\rm trial}=\frac{N\mu}{\kappa_Bc^2}Q^{-2}. \tag{5}$$

The positive tensor is the response of the restricted Gaussian family. It is not established complete-field inertia of GC-R1.

Why the Gaussian Phase Tangent Does Not Translate

Let $L$ be the real fixed-direction linearization of $G_{\rm GC}-\mu I$ at $a_0$. Put

$$\mathcal A(y)=C_\chi^{-1}\int\chi_x(y)^2e(x)dx,\qquad \mathcal V(y)=By\mathcal A(y)-\tfrac12\nabla\mathcal A(y).$$

Differentiation of the complete residual (3) at $p=0$ gives

$$L[i(p\cdot y)a_0]=i\frac{c^2}{\mu}J_\sigma[(Qp)\cdot\mathcal V\,u_0]. \tag{6}$$

Terms proportional to $h-\mu$ start at second order, and $C_\chi^{-1}\int\chi_x^2 e B'(y-X_x)dx=By\mathcal A-\nabla\mathcal A/2$. The right side of (6) is not a constant translation tangent.

A tail comparison proves this even when quadratic Gaussian exponents coincide. In coordinates diagonalizing $B$, set $C=B+2\alpha I$, $D_0=(I+\sigma^2C)^{-1}$, $R=CD_0$. For each anchor,

$$J_\sigma[u_0\chi_x^2](y)=\text{positive constant}\, e^{-y^TRy/2}e^{2\alpha y^TD_0x} e^{-\alpha|x|^2+2\sigma^2\alpha^2x^TD_0x}.$$

Inserting $B'(z-X_x)$ multiplies this by $B'D_0y+\alpha(2\sigma^2B'D_0-I)x$. Choose a coordinate $j$ with $(Qp)_j\ne0$, and take $y=t e_j$, $t\to+\infty$. Its inserted leading polynomial has fixed sign uniformly on the compact positive anchor support. Elementary upper/lower bounds on the positive anchor integral give

$$\log|J_\sigma[(Qp)\cdot\mathcal V\,u_0](te_j)| =-R_jt^2/2+\ell_jt+o(t),\qquad \ell_j=2\alpha(D_0)_j\sup\{x_j:e(x)>0\}>0. \tag{7}$$

The lower bound integrates over an open positive-measure set arbitrarily near that support supremum. A nonzero translation tangent on the same ray has logarithm $-B_{{\rm in},j}t^2/2+O(\log t)$; if its $j$-velocity vanishes, the tangent vanishes there. Either the quadratic rates differ or the strictly positive linear rate in (7) still prevents equality. For the common check parameters $b_1=1,b_0=4,\alpha=1/4,\sigma^2=1/16$, input widths are $48/61,240/49$, while response widths are $80/69,272/81$.

This excludes a differentiable traveling Gaussian family generated by uniform linear phase. Chirp tangents are even, width changes are real, and symmetry tangents cannot repair the odd imaginary equation. It does not exclude an odd non-Gaussian phase/shape deformation. There is no exceptional depth-five result, including in comparison with depth ten.

For a genuine differentiable branch $U_v=a_0+\eta_v+o(|v|)$, equation (1) requires

$$L\eta_v=\delta\omega\,a_0-i v\cdot\nabla a_0. \tag{8}$$

Since $La_0=0$, self-adjointness gives $\delta\omega=0$. The norm condition is $2\operatorname{Re}\langle a_0,\eta_v\rangle=0$; phase, translation and orientation zero tangents can be fixed by orthogonality. If constrained solutions $\eta_j$ of $L\eta_j=-i\partial_ja_0$ exist, their full momentum response would be $M_{ij}=2\operatorname{Re}\langle-i\partial_i a_0,\eta_j\rangle$. It is symmetric and nonnegative because $L$ is self-adjoint and nonnegative at a homogeneous minimizer. These are conditional statements, not a solved inverse problem.

At this Gaussian, $L$ is a bounded nonnegative self-adjoint directional Hessian. Its nonzero coefficients have compact anchor support, where the base window norm has a positive lower bound. Gaussian-smoothed differential kernels then have square-integrable spatial kernels, making $L$ compact. This supplies neither a gap nor a bounded inverse on the infinite-dimensional symmetry complement. Moreover the GC $C^{1,1}$ theorem asserts a Lipschitz source, not a Fréchet derivative. Translated small packets explain the distinction: homogeneity and source clustering can leave an order-$\epsilon$ nonlinear response near $\epsilon\tau_Lb$, while compact $L$ sends $\tau_Lb$ to zero. The remainder need not be $o(\epsilon)$. Ordinary implicit-function arguments with a Fréchet derivative are therefore unavailable. A deformed moving GC branch remains open.

A Changed Phase Action with Exact Gaussian Motion

Call the first changed law PM-R1. With $j=\operatorname{Im}(\psi^*\nabla\psi)=\rho\nabla\theta$, define

$$H_{\rm PM}[\psi]=H_{\rm GC}[a]-\mu N+ \int\sqrt{\mu^2\rho^2+c^2|j|^2}\,dy. \tag{9}$$

The current integrand is zero at $\rho=j=0$. It is finite on $H^1$, since its excess above $\mu\rho$ is at most $c|j|$ and $\int|j|\le\|\psi\|_2\|\nabla\psi\|_2$. The differential equation is asserted on admissible smooth nowhere-zero profiles. Arbitrary-node evolution needs a separate proof.

This law retains phase, translation, rotation and amplitude symmetries. It replaces GC's phase dependence, including phase-sensitive smoothing, while agreeing with its energy on real nonnegative amplitudes. It uses no extra interaction field. Write $g_a=G_{\rm GC}[a]$, real on real amplitudes, and $h(q)=\sqrt{\mu^2+c^2|q|^2}$, $v(q)=c^2q/h(q)$, $q=\nabla\theta$. Complete variation gives

$$\rho_t+\nabla\cdot(\rho v(q))=0,\qquad \theta_t=-g_a/a-[h(q)-\mu],$$ $$G_{\rm PM}[\psi]=e^{i\theta}\left\{g_a+a[h(q)-\mu] -\frac{i}{2a}\nabla\cdot[a^2v(q)]\right\}. \tag{10}$$

The current variation supplies the advection term. Since $g_a=\mu a_0$, this complete equation has exact solutions

$$\psi(y,t)=e^{ip\cdot y-i\Omega t}a_0(y-X_0-vt),\qquad \Omega=\sqrt{\mu^2+c^2|p|^2},\quad v=c^2p/\Omega. \tag{11}$$

Their complete energy, momentum and rest response are

$$\mathcal E=N\Omega,\qquad\mathcal P=Np,\qquad \mathcal E^2-c^2|\mathcal P|^2=N^2\mu^2,\qquad \left.\frac{\partial\mathcal P}{\partial v}\right|_{v=0}=\frac{N\mu}{c^2}I. \tag{12}$$

At finite speed the positive tensor is $(N\mu/c^2)[\gamma I+\gamma^3vv^T/c^2]$, $\gamma=(1-|v|^2/c^2)^{-1/2}$. The comoving frequency in (1) is $\omega_v=\Omega-p\cdot v=\mu^2/\Omega$; the laboratory phase factor in (11) uses $\Omega$.

These are full-field quantities for the changed law, whose prescribed square-root current term supplies the relation. They are not a derivation from original GC internal oscillation dynamics. No profile contraction has been imposed; the energy relation alone proves no local Lorentz covariance. This gives count-one motion at all maintained depths, without a depth-five exception, excited clocks or perturbative persistence.

A General Frequency-Following Mobility Law

Let $E[\rho]$ be real, translation and rotation invariant, and homogeneous of degree one in a positive classical density. Assume an admissible first derivative $w[\rho]=\delta E/\delta\rho$, a differentiable derivative in the variations being used, and the required integrability and adjoints. An admissible nodeless stationary branch obeys

$$w[\rho_*](y)=\omega>0.$$

Define the changed frequency-following law FL by

$$H_{\rm FL}[\rho,\theta]=E[\rho]+ \int\rho\left[\sqrt{w[\rho]^2+c^2|\nabla\theta|^2}-|w[\rho]|\right]dy. \tag{13}$$

On a positive-$w$ domain, write $q=\nabla\theta$, $h=\sqrt{w^2+c^2|q|^2}$. Its complete variations are

$$\frac{\delta H_{\rm FL}}{\delta\rho}=h+(Dw)^*[\rho(w/h-1)],\qquad \frac{\delta H_{\rm FL}}{\delta\theta}=-\nabla\cdot(\rho c^2q/h). \tag{14}$$

The adjoint term cannot be dropped by holding $w$ fixed. Euler homogeneity and its full differential give

$$E=\int\rho w,\quad \delta E=\int w\,\delta\rho=\int w\,\delta\rho+\int\rho Dw[\delta\rho], \quad (Dw)^*\rho=0. \tag{15}$$

At $\rho_*$ and uniform $q=p$, both $w=\omega$ and $\Omega=\sqrt{\omega^2+c^2|p|^2}$ are spatially constant. The coefficient $\omega/\Omega-1$ in (14) is constant, so (15) cancels the entire Hessian-response term. The density derivative is exactly $\Omega$, and translation covariance supplies

$$\psi(y,t)=e^{ip\cdot y-i\Omega t}\sqrt{\rho_*(y-X_0-vt)},\quad v=c^2p/\Omega,\quad \mathcal E=N\Omega,\quad\mathcal P=Np, \quad M_{\rm rest}=\frac{N\omega}{c^2}I. \tag{16}$$

Thus one changed functional follows each admitted positive stationary frequency without separate inertia tuning. Homogeneity retains amplitude freedom. It does not construct the stationary branches or derive their count spectrum. The underlying $w$ may be nonlocal. Nodes, zeros or sign changes of $w$, and functional differentiability require their own analysis; the absolute value in (13) does not prove differentiability at $w=0$. A signed nodal shell cannot be replaced by its modulus while retaining its branch equation.

All Compact Field Variations for the Actual GC Gaussian

The Gaussian tends to zero at infinity, and a local perturbation can change the nonlocal GC source's sign in distant tails. There is no asserted positive-$w$ open $L^2$ neighborhood. Nevertheless the actual positive GC Gaussian gives an exact variational weak solution of FL against every compact smooth complex field variation. The following amplitude proof establishes that scope directly.

On real amplitudes, write

$$E_a[a]=H_{\rm GC}[a],\quad g[a]=G_{\rm GC}[a],\quad dE_a[\eta]=2\int g[a]\eta,\quad E_a[a]=\int a g[a].$$

For $a>0$, $E[\rho]=H_{\rm GC}[\sqrt\rho]$ has $w=g[a]/a$. Therefore (13) is algebraically identical to

$$H_{\rm FL}[a,\theta]=E_a[a]+\mathcal C(a,g[a],a\nabla\theta),\qquad \mathcal C(a,g,b)=\int a[\sqrt{g^2+c^2|b|^2}-|g|]. \tag{17}$$

This introduces no change where $a>0$, and removes division by $a$ from the energy. Products are integrable for $a,g,b\in L^2$ by Cauchy–Schwarz; for nonnegative $a$, $0\le\mathcal C\le c\int a|b|$. Bounded phase gradient is sufficient, including the Gaussian and its small compact smooth variations. Defining an energy at $a=0$ would not itself prove differentiability through nodes.

At $a_0$, GC has a strong fixed-direction derivative

$$g[a_0+\epsilon\eta]=\mu a_0+\epsilon DG[a_0]\eta+o_{L^2}(\epsilon). \tag{18}$$

To justify it, use the bounded window analysis map $\mathcal A$ from the GC regularity proof. Every base Gaussian window is nonzero, so its local perspective functional is smooth near $\mathcal A_xa_0$. Its Hessian applied to $\mathcal A_x\eta$ converges pointwise in anchor and is dominated by $2C_*\|\mathcal A_x\eta\|$. Dominated convergence in squared anchor norm, followed by bounded synthesis $\mathcal A^*$, gives (18). This is a strong derivative for each fixed direction, not a Fréchet derivative uniform over all $L^2$ directions.

The integrated Hessian is self-adjoint. Degree-two amplitude homogeneity gives $DG[a_0]a_0=g[a_0]=\mu a_0$. Thus, for $L=DG[a_0]-\mu I$,

$$La_0=0,\qquad \int a_0L\eta=0. \tag{19}$$

Consider the candidate (16) at $\omega=\mu$ and a smooth compact spacetime perturbation $\zeta$. The Gaussian has a positive minimum on the compact moving test region over its finite time interval. For sufficiently small $\epsilon$, $\psi_0+\epsilon\zeta$ is nowhere zero and admits smooth amplitude/phase variables. Their tangents are

$$\eta=\operatorname{Re}(e^{-i\theta_0}\zeta),\qquad \xi=\operatorname{Im}(e^{-i\theta_0}\zeta)/a_0.$$

Both are compact smooth functions, and conversely arbitrary such $\eta,\xi$ arise from $\zeta=e^{i\theta_0}(\eta+i a_0\xi)$. These are all local field directions, not only Gaussian collective-coordinate variations.

The amplitude path satisfies $a_\epsilon=a_0+\epsilon\eta+o_{L^2}(\epsilon)$. GC Lipschitzness and (18) imply $g[a_\epsilon]=\mu a_0+\epsilon k+o_{L^2}(\epsilon)$, $k=DG[a_0]\eta$. The tangent of $b_\epsilon=a_\epsilon\nabla\theta_\epsilon$ is $\eta p+a_0\nabla\xi$. The Euclidean norm and absolute value in (17) are Lipschitz. At the base, $a_0>0$ and $g=\mu a_0>0$ almost everywhere, so their directional derivatives exist almost everywhere. Difference quotients are controlled by integrable $L^2$ products, including $a_0(|k|+c|\eta p+a_0\nabla\xi|)$. Dominated convergence handles exact tangent lines, and Lipschitz/product bounds handle the $o_{L^2}$ remainders. Sign changes of the perturbed source in distant tails therefore cause no problem for this derivative.

The complete correction variation is

$$\delta\mathcal C=2(\Omega-\mu)\int a_0\eta +(\mu/\Omega-1)\int a_0(k-\mu\eta) +\int a_0^2\frac{c^2p}{\Omega}\cdot\nabla\xi.$$

Equation (19) cancels the middle term. Adding $\delta E_a=2\mu\int a_0\eta$ gives

$$\delta H_{\rm FL}=2\Omega\int a_0\eta+\int\rho_0v\cdot\nabla\xi, \qquad v=c^2p/\Omega. \tag{20}$$

Since $\delta\rho=2a_0\eta$, this is the full density/phase derivative at the branch. The traveling history obeys $\theta_t=-\Omega$, $\rho_t+\nabla\cdot(\rho v)=0$; spacetime integration by parts makes $\delta S=0$ for every compact smooth complex field variation. Translation covariance and the Gaussian's positive minimum on the compact test region make the estimates uniform over the test's finite time interval.

Every positive GC Gaussian at depths two, three, four, five, six and ten therefore gives a global explicit traveling history that is a variational weak solution of the changed FL action, with arbitrary positive amplitude, translation and orientation, finite complete energy and momentum, and positive rest response (12). PM has the same compact-variation property by its simpler direct amplitude variation.

This exceeds ansatz stationarity, but is weaker than a neighborhood evolution theorem. Neither PM nor FL inherits GC's global $L^2$ $C^{1,1}$ or Cauchy result. Continuity of their vector fields on an open field domain, uniqueness, perturbative persistence, arbitrary-node evolution, clustering for their changed laws, excited positive branches and occupied-state selection remain open. The exact Gaussian ground solutions do not establish inherited signed or nodal Hermite shells, observed charge or spin, or the maintained mass family.

The mobility checker and independent review checker retain finite algebraic controls for modulation, complete PM residuals, canonical conventions and homogeneous Hessian cancellation. Their executed source calculations reported 76 and 20 exact checks respectively. A separate finite two-density example has stationary frequencies two and three under one homogeneous shape law and positive ratio-mode curvatures; it illustrates the full Euler cancellation without constructing a spatial count spectrum. The continuum and compact-variation arguments above carry the actual existence scope.