Normalized Spatial Contact and a Supported Profile

A spatially normalized interaction can preserve amplitude freedom while allowing separated patterns to retain their own energies. Combined with a gradient term and competing attractive and saturating contacts, it gives a converging numerical localized profile without an external trap or a prescribed finite band. This is a candidate classical spatial object, not an identified particle or a derived mass family.

Common Action and Complete Variation

For two real canonical fields on ordinary $\mathbb R^{10}$, put $\rho=q^2+p^2$, $J=\int\rho$, and

$$T=\epsilon I+(1-\epsilon)G_\ell*,\quad 0<\epsilon<1,\qquad F_j=\int\frac{\rho^j}{(T\rho)^{j-1}},\quad j=2,3,$$ $$H=\frac a2\int(|\nabla q|^2+|\nabla p|^2)-\frac b4F_2+\frac c6F_3, \qquad a,b,c>0.$$

The Gaussian has covariance $\ell^2I$. Its length is a common spatial interaction scale, not compactification. All terms are degree two in field amplitude, and no amplitude is selected to fit a mass.

Set $R=\rho/(T\rho)$. The complete density derivatives are

$$v_j=jR^{j-1}-(j-1)T(R^j),\qquad V=-bv_2/2+cv_3/3,$$ $$q_t=-a\Delta p+Vp,\qquad p_t=a\Delta q-Vq.$$

The convolved terms arise from varying the denominator. Omitting them changes both the equation and the energy-frequency relation. Any actual stationary rotating profile satisfies $2H=J\Omega$.

Using $J$ as one global denominator instead would make two distant identical copies have the contact energy of one and change their clocks. The spatial denominator clusters at fixed finite $\ell$. The positive local fraction $\epsilon$ also controls the vacuum limit and gives $J\le F_j\le\epsilon^{1-j}J$; it caps the enhancement rather than preserving an arbitrarily large contact factor.

Global Evolution for This Action

For a real field variation $h$, write $z=(q,p)$, $s=T\rho$, and $\eta=z\cdot h$. Direct differentiation gives

$$D^2F_j[h,h]=4j(j-1)\int\frac{R^{j-2}}s [\eta-RT\eta]^2+2\int v_j|h|^2.$$

Positivity of $T$ yields $(T\eta)^2\le sT|h|^2$, and hence a globally Lipschitz gradient with bound

$$C_j=2j(4j-3)\epsilon^{1-j} +2(j-1)(4j+1)\epsilon^{-j}.$$

The contact energy is $C^{1,1}$ on real $L^2$, without a claimed second Fréchet derivative at vacuum. The linear gradient propagator and this Lipschitz nonlinearity give unique global $L^2$ evolution. Translation difference quotients propagate $H^1$. Bounded kinetic regularizations, lower semicontinuity and reversibility establish exact energy conservation and strong $H^1$ continuity. Norm and canonical momentum are conserved. In particular,

$$H\ge\frac a2\|\nabla z\|_2^2-\frac b{4\epsilon}J$$

gives a uniform gradient bound at fixed energy and norm. Global existence alone does not establish a persistent localized object.

Support Test and Full Radial Calculation

For Gaussian density with principal variances $v_i$, let $t_i=\ell^2/v_i$. At $\epsilon=0$, exact integration gives

$$\frac{F_j}J=\prod_{i=1}^{10} \frac{(1+t_i)^{j/2}}{\sqrt{1+jt_i}},\qquad \frac HJ=\frac a{8\ell^2}\sum_i t_i-\frac b4\frac{F_2}J +\frac c6\frac{F_3}J.$$

One common example is $\ell=1$, $a=4765923522/1838265625$, $b=4$, $c=27/500$. It has a strict local Gaussian covariance minimum at all $t_i=2$, with positive length and aspect curvatures and energy below the spreading threshold. The minimum continues for sufficiently small positive $\epsilon$. Its Gaussian tail fails the full stationary equation, so the actual profile must be solved.

At $\epsilon=10^{-4}$, solve instead

$$-a(u''+9u'/r)+V[u^2]u=\Omega u,\qquad \int u^2=1.$$

Radial Gaussian convolution is evaluated with its exact modified-Bessel angular kernel. Finite-volume and independent fourth-order transformed kinetic discretizations converge toward the same positive localized shape:

The finest fourth-order root has algebraic residual $2.74\times10^{-12}$. An independent seven-point differential evaluation over every grid node gives continuum residual $4.33\times10^{-4}$, about $2.19\times10^{-5}$ relative to its source scale. Refinement lowers it; domain enlargement has negligible influence at the tested accuracy. Its RMS radius is about $1.86286$. These are numerical convergence results, not a rigorous continuum existence certificate.

The common positive clock shift $\nu J/2$, with $\nu=56$, leaves shape unchanged and gives $\Omega_0\approx36.21254$. Since $F_3/J\ge(F_2/J)^2$, the shifted complete energy obeys $H+\nu J/2\ge2J/9$ for these coefficients. This offset is an extra constitutive premise, not a measured mass calibration.

Nonradial Response and Tails

In angular degree $l$, define $T_l$ by the corresponding radial Gaussian kernel and let $M_f$ denote multiplication by $f$. The full rotating-frame blocks are

$$L_{-,l}=-a\left[\partial_r^2+\frac9r\partial_r -\frac{l(l+8)}{r^2}\right]+V-\Omega,$$ $$L_{+,l}=L_{-,l}+2M_u(I-T_lM_R) M_{(-b+2cR)/s}(I-M_RT_l)M_u.$$

The tested degrees $l=0,\ldots,15$ have positive remaining spectra after fixing norm and removing the phase/translation directions. The smallest non-symmetry amplitude margin is about $3.2654$, in degree two. The numerical translation eigenvalue decreases from $0.00570$ to $0.00141$ and then $8.48\times10^{-6}$ across the two schemes, consistent with its exact continuum zero. It is not a restoring force.

For $A_-=\max(b-2cR,0)$, the analytic bound

$$L_+\ge -a\Delta+V-\Omega-4[A_-R+T(A_-R^2)]$$

provides a centrifugal test for higher angular degrees. Its sampled sufficient cutoff is $l\ge15$; exact profile and exterior bounds would be needed for a continuum certificate. Thus the angular calculation is numerical linear-persistence evidence, not a nonlinear stability theorem.

The nonlocal potential need not vanish in an exponential tail. If $u\sim r^{-\gamma}e^{-kr}$ with the required derivative control, then

$$R_\infty=[\epsilon+(1-\epsilon)e^{2k^2\ell^2}]^{-1},\qquad \Omega=-bR_\infty+(b/2+c)R_\infty^2 -\frac{2c}{3}R_\infty^3-ak^2.$$

For $b\ge2c$ and $a>b\ell^2/2$, this tail frequency decreases strictly with $k^2$. The tested coefficients satisfy both conditions.

Conditional Complete Motion

The original gradient action has inertia $J/(4a)$, which does not automatically follow a branch's rest frequency. The existing frequency-following mobility law supplies an explicit changed action. Let $E[\rho]$ be the amplitude energy including the common offset, and $w=\delta E/\delta\rho$. Set

$$H_{\rm mov}=E+\int\rho \left[\sqrt{w^2+c_{\rm phys}^2|\nabla\theta|^2/4}-|w|\right].$$

For an actual smooth positive $H^2$ stationary profile with $w=\Omega_0/2>0$, full variation and homogeneity $(Dw)^*\rho=0$ give exact traveling histories with

$$\Omega_{\rm lab}=\sqrt{\Omega_0^2+c_{\rm phys}^2|k|^2},\quad v=c_{\rm phys}^2k/\Omega_{\rm lab},\quad E=J\Omega_{\rm lab}/2,\quad P=Jk/2, \quad M_{\rm rest}=E_{\rm rest}/c_{\rm phys}^2.$$

The result holds against all compact smooth field variations, including the nonlocal source response. It is conditional on the continuum profile. At rest the amplitude block stays unchanged and the phase block is multiplied by $c_{\rm phys}^2/(2a\Omega_0)>0$. The corresponding formal spectra scale by the square root of that factor, preserving their stability signs. The changed action does not inherit the original global Cauchy theorem, and no open-domain nonlinear persistence or local Lorentz covariance has been proved.

The useful connection is between spatial support, full variation, amplitude-independent clocks and complete inertial motion. The isotropic candidate selects no particle depth or simplex band, and it supplies no observed mass, spin or charge. The fourteen fixed-band certificates belong to a different action and are not silently transferred here.