Entropy-Difference Field Action with Exact Localized Patterns and Global Evolution

A second-spatial-order classical field equation can sustain exact finite-norm Gaussian patterns with two preferred widths, continuous spatial orientations and free center motion. Its full nonlinear Cauchy problem has a unique global finite-energy flow. Nearby Gaussian initial fields remain localized within their invariant family. These statements coexist with growing non-Gaussian linear disturbances: persistence within that family is a narrower result than stability of the complete field.

The construction is a candidate same-field support law. The filter lengths, coefficients and preferred widths are inverse inputs. Every maintained width multiplicity has the same support clock; neither the unequal observed mass-frequencies nor their simplex selection is derived. The flat-space count action supplies a different inverse construction with exact count clocks. Its action and localization theorem must be kept distinct from those below.

Classical Field, Action and Complete Variation

Let $\psi(x,t)$ be a complex classical scalar field on ordinary $\mathbb R^D$. For comparison of patterns the ambient space is $D=10$; the evolution theorem also holds in every finite positive dimension. Define

$$\rho=|\psi|^2,\qquad N=\int\rho,\qquad T=\int|\nabla\psi|^2, \qquad \operatorname{Ent}(f)=\int f\log f.$$

The integrals use ordinary spatial coordinates. Squared field amplitude, normalized integration weights, mode coefficients and physical particle position are distinct. This scalar support model supplies no Born law and does not replace an electron's literal deterministic six-dimensional orbit-plus-precession trajectory. The ten dimensions are ordinary spatial dimensions; width multiplicity classifies a pattern rather than a separate sector manifold. No compactification or particle-exchange mechanism enters.

Let $K_s$ be convolution by the normalized Gaussian of covariance $s^2I$, and put

$$\mathcal D_s(\rho)=\operatorname{Ent}(\rho) -\operatorname{Ent}(K_s\rho),\qquad H=\mu_0N+\alpha T-\sum_{j=1}^J\beta_j\mathcal D_{s_j}(\rho), \quad \alpha>0, \tag{1}$$ $$S=\int dt\left[\frac i2\int (\psi^*\dot\psi-\dot\psi^*\psi)-H\right]. \tag{2}$$

Coefficients $\beta_j$ may have either sign. Self-adjointness and mass preservation of each convolution give, first on smooth positive fields,

$$\delta\mathcal D_s =\int[\log\rho-K_s\log(K_s\rho)]\,\delta\rho.$$

The two constants from differentiating $f\log f$ cancel. Thus the full equation is

$$i\psi_t=\mu_0\psi-\alpha\Delta\psi -\sum_j\beta_j[\log\rho-K_{s_j}\log(K_{s_j}\rho)]\psi. \tag{3}$$

Both density arguments vary. In particular the second term is the convolution of the logarithm of the convolved density. Replacing it by $\log(K_s\rho)$ would change the variational equation. At an individual field zero, $\psi\log|\psi|^2$ has its continuous value zero. At the identically zero field the entire nonlinear source is defined to be zero.

For every constant complex $a$, the two norm-logarithm terms cancel:

$$\mathcal D_s(|a|^2\rho)=|a|^2\mathcal D_s(\rho),\qquad H(a\psi)=|a|^2H(\psi).$$

The source scales by $a$. Amplitude differentiation and phase invariance give $\int\psi^*\delta H/\delta\psi^*=H$, so any stationary field $\psi=e^{-i\omega t}\phi$ obeys $H(\phi)=\omega N$. The overall physical amplitude remains free and uncalibrated.

Exact Gaussian Width Equation

Take

$$\phi(x)=a\exp[-(x-X)^TB(x-X)/2],\qquad B=B^T>0, \quad t_j=2s_j^2,\quad C_j=B(I+t_jB)^{-1}.$$

Completing the Gaussian convolution gives

$$\log\rho-K_{s_j}\log(K_{s_j}\rho) =\frac12\log\det(I+t_jB)+\frac{t_j}{2}\operatorname{tr}C_j -(x-X)^T(B-C_j)(x-X).$$

Since $-\Delta\phi=[\operatorname{tr}B-(x-X)^TB^2(x-X)]\phi$, all position-dependent coefficients cancel exactly if

$$\alpha B^2=\sum_j\beta_j(B-C_j).$$

For each eigenvalue $b>0$ of $B$, this is

$$F(b)=\alpha,\qquad F(b)=\sum_j\frac{\beta_jt_j}{1+t_jb}. \tag{4}$$

Define the scalar Gaussian shape energy

$$h(b)=\frac{\alpha b}{2} -\frac12\sum_j\beta_j\log(1+t_jb),\qquad h'(b)=\frac{\alpha-F(b)}2.$$

Then

$$\frac HN=\mu_0+\sum_{i=1}^D h(b_i),\quad \frac TN=\frac12\operatorname{tr}B,\quad N=\frac{|a|^2\pi^{D/2}}{\sqrt{\det B}}, \tag{5}$$ $$\omega=\mu_0+\alpha\operatorname{tr}B -\sum_j\beta_j\left[\frac12\log\det(I+t_jB) +\frac{t_j}{2}\operatorname{tr}C_j\right].$$

At a width root the identity $\sum_j\beta_jt_jb/(1+t_jb)=\alpha b$ makes $\omega=H/N$ explicit. If all $\beta_j>0$, $F$ is strictly decreasing and has at most one positive root. Signed filters permit the two-well construction below.

Two Equal Gaussian Energy Wells

Use inverse-length and time units in which $\alpha=1$, and choose

$$(t_1,t_2,t_3)=(1,2,3),\qquad (\beta_1,\beta_2,\beta_3) =\left(30(m+1),-81m-\frac{81}{2},52m+\frac{52}{3}\right). \tag{6}$$

The exact rational identity is

$$F(b)-1=-\frac{6(b-1)(b-m)(b-4)}{(1+b)(1+2b)(1+3b)}. \tag{7}$$

Equal energies $h(1)=h(4)$ give

$$m=\frac{3-B_0}{A_0},\quad A_0=30\log\frac52-81\log3+52\log\frac{13}{4},$$ $$B_0=30\log\frac52-\frac{81}{2}\log3 +\frac{52}{3}\log\frac{13}{4}. \tag{8}$$

There is an analytic existence argument. On $(1,4)$, equation (7) is $6w(b)(b-m)$, where $w(b)=(b-1)(4-b)/[(1+b)(1+2b)(1+3b)]>0$. The equal-well condition $\int_1^4(F-1)\,db=0$ therefore makes $m=\int bw/\int w$, strictly between 1 and 4. Integrating by partial fractions yields (8). The derivative signs from (7), together with $h(0)=0$ and $h(b)\to\infty$, show that 1 and 4 are the two global minima of the Gaussian shape energy. This concerns that restricted energy, not the unrestricted field Hamiltonian.

Rigorous logarithmic enclosures give

$$2.03504067233352672920<m<2.03504067233352672922, \qquad h_*=h(1)=h(4)\simeq-3.627213216329755.$$

The intermediate barrier is $h(m)-h_*\simeq0.027152951202>0$. For any rank-$d$ orthogonal projector $\Pi$ in the common ordinary $\mathbb R^{10}$, set

$$B_d=\Pi+4(I-\Pi),\qquad d\in\{2,3,4,5,6,10\}.$$

All six branches solve the same equation, with $\omega_d=\mu_0+10h_*$ and $H_d=N\omega_d$.

Rotating $\Pi$ rotates an actual spatial pattern continuously. Every profile has noncompact Gaussian tails in all ten directions. This is a finite-norm localized alternative to exact uniformity through unused infinite directions; the narrow directions are still occupied. Broad-width rank does not yet derive literal dimensional absence. The orientation count at five supplies no fixed-axis dynamics or coupling suppression. There is no depth-five-specific coefficient or odd-depth stability rule.

Gaussian Dynamics and Restricted Persistence

For a centered complex Gaussian write $\psi=A(t)e^{-x^TQx/2}$, with $Q=B-iC$, $B>0$, and $C=C^T$ real. The exact nonlinear bracket remains quadratic, so this family is invariant. Put $V(B)=\sum_j\beta_j[B-B(I+t_jB)^{-1}]$. Without assuming that $B$ and $C$ commute,

$$i\dot Q=2\alpha Q^2-2V(B),\qquad \dot B=-2\alpha(BC+CB),\qquad \dot C=2\alpha(B^2-C^2)-2V(B), \tag{9}$$ $$i\frac{\dot A}{A}=\mu_0+\alpha\operatorname{tr}Q -\sum_j\beta_j\left[\frac12\log\det(I+t_jB) +\frac{t_j}{2}\operatorname{tr}B(I+t_jB)^{-1}\right].$$

The amplitude equation conserves $N$. The restricted Hamiltonian is

$$\frac HN=\mu_0+\sum_i h(b_i) +\frac\alpha2\operatorname{tr}(C^2B^{-1}). \tag{10}$$

The chirp term is nonnegative. Let $\varepsilon=H/N-\mu_0-10h_*$. Each $h(b_i)-h_*\le\varepsilon$. Choose $\varepsilon>0$ smaller than the barrier and than the positive energy gaps on the boundaries of chosen small neighborhoods of 1 and 4. Continuity prevents eigenvalues from leaving those neighborhoods or exchanging well multiplicity. Thus $0<\ell\le b_i\le L<\infty$. Also $\operatorname{tr}(C^2B^{-1})\ge\operatorname{tr}(C^2)/L$, so the chirp is bounded. The smooth matrix ODE remains in a compact subset of its domain and extends for all times. The centered variance obeys $\operatorname{tr}(B^{-1})/2\le10/(2\ell)$.

This proves bounded localization of nearby Gaussian fields at fixed norm and zero total momentum, modulo phase, center and orientation. Scalar commuting width disturbances satisfy

$$\delta\ddot b=8\alpha b^3F'(b)\delta b =-16\alpha b^3h''(b)\delta b.$$

The squared width-oscillation frequencies at 1 and 4 are approximately 6.210244034 and 30.955666947. These describe shape disturbances, not the particle mass-frequency. Arbitrary non-Gaussian fields require the full linearization below and the Cauchy theorem later in this note.

Complete Hermite Spectrum

Let $u>0$ be a stationary real Gaussian and write $\psi=e^{-i\omega t}(u+a+iz)$. Differentiating the full source, including the smoothed density, on Gaussian times polynomial disturbances gives

$$a_t=L_-z,\qquad z_t=-L_+a,\qquad L_-=\alpha(-\Delta+x^TB^2x-\operatorname{tr}B),$$ $$L_+=L_--2\sum_j\beta_j(I-R_j),\qquad R_ja=uK_{s_j}\left[\frac{K_{s_j}(ua)}{K_{s_j}(u^2)}\right]. \tag{11}$$

To compute $R_j$, put $a=uf$ and diagonalize $B$. In one direction of width $b$, the inner Gaussian integration has mean $ry$ and variance $s^2r$, where $r=(1+2s^2b)^{-1}$. The outer integration has mean $x$ and variance $s^2$. Their composition has mean $rx$ and variance $s^2r+s^2r^2=(1-r^2)/(2b)$. Applying it to the Hermite generating function yields

$$e^{2\sqrt b\,xz-z^2} \longmapsto e^{2\sqrt b\,rxz-r^2z^2}.$$

Consequently $uH_n(\sqrt b\,x)$ has multiplier $r^n$. Tensor products give the multiplier $\prod_i(1+t_jb_i)^{-n_i}$. These integrations are classical-field calculus, with no physical position-probability interpretation.

The Gaussian-weighted Hermite polynomials are complete; multiplication by $u$ identifies their weighted space with spatial $L^2$. Each $R_j$ is a bounded positive self-adjoint contraction. Thus $L_+$ is a bounded self-adjoint perturbation of $L_-$ on the oscillator domain, and both have this complete common eigenbasis. This realizes the whole spatial linear operator, with no omitted continuous spectrum. It does not assert Fréchet differentiability of the nonlinear flow near vacuum.

For broad degree $k$ and narrow degree $l$,

$$\lambda_-(k,l)=2\alpha(k+4l),\qquad \lambda_+(k,l)=2G(k,l),$$ $$G(k,l)=\alpha(k+4l)-\sum_j\beta_j \left[1-(1+t_j)^{-k}(1+4t_j)^{-l}\right]. \tag{12}$$

The real dynamical block is $\bigl(\begin{smallmatrix}0&\lambda_-\\-\lambda_+&0\end{smallmatrix}\bigr)$. A negative product produces exponents $\pm\sqrt{-\lambda_-\lambda_+}$; a positive product produces imaginary exponents. For the three-filter equal-well law, the complete negative list is

There is a rigorous finite verification plus an analytic tail. Each bracket in (12) lies in $[0,1]$, so $G(k,l)\ge\alpha(k+4l)-\sum_j\beta_j^+$, where $\beta_j^+=\max(\beta_j,0)$. Here $\sum_j\beta_j^+=82m+142/3<215$. All $k+4l\ge215$ are therefore strictly positive. For every pair with $k+4l\le215$, the expression is affine in $m$; exact rational interval evaluation on the enclosure above classifies all 5,940 pairs. The only zero pairs are $(0,0),(1,0),(0,1),(1,1)$. For $d=10$, only $l=0$ is allowed.

For $d<10$, the multiplicity is

$$M_d(k,l)=\binom{k+d-1}{d-1}\binom{l+9-d}{9-d}.$$

For $d=10$, it is $\binom{k+9}{9}$ for $l=0$, and zero otherwise. The resulting comparison is

Each negative spatial mode supplies one growing and one decaying real direction. In particular the broad cubic mode has $\lambda_-=6$ and $\lambda_+=3(3m-7)/8<0$, at every maintained multiplicity. It lies outside the Gaussian tangent family, whose shape directions have degree at most two.

The nonconstant zero pairs represent translations and rotations mixing the unequal widths. Since $\lambda_->0$ there, each has a size-two zero Jordan block and can produce linear drift. Degree zero has $\lambda_-=\lambda_+=0$, supplying amplitude and phase. Thus $\dim\ker L_+=1+10+d(10-d)$, the real dynamical kernel has one extra dimension, and the algebraic zero multiplicity is twice $\dim\ker L_+$. Fixed norm removes amplitude. These are complete linear spectral statements. A growing linear mode alone does not establish nonlinear orbital instability, an outgoing decay channel or a measured lifetime, even when global nonlinear existence is available.

Why Finitely Many Filters Cannot Stabilize Every Mixed Branch

The obstruction holds for arbitrary signed coefficients and any finite collection of strictly positive isotropic Gaussian lengths, under the same two width roots $F(1)=F(4)=\alpha$. Set $r_j=(1+t_j)^{-1}$, $v_j=(1+4t_j)^{-1}$. The identity $(4-3r_j)v_j=r_j$ gives

$$3G(k,l)-4G(k-1,l)+G(k,l-1)=0,\qquad k,l\ge1. \tag{13}$$

The affine $\alpha(k+4l)$ part cancels too. The width equations give $G(1,0)=G(0,1)=G(1,1)=0$, and hence $G(2,1)=-G(2,0)/3$. Strict positive broad-width curvature already forces a negative mixed mode. Indeed the stable Gaussian width oscillation and this mixed instability have equal squared rates, because $\lambda_-(2,1)=3\lambda_-(2,0)$ and $\lambda_+(2,1)=-\lambda_+(2,0)/3$.

One can also rule out a repair in which all these curvatures are merely nonnegative, including degenerate cases. Combine repeated lengths, let $J$ be the number of remaining filters, and set

$$P(z)=\prod_{j=1}^J(z-r_j),\qquad Q(z)=(z-1)P(z)=\sum_{i=0}^{J+1}c_i z^i,\qquad K=J+1.$$

The difference $G(k+1,0)-G(k,0)=\alpha-\sum_j\beta_j(1-r_j)r_j^k$ is a constant plus finitely many exponentials. Applying $P$ to the forward shift annihilates those exponentials and gives

$$\sum_{i=0}^{K}c_iG(i,0)=\alpha P(1)>0. \tag{14}$$

On the other hand, iterating (13) gives

$$G(K,1)=-\sum_{i=2}^{K}w_iG(i,0),\qquad w_i=\frac{4^{K-i}}{3^{K-i+1}}>0.$$

Take $M=6\prod_j(1+r_j)$. The coefficient bound $|c_i|\le2\prod_j(1+r_j)$ and $w_i\ge1/3$ ensure $a_i=Mw_i-c_i\ge0$. Equations (14) and the recurrence yield the positive-weight certificate

$$M G(K,1)+\sum_{i=2}^{K}a_iG(i,0)=-\alpha P(1)<0. \tag{15}$$

At least one required mode is negative. For the explicitly checked ten lengths $(1/16,1/8,1/4,1/2,1,2,3,4,8,16)$, the constants are $K=11$, $M=3960/17$ and $P(1)=128/1755675$. This exact certificate uses modes only through $k+4l=15$ and needs no equal-energy condition or numerical optimization assumption.

The result excludes full Hermite stabilization of the common mixed-width branches by adding finitely many filters of precisely (1)'s type. The isotropic $d=10$ branch alone has no narrow degree and is not separately excluded by this mixed-mode argument. Other constitutive actions and profiles remain outside its hypotheses.

Finite Energy and the Global Cauchy Theorem

Use the weighted energy space

$$\Sigma=H^1(\mathbb R^D;\mathbb C) \cap\{u:|x|u\in L^2\}.$$

For every $u_0\in\Sigma$, every finite signed filter list, $\alpha>0$ and real $\mu_0$, equation (3) has a unique global solution

$$u\in C(\mathbb R;\Sigma),\qquad u_t\in L^\infty_{\mathrm{loc}}(\mathbb R;\Sigma^*), \tag{16}$$

with the equation understood in $\Sigma^*$. It conserves $N$ and $H$, the gradient is bounded for all time, and

$$\|u(t)-v(t)\|_2^2\le e^{C_\beta|t|}\|u_0-v_0\|_2^2,\qquad C_\beta=4\sum_j|\beta_j|, \tag{17}$$ $$\|xu(t)\|_2\le\|xu_0\|_2 +2\alpha|t|\sup_\tau\|\nabla u(\tau)\|_2. \tag{18}$$

The data map is continuous from $\Sigma$ to $C([-T,T];\Sigma)$; no Lipschitz estimate in that stronger norm is asserted. Equation (18) permits linear radius growth. Global finite-energy evolution is therefore a different statement from the Gaussian family's uniform localization. The following proof constructs the flow rather than assuming a local PDE existence theorem.

Entropy bounds and the source domain

For $p=|u|^2/N$, let $p_v$ denote convolution with covariance $vI$ and $I(p)=\int|\nabla p|^2/p$. Direct heat differentiation gives

$$\frac{\mathcal D_s(|u|^2)}N=\frac12\int_0^{s^2}I(p_v)\,dv, \qquad I(p)\le\frac{4T}{N},$$ $$I'(v)=-\int p_v|\operatorname{Hess}\log p_v|^2 \le-\frac{I(p_v)^2}{D}.$$

The last inequality uses the matrix trace inequality, Cauchy–Schwarz and $\int p_v\Delta\log p_v=-I(p_v)$. Integrating yields

$$0\le\mathcal D_s(|u|^2)\le \frac{ND}{2}\log\left(1+\frac{4s^2T}{ND}\right). \tag{19}$$

Thus, for $y=T/N$,

$$\frac HN\ge\mu_0+\alpha y -\frac D2\sum_j\beta_j^+ \log\left(1+\frac{4s_j^2y}{D}\right). \tag{20}$$

Linear growth dominates this finite sum of logarithms. At fixed norm the energy is bounded below and coercive in $T$, including for signed coefficients. A sufficiently large $\mu_0$ makes the lower bound positive without changing any width equation. The heat integral also defines $\mathcal D_s$ on unweighted $H^1$, even when separate entropy integrals would be infinite. The theorem here uses $\Sigma$, where both integrals are finite.

Weighted Hölder gives $\Sigma\hookrightarrow L^q$ for $2D/(D+2)<q<2$. Sobolev embedding supplies exponents above two. For example $\delta=1/[2(D+2)]$ leaves sufficient margins to bound $u\log|u|^2$ in $L^2$, using $|z\log|z|^2|\le C_\delta(|z|^{1-\delta}+|z|^{1+\delta})$, with a spare subcritical exponent. The same bounds hold uniformly for $u\log(|u|^2+\varepsilon g)$, $0<\varepsilon\le1$, with any fixed strictly positive bounded Gaussian $g$.

For a nonnegative density $f$ of mass $M>0$ and second moment $V$, choose $L^2\ge2V/M$. At least $M/2$ lies in $B_L$, so

$$\frac M2(2\pi s^2)^{-D/2} e^{-(|x|+L)^2/(2s^2)}\le K_sf(x) \le M(2\pi s^2)^{-D/2}. \tag{21}$$

Both $|\log K_sf|$ and $|K_s\log K_sf|$ are bounded by $C(1+|x|^2)$. Multiplication by this real potential maps $\Sigma$ into $\Sigma^*$, since $\int(1+|x|^2)|uv|\le C\|u\|_\Sigma\|v\|_\Sigma$. The positive lower mass needed for (21) is available for nonzero solutions on each bounded time interval; zero data are treated directly.

Entropy continuity under the compact limit

If $u_n\to u$ in $L^2$ and the sequence is bounded in $\Sigma$, then both $\operatorname{Ent}(|u_n|^2)$ and $\operatorname{Ent}(K_s|u_n|^2)$ converge. Positive entropy is controlled by a small superquadratic power, with convergence supplied by interpolation below a strictly higher bounded Sobolev exponent. Negative entropy requires a tail estimate: for any $a>0$ and $0\le f\le1$,

$$f\log(1/f)\le a|x|^2f+e^{-a|x|^2-1}. \tag{22}$$

This follows by maximizing $y\log(1/y)-a|x|^2y$. Outside $B_R$ its integral is at most $aV+\int_{|x|>R}e^{-a|x|^2-1}$. Choosing $a$ first and $R$ second proves uniform smallness. A second-moment bound alone does not assert uniform integrability of the second moments; (22) supplies precisely the entropy tail needed. Convolved densities converge in $L^1$, are uniformly bounded above and have second moment $V+Ds^2M$, so the same proof applies. It also applies to $|u_n|^2+\varepsilon_ng$ when $\varepsilon_n$ converges, and is uniform in time under uniform bounds and uniform $L^2$ convergence. Every signed entropy term can therefore pass to the limit separately.

Logarithmic difference estimate

The essential elementary inequality is, for $a,b>0$ and $|z|^2\le ab$,

$$|\operatorname{Im}z|\,|\log(a/b)|\le a+b-2\operatorname{Re}z. \tag{23}$$

Write $z=\sqrt{ab}\,r e^{i\theta}$, $0\le r\le1$. For fixed angle the ratio of the imaginary numerator to the denominator increases with $r$, so the boundary circle suffices. Maximizing there gives $\sqrt{ab}/|a-b|$. Multiplication by $|\log(a/b)|$ is at most one, because $|t|\le2\sinh(|t|/2)$ with $t=\log(a/b)$. The equal-value and zero cases follow by limits.

Locally use $a=|u|^2,b=|v|^2,z=u\bar v$. For the nonlocal term use $a=K_s|u|^2,b=K_s|v|^2,z=K_s(u\bar v)$; convolution Cauchy–Schwarz supplies the disk condition. Move the outer convolution through the pairing using self-adjointness, and integrate (23), using mass preservation. Each local or nonlocal term is bounded by $\|u-v\|_2^2$. The factor two in differentiating that norm gives $|d\|u-v\|_2^2/dt|\le C_\beta\|u-v\|_2^2$. Quadratic potential bounds ensure absolute integrability of these pairings in the $\Sigma,L^2,\Sigma^*$ Hilbert triple. Gronwall yields (17) and uniqueness in the constructed weak class.

Hamiltonian approximation and existence

Regularize inside both entropy arguments:

$$q_\varepsilon=|u|^2+\varepsilon g,\qquad H_\varepsilon=\alpha T+\mu_0N -\sum_j\beta_j\mathcal D_{s_j}(q_\varepsilon).$$

Its exact gradient is (3) with $\rho$ replaced inside the logarithms by $q_\varepsilon$. The fixed $g$ does not add to the conserved wave norm. Fisher convexity gives $I(q_\varepsilon)\le4T+\varepsilon I(g)$, where here $I(f)=\int|\nabla f|^2/f$ for unnormalized densities. With $M=N+\varepsilon\int g$,

$$H_\varepsilon\ge\alpha T+\mu_0N -\frac{MD}{2}\sum_j\beta_j^+ \log\left(1+\frac{s_j^2[4T+\varepsilon I(g)]}{DM}\right). \tag{24}$$

The bound is uniform for bounded mass and $0<\varepsilon\le1$, including as $M$ tends to zero: use $\log(1+y)\le\eta y+C_\eta$ with $\eta$ small enough.

First fix $\varepsilon>0$ and a ball $\Omega=B_R$. Extend $H^1_0(\Omega)$ fields by zero, retaining full-space Gaussian convolution and entropy integrals. Restrict $H_\varepsilon$ to the first $m$ Dirichlet Laplacian modes. The finite-dimensional Hamiltonian ODE is smooth on bounded coefficient sets: $\varepsilon g$ is bounded below on the ball, and Gaussian tail domination justifies differentiating the convolved terms. It conserves wave norm and energy. Norm bounds the coefficient vector, so these ODE solutions are global. Equation (24) bounds their gradients independently of $m$.

On the fixed ball the nonlinear source is uniformly $L^2$, and the spectral projections are bounded in $H^1_0$ and $H^{-1}$. Thus time derivatives are bounded in $H^{-1}$. Rellich compactness and time equicontinuity give a subsequence converging in $C([-T,T];L^2(\Omega))$, weakly in $H^1_0$ at every time. The continuous regularized local source passes by uniform integrability. Strong density $L^1$ convergence makes $K_sq_m$ converge uniformly. The lower Gaussian bound permits logarithms on compact sets; a common quadratic logarithmic bound then supplies Gaussian dominated convergence in the second convolution. This identifies the complete ball PDE.

Mass passes strongly. Entropy continuity and kinetic lower semicontinuity give $H_\varepsilon(u(t))\le H_\varepsilon(u_0)$ at every time. Inequality (23), with the same added background, proves uniqueness. Since the construction works for every datum and both time directions, restart at $u(t)$ and apply its inequality at time $-t$. Uniqueness identifies its value there with $u_0$ and gives the reverse energy inequality. Hence energy is conserved. This argument does not infer equality from lower semicontinuity or assume a Hamiltonian chain rule for weak solutions.

For the limiting ball PDE the variance test is legitimate and gives

$$\frac d{dt}\|xu\|_2^2 =4\alpha\operatorname{Im}\int x\cdot\nabla u\,\bar u, \qquad \left|\frac d{dt}\|xu\|_2\right| \le2\alpha\|\nabla u\|_2. \tag{25}$$

It is applied after the Galerkin limit, avoiding an unproved projection commutator. Approximate general $\Sigma$ data by compactly supported data in expanding balls. Equations (24)–(25) give uniform $\Sigma$ bounds on bounded time intervals. A zero extension has boundary distributions, so no global derivative bound is inferred from it. Instead multiply by a fixed interior cutoff on a smaller ball; the local PDE gives an $H^{-1}$ derivative bound there. Local compactness followed by the uniform tail bound $\int_{|x|>L}|u|^2\le L^{-2}\|xu\|_2^2$ yields full-space $C(L^2)$ convergence. Source identification, entropy limits, uniqueness and restart now produce a whole-space regularized flow.

Removing the background and recovering strong energy continuity

For backgrounds $\varepsilon,\eta$, apply (23) to the Gram vectors $(u_\varepsilon,\sqrt{\varepsilon g})$ and $(u_\eta,\sqrt{\eta g})$. Their added cross term is real, while their squared difference adds $(\sqrt\varepsilon-\sqrt\eta)^2g$. The same construction survives convolution. For equal initial data this proves

$$\|u_\varepsilon(t)-u_\eta(t)\|_2^2 \le\left(\int g\right)(e^{C_\beta|t|}-1) (\sqrt\varepsilon-\sqrt\eta)^2. \tag{26}$$

The regularized flows are Cauchy in $C([-T,T];L^2)$; no subsequence ambiguity remains. For nonzero data, mass and the uniform variance bound retain positive mass in a fixed ball. Bound (21) is therefore uniform in the background parameter. It gives convergence of $K_s\log(K_sq_\varepsilon)$ uniformly on compact spacetime sets: split its integral into a compact part, where logarithms converge uniformly, and a common Gaussian-weighted quadratic tail. Local logarithmic products converge in measure, including at zero, and uniform integrability identifies their distributional limit. Uniform $\Sigma^*$ bounds extend compact-test identification by density. This constructs the asserted global weak equation for the original action.

Mass conservation passes directly; entropy continuity gives an energy inequality. Uniqueness from (23) and the same restart argument upgrade it to exact conservation. Weak $H^1$ continuity follows from strong $L^2$ continuity and local energy bounds. The conserved energy and continuous entropy make the gradient norm continuous, hence give strong $H^1$ continuity. For the weighted part use the bounded tests $w_R(x)=|x|^2/[1+|x|^2/R^2]$, whose gradients satisfy $|\nabla w_R|\le2|x|$. Dominated convergence gives the integrated version of (25). Weighted norm continuity and weak continuity of $xu$ then give strong weighted continuity, proving (16) and (18).

Finally, for data converging in $\Sigma$, equation (17) gives uniform $L^2$ convergence on bounded time intervals. Uniform energy and moment bounds make the entropy limit uniform. Conserved energies give uniform convergence of gradient norms. Gradients converge weakly uniformly against each fixed test; compactness of the limiting continuous gradient path upgrades this to convergence against that moving path. Weak convergence plus norm convergence gives strong $C(H^1)$ convergence. Use (25) next: pair weakly converging $xu_n$ against the strongly converging gradients to obtain uniform weighted norm convergence, and apply the same compact-path argument. This proves strong $C(\Sigma)$ data dependence. Near zero no uniform positive-mass logarithm bound is needed for this last step: (24) supplies a preliminary uniform gradient bound, then energy and entropy convergence force that gradient norm to zero. The zero solution is included.

Complete Translation Response and Galilean Inertia

Equation (3) is translation invariant and its nonlinear multiplier depends only on density. Every regular solution has the exact Galilean transform

$$\psi_v(x,t)= e^{i[v\cdot x/(2\alpha)-|v|^2t/(4\alpha)]} \psi_0(x-vt,t). \tag{27}$$

Gaussian convolution commutes with translations, and the boost phase does not alter density, so this holds beyond the Gaussian family. Define the complete canonical momentum $P=\operatorname{Im}\int\psi^*\nabla\psi$. For a resting real profile,

$$P_v=\frac{Nv}{2\alpha},\qquad H_v=H_0+\frac{N|v|^2}{4\alpha},\qquad M_{\mathrm{inertia}}=\frac{N}{2\alpha}>0. \tag{28}$$

For general initial momentum $P_0$, the energy shift is $v\cdot P_0+N|v|^2/(4\alpha)$. The current is $j=2\alpha\operatorname{Im}(\psi^*\nabla\psi)$, and the center obeys $\dot X=2\alpha P/N$. It moves in every ambient direction, independently of broad-width multiplicity. Translation symmetry conserves the complete field momentum. During overlap, a decomposition into separately conserved particle momenta is not assumed.

Amplitude homogeneity gives $H_0=N\omega$, but inertia in (28) is independent of that clock. The comparison $M_{\mathrm{inertia}}=H_0/c^2$ would require $\omega=c^2/(2\alpha)$. One $\mu_0$ can impose it for the common support clock. Fixed $\alpha$ cannot automatically impose it for several unequal particle clocks, and changing amplitude cannot remove the discrepancy because $N$ cancels. The equation is Galilean, with spatially nonlocal Gaussian smoothing. It supplies neither Lorentz covariance nor a finite-speed causal propagation theorem.

Separation and Reciprocal Interaction Scope

For nonnegative densities $a,b$, put $J(a,b)=\operatorname{Ent}(a+b)-\operatorname{Ent}(a) -\operatorname{Ent}(b)$. Algebra gives

$$\mathcal D_s(a+b)-\mathcal D_s(a)-\mathcal D_s(b) =J(a,b)-J(K_sa,K_sb), \qquad 0\le J(a,b)\le4\int\sqrt{ab}. \tag{29}$$

The bound uses $\log(1+t)\le2\sqrt t$. Fixed Gaussian density packets have exponentially vanishing overlap under increasing separation, both before and after smoothing. Their entropy-difference cross energy therefore vanishes; there is no surviving global norm-logarithm cross term.

Equation (29) applies to additive densities, including orthogonal components. A single coherent scalar sum has $\rho=|\psi_1+\psi_2|^2$, with its relative-phase interference term. That term must be retained in (1) and (3). Gaussian overlap makes it small at large separation, but (29) alone is not a complete quantitative coherent source estimate. The present result does not derive force range, capture, phase acceptance, depth-five suppression or a reciprocal two-particle trajectory. Gaussian filters have no compact interaction cutoff. Any such interaction must follow from the complete same-field equation, with each pattern's own isolated frequency as the initial baseline.

What the Construction Establishes

One common action has exact moving localized classical profiles, complete energy and positive momentum response, two equal Gaussian shape wells, Gaussian-family persistence and a global finite-energy nonlinear flow for arbitrary $\Sigma$ data. Its full spatial linear spectrum is classified, including all neutral Jordan blocks and growing channels, with a rigorous tail. The finite-filter obstruction identifies a precise limitation of this particular two-width support mechanism.

The common support clock is not the maintained unequal mass-frequency family. Hermite multiplicities count polynomial distributions among spatial directions; they do not convert this support spectrum into the simplex particle clock law. Observed spin, charge, photon transversality, physical depth selection, relativistic inertia and coherent encounter dynamics remain open. The growing modes materially limit an electron persistence interpretation. Their nonlinear fate, outgoing energy channels and timescales would need analysis before connecting them to short-lived particles. Neither complete stability nor a measured lifetime follows from the present spectral calculation.

The entropy/Fisher mathematics is related to Dembo, Cover and Thomas, Information theoretic inequalities (1991), especially its Fisher-information and de Bruijn identities. Here the normalized density is an integration tool for classical field amplitude. The explicit heat-flow dissipation argument above supplies the sharper logarithmic bound used in the proof. Carles and Gallagher, Universal dynamics for the defocusing logarithmic Schrödinger equation, sections 2–3, provides a mathematical comparison for logarithmic regularization, weighted compactness and Gaussian evolution. Its local equation does not imply the signed nonlocal Cauchy theorem proved here.

The supporting finite checks include 75 symbolic support identities, an independent 15-check support review, 5,940 exact rational-interval spectral classifications, an exact rational finite-filter certificate and a 19-check independent Cauchy review. Rigorous logarithmic enclosures fix the well parameter and all spectral signs. These checks support the displayed analytical arguments; finite identities alone cannot verify an infinite-dimensional PDE theorem or identify a physical particle.

Portable checks: support and Cauchy identities (103 checks) and complete spectral classification (5,998 checks, including all 5,940 finite mode pairs). These executed companions retain exact parameter enclosures and historical provenance separately from their own source hashes.