A Smoothed Count Action with Global Finite-Energy Histories

A positive spatial kinetic term and a Gaussian-smoothed density energy can coexist with exact localized patterns at the supplied simplex frequencies. The construction below proves a continuous source on every finite-norm field, a real continuously differentiable density energy, and at least one global finite-energy weak and mild history for every finite-energy initial field. It also gives the complete nonnegative Gaussian shape Hessian, with an all-degree proof. These properties belong to one explicitly stated action.

This is an inverse construction: the count function, positive sector units, width sequence, smoothing length and restoring penalties are constitutive inputs. The local count action and the frequency-following mobility action are distinct constructions. The present completion regularizes density nodes and adds kinetic coercivity, but its translational inertia is $N/(2\alpha)$, independent of the rest frequency at fixed norm.

All coordinates are ordinary physical coordinates in shared $\mathbb R^{10}$. Depth labels a pattern's broad directions, rather than an additional physical space. The field is classical; squared amplitude and normalized integration measures are not position probabilities. A pattern's internal profile, its center, and an electron's literal six-dimensional orbit and precession remain distinct. This scalar candidate does not derive that trajectory, spin or charge. It introduces no compact spatial dimension, mediator particle or particle-exchange mechanism.

One explicit finite-band constitutive law

Let $b,\alpha,\Lambda,\eta>0$, and let $C$ be a real symmetric $10\times10$ matrix. Define the descriptors

$$L=C-bI,\quad t=\operatorname{tr}L,\quad z=\operatorname{tr}L^2, \quad a=b+\frac zt,\quad K=\frac ba-2,\quad D=\frac{t^2}{z}. \tag{1}$$

Initially use the domain $t<0,z>0,a>0$. Then $a<b$, $K>-1$, and $0<D\le10$. Put

$$g(s)=s-\frac{\sin(2\pi s)}{2\pi},\qquad \mathcal F(u,v)=\frac{\Gamma(u+v+1)}{\Gamma(u+1)\Gamma(v+1)}, \qquad R=L^2-(a-b)L,$$ $$V(C)=m(g(D))\mathcal F(g(K),g(D)) +\Lambda[\sin^2(\pi K)+\sin^2(\pi D)] +\eta\operatorname{tr}R^2. \tag{2}$$

Since $g(K)>-1$ and $g(D)>0$, every Gamma argument is positive, so (2) is smooth on the descriptor domain. Here $m$ is one fixed positive smooth function. For a common-unit comparison take $m=\mu>0$. To preserve any supplied positive depth units $\mu_d$, assign positive values for all ten integer depths and set

$$\ell_d(v)=\prod_{\substack{1\le e\le10\\e\ne d}}\frac{v-e}{d-e}, \qquad m(v)=\mu_{\mathrm{ref}}\exp\left[ \sum_{d=1}^{10}\ell_d(v)\log\frac{\mu_d}{\mu_{\mathrm{ref}}}\right], \quad\mu_{\mathrm{ref}}>0. \tag{3}$$

The logarithms are dimensionless and $m(d)=\mu_d$. Values at depths without supplied units are common constitutive choices. This interpolates already supplied scales; it does not derive them, resolve the neutrino absolute unit, or independently adjust members of a family.

For an orthogonal rank-$d$ projector $P$, write $Q=I-P$. The target matrices are

$$C_*=aP+bQ,\qquad a=\frac b{k+2},\qquad 1\le d\le10,\quad 0\le k\le95. \tag{4}$$

They have $K=k,D=d,R=0$. Since $g'$ and $g''$ vanish at integers,

$$V(C_*)=\omega_{d,k}=\mu_d\binom{k+d}{d},\qquad V_C(C_*)=0. \tag{5}$$

The entire first term of (2) has zero first and second matrix derivative there, including with the variable $m$ in (3). Thus arbitrary supplied positive depth units preserve the critical jets used below.

We now specify a globally smooth finite-band extension. Choose scalar smooth cutoff functions in $[0,1]$ with the following plateaus and supports.

For complete explicitness, put $e(s)=e^{-1/s}$ for $s>0$, zero otherwise, and $s_0(s)=e(s)/(e(s)+e(1-s))$. For an outer interval $(A,B)$ and inner interval $[a_0,b_0]$, the product $s_0((x-A)/(a_0-A))s_0((B-x)/(B-b_0))$ supplies the required cutoff. Let $\chi_0$ be the product of the three table cutoffs, zero off its descriptor domain. Its vanishing neighborhoods of $t=0,z=0,a=0$ make this extension smooth. Introduce

$$r(C)=\frac{\operatorname{tr}(C-bI/2)^4}{b^4},\qquad h(r)=1-s_0\!\left(\frac{r-7/10}{4/5-7/10}\right),\qquad \chi(C)=\chi_0(C)h(r(C)). \tag{6}$$

Every target obeys $r(C_*)\le10/16=5/8<7/10$ and lies strictly inside each plateau. Hence $\chi=1$ on neighborhoods of all 960 full rotation orbits. Its compact support avoids the inverse pole: every eigenvalue $\lambda$ there satisfies

$$\left|\frac\lambda b-\frac12\right|\le(4/5)^{1/4}<1, \qquad \lambda<\frac{3b}{2}<2b. \tag{7}$$

Set $\sigma^2=1/(4b)$, choose a common real $Z_0$, and define

$$\Phi(C)=-\frac\alpha2\operatorname{tr}[C(I-2\sigma^2C)^{-1}], \qquad Z(C)=Z_0+\chi(C)[V(C)+\Phi(C)-Z_0]. \tag{8}$$

Outside the supported descriptor region set $Z=Z_0$. The inverse is used only on a neighborhood of the support, where it is regular. Thus $Z$ is globally smooth, real and rotation invariant, with bounded values and bounded derivatives of every order. It need not be positive. The finite band is a common input. This extension does not preserve the infinite count ladder: at fixed depth $a\to0$ approaches a finite matrix while the count in (5) diverges, incompatible with a continuous bounded extension at that matrix.

Action, complete source and exact Gaussian closure

Let $G_\sigma$ be the normalized ten-dimensional Gaussian of covariance $\sigma^2I$. For $\psi\ne0$ in $L^2$, define

$$\rho=|\psi|^2,\quad N=\|\psi\|_2^2,\quad u=G_\sigma*\rho,\quad C=-\frac12D^2\log u,$$ $$J(\psi)=\int uZ(C)\,dx,\qquad E(\psi)=\alpha\|\nabla\psi\|_2^2+J(\psi),$$ $$S=\int dt\left[\frac i2\int(\bar\psi\psi_t-\psi\bar\psi_t)\,dx-E(\psi)\right]. \tag{9}$$

Here and below $u=G_\sigma*\rho$; it is strictly positive for every nonzero finite-norm field, even when $\psi$ has nodes. Set $J(0)=0$. For a density variation $v$, $\delta C=-D^2((G_\sigma*v)/u)/2$. Two integrations by parts and the symmetry of the kernel give

$$W=Z-\frac1{2u}\partial_i\partial_j(uZ_{C,ij}),\quad H=G_\sigma*W,\quad F(\psi)=\psi H, \qquad i\psi_t=-\alpha\Delta\psi+F(\psi). \tag{10}$$

Matrix derivatives use the Frobenius pairing and repeated physical indices are summed. Equation (10) is the full unrestricted field variation. It is not an equation obtained only by varying Gaussian parameters.

For a target (4), set

$$B=C_*(I-2\sigma^2C_*)^{-1},\qquad \psi_*(x)=A\exp[-(x-X)^TB(x-X)/2],\quad A\ne0. \tag{11}$$

The physical density has curvature $B$; its convolution has curvature $C_*$, since $C_*=B(I+2\sigma^2B)^{-1}$. Put $M=(I-2\sigma^2C_*)^{-2}$. On this branch,

$$Z=\omega-\frac\alpha2\operatorname{tr}B, \qquad Z_C=-\frac\alpha2M,$$ $$W=Z-\frac\alpha2\operatorname{tr}(C_*M) +\alpha(x-X)^TB^2(x-X),$$ $$G_\sigma*W=Z-\frac\alpha2\operatorname{tr}(C_*M) +\alpha[(x-X)^TB^2(x-X)+\sigma^2\operatorname{tr}B^2],$$ $$-\alpha\frac{\Delta\psi_*}{\psi_*} =\alpha\operatorname{tr}B-\alpha(x-X)^TB^2(x-X). \tag{12}$$

The quadratic terms cancel. The remaining constant equals $\omega$ because $C_*M=B+2\sigma^2B^2$. Therefore

$$\psi(x,t)=e^{-i\omega_{d,k}t}\psi_*(x),\qquad \frac{\|\nabla\psi_*\|_2^2}{N}=\frac12\operatorname{tr}B, \qquad \frac{E(\psi_*)}{N}=\omega_{d,k}. \tag{13}$$

The norm is $N=|A|^2\pi^5/\sqrt{\det B}$. This holds for arbitrary amplitudes, centers and orientations. All ten directions are localized. Their physical widths are $B_P=2b/(2k+3)$ and $B_Q=2b$, differing from those of the unsmoothed action. The label $k$ describes curvature; it is not a disturbance Hermite degree. In the assignment convention $n=k+1$, the frequency multiplier is $\binom{k+d}{d}$.

These fourteen controls are retained targets of the chosen potential, not independent empirical predictions. The construction supplies no selection of occupied branches.

A source estimate without field derivatives or moments

For a multi-index $\beta$, define normalized derivatives and Gaussian scores

$$R_\beta(x)=\frac{\partial^\beta u(x)}{u(x)},\qquad \mathcal H_\beta(z)=\frac{\partial^\beta G_\sigma(z)}{G_\sigma(z)}.$$

At each $x$, the measure $G_\sigma(x-y)\rho(y)\,dy/u(x)$ has total mass one. It is only an integration device for a classical field. Jensen and Fubini imply, for every finite $p\ge1$,

$$\int u|R_\beta|^p\,dx \le N\int G_\sigma|\mathcal H_\beta|^p\,dx =:NM_{\beta,p}<\infty. \tag{14}$$

Each score is a Gaussian Hermite polynomial. Every $D^j\log u$ is a polynomial in the $R_\beta$ of total derivative weight $j$. Writing $A_{ij}=Z_{C,ij}$, the full source expansion is

$$W=Z-\frac12\big[ R_{ij}A_{ij} +2R_i Z_{C,ij;C,ab}\partial_jC_{ab} +Z_{C,ij;C,ab}\partial_{ij}C_{ab} +Z_{C,ij;C,ab;C,cd}\partial_iC_{ab}\partial_jC_{cd}\big]. \tag{15}$$

All coefficients are bounded. The remaining factors form finitely many score monomials of total weight at most six. Products require higher moments: for a monomial with $m$ factors, Hölder gives

$$\int u\prod_{\ell=1}^m|R_{\beta_\ell}|^p \le\prod_{\ell=1}^m \left(\int u|R_{\beta_\ell}|^{pm}\right)^{1/m} \le N\prod_{\ell=1}^m M_{\beta_\ell,pm}^{1/m}.$$

Finite sums, followed by kernel Jensen and Fubini, prove

$$\int u|W|^p\le K_pN,\qquad \int\rho|H|^p\le\int u|W|^p\le K_pN, \qquad \|F(\psi)\|_2\le\sqrt{K_2}\|\psi\|_2. \tag{16}$$

The constants depend on the fixed constitutive law, kernel and dimension, not on the profile or amplitude of $\psi$. No derivative or spatial moment of $\psi$ is required. Define $F(0)=0$.

The convolution defining $H$ exists pointwise as well. Choose a ball with $\int_{|y|\le R}\rho\ge m>0$; then $u(x)\ge mG_\sigma(|x|+R)$. Split conditional displacements at $|x-y|=2(|x|+R)+1$. Inside, their $j$-th moment is bounded by $[2(|x|+R)+1]^j$. Outside, the Gaussian quotient is bounded by a constant times $e^{-|x-y|^2/(4\sigma^2)}$, yielding

$$\left\langle|x-y|^j\right\rangle_x \le[2(|x|+R)+1]^j+C_{j,\sigma}\frac Nm. \tag{17}$$

Thus $R_\beta$ has at most degree-$|\beta|$ polynomial growth, $W$ at most degree-six growth, and $G_\sigma*W$ is smooth. The estimate does not assert that $W$ is bounded.

Strong source continuity and a real C1 energy

Suppose $\psi_j\to\psi\ne0$ in $L^2$. Then $\rho_j\to\rho$ in $L^1$, with a common positive mass ball. Bounded Gaussian derivatives give uniform convergence of every derivative of $u_j$. Positive lower bounds on compact sets then give local uniform convergence of $C_j,W_j$. Equation (17) supplies a common polynomial bound on $W_j$, so Gaussian domination gives local uniform convergence of $H_j$ and its spatial derivatives. For any $p>2$, (16) gives

$$\int_A|F(\psi_j)|^2 \le\left(\int_A\rho_j\right)^{1-2/p}(K_pN_j)^{2/p}. \tag{18}$$

Strong $L^1$ convergence makes the densities uniformly tight. On a fixed ball, convergence of $H_j$ and $\psi_j$ gives strong source convergence; (18) removes the tails. At zero use the linear growth bound. Hence $F:L^2\to L^2$ is continuous everywhere, including nodal fields. The same compact-and-tail argument proves continuity of $J$, since $|J(\psi)|\le\|Z\|_\infty N$.

The complete derivative is

$$DJ(\psi)[h]=2\operatorname{Re}\int\bar hF(\psi)\,dx. \tag{19}$$

To justify it beyond formal integrations by parts, use the straight line $\psi_s=\psi+sh$, $v_s=2\operatorname{Re}(\bar\psi_s h)$. Set $v_s/\rho_s=0$ on its zero set; then $\int|v_s|^2/\rho_s\le4\|h\|_2^2$. Derivatives of $\delta u=G_\sigma*v_s$ are conditional score averages with an extra factor $v_s/\rho_s$. Conditional Cauchy–Schwarz gives, for example,

$$\int u|R_\beta|\left\langle|\mathcal H_\gamma v_s/\rho_s|\right\rangle \le\left(\int u|R_\beta|^2 \langle|\mathcal H_\gamma|^2\rangle\right)^{1/2} \left(\int u\langle|v_s/\rho_s|^2\rangle\right)^{1/2} \le C\|\psi_s\|_2\|h\|_2. \tag{20}$$

Higher products use the higher score moments in (14). These bounds control the differentiated integrand and all integration-by-parts terms. Integrating the absolute bounds over a compact $s$-interval permits Fubini and the pointwise fundamental theorem; spatial cutoff errors vanish by integrability. A possible value with $\psi_s\equiv0$ is handled by the quadratic bound on $J$. Consequently

$$J(\psi+h)-J(\psi) =2\operatorname{Re}\int_0^1\langle h,F(\psi+sh)\rangle\,ds. \tag{21}$$

Continuity of $F$ makes the remainder after subtracting (19) $o(\|h\|_2)$, and makes the derivative continuous. Thus $J$ is real Fréchet $C^1$ on $L^2$. This proves neither a locally Lipschitz gradient nor a Fréchet $C^2$ energy or differentiable flow.

Global finite-energy weak and mild existence

For every $\psi_0\in H^1(\mathbb R^{10})$, there exists at least one two-sided global solution of (10) satisfying

$$\psi\in C(\mathbb R;L^2)\cap L^\infty(\mathbb R;H^1), \quad \|\psi(t)\|_2=\|\psi_0\|_2, \quad E(\psi(t))\le E(\psi_0)\quad(t\in\mathbb R). \tag{22}$$

It is weakly continuous in $H^1$, solves the distributional equation, and satisfies the $L^2$ Duhamel identity below. The inequality in (22) compares every time with the initial time; it is not an energy monotonicity law. The energy argument below strengthens this to energy equality and strong $H^1$ time continuity. General uniqueness and a continuous solution map remain unproved.

First temporarily add a confining term $\varepsilon|x|^2$, and use

$$E_\varepsilon=\alpha\|\nabla\psi\|_2^2 +\varepsilon\|x\psi\|_2^2+J(\psi),\qquad Q_\varepsilon=H^1\cap L^2(|x|^2dx).$$

Let $P_n$ project onto the first $n$ eigenfunctions of $-\alpha\Delta+\varepsilon|x|^2+1$. The finite-dimensional equation

$$i\dot\psi_n=(-\alpha\Delta+\varepsilon|x|^2)\psi_n+P_nF(\psi_n) \tag{23}$$

has a local solution by continuity of its vector field. Phase invariance gives exact norm conservation; the $C^1$ chain rule gives exact $E_\varepsilon$ conservation. Bounded norm prevents finite-dimensional escape, so a solution extends in both time directions. Initial projections converging in $Q_\varepsilon$ and the coercive bound

$$E_\varepsilon\ge\alpha\|\nabla\psi\|_2^2 +\varepsilon\|x\psi\|_2^2-\|Z\|_\infty N \tag{24}$$

supply a uniform form bound. Spectral projections contract the oscillator form norm and its dual; (16) and (23) bound time derivatives in $Q_\varepsilon^*$. The dual interpolation inequality $\|f\|_2^2\le\|f\|_{Q_\varepsilon}\|f\|_{Q_\varepsilon^*}$ gives $L^2$ equicontinuity. The embedding $Q_\varepsilon\to L^2$ is compact: its squared spectral tail beyond eigenvalue $L$ is at most $L^{-1}\|f\|_{Q_\varepsilon}^2$. Finite-mode Arzelà–Ascoli and this tail estimate yield a subsequence converging strongly in $C([-T,T];L^2)$ for every finite $T$. Source continuity passes the nonlinearity, uniformly along these compact time paths. Lower semicontinuity of the positive quadratic form and continuity of $J$ give a trapped solution conserving norm with $E_\varepsilon(t)\le E_\varepsilon(0)$ at every time.

The exact trapped limit satisfies the local mass identity, for smooth bounded real $\chi$ with bounded gradient,

$$\frac d{dt}\int\chi|\psi|^2 =2\alpha\operatorname{Im}\int\nabla\chi\cdot\bar\psi\nabla\psi. \tag{25}$$

Multiplication by $\chi$ preserves the form domain, and time regularization justifies the test. Both the real trap and real $H$ cancel; $F\in L^2$ gives the needed integrability. Choose $\chi_R$ zero on $B_R$, one outside $B_{2R}$, with $|\nabla\chi_R|\le C/R$. Norm and kinetic bounds give

$$\int_{|x|>2R}|\psi(t)|^2 \le\int_{|x|>R}|\psi(0)|^2+\frac{C|t|}{R}. \tag{26}$$

For bounded initial norms and energies this constant is independent of $\varepsilon$. This is the tightness estimate that permits removing the auxiliary trap.

For general $\psi_0\in H^1$, choose smooth compact $\phi_j\to\psi_0$ in $H^1$, and $\varepsilon_j\downarrow0$ so $\varepsilon_j\|x\phi_j\|_2^2\to0$. Their initial trapped energies converge to $E(\psi_0)$, and their solutions have common $H^1$ bounds. To prove local compactness precisely, fix a smooth compact $\zeta$ equal to one on the desired ball. Then $v_j=\zeta\psi_j$ is bounded in global $H^1$, and $\partial_tv_j$ is bounded in global $H^{-1}$: localization of $\Delta\psi_j$ gives only the usual first-order commutator terms, $\zeta F_j$ is bounded in $L^2$, and $\varepsilon_j\zeta|x|^2\psi_j$ is bounded and tends to zero. Global dual interpolation for $v_j(t)-v_j(s)$, and compactness for fixed support, give local $C_tL^2$ compactness. This avoids confusing $H^1(B)$ with the dual pairing for $H_0^1(B)$.

Uniform tails from (26), and tightness of the initial $\phi_j$, upgrade local convergence to $C([-T,T];L^2(\mathbb R^{10}))$. A diagonal subsequence treats all finite $T$. The trap vanishes against compact tests, and source continuity passes (10). Norm conservation survives the strong convergence. Dropping the positive trapped moment term and using kinetic lower semicontinuity gives (22), separately at every time. Strong $L^2$ continuity with the uniform $H^1$ bound gives weak $H^1$ continuity.

Finally $F(\psi(t))$ is continuous in $L^2$. Fourier transformation of the linear distributional equation and integration in time give

$$\psi(t)=e^{i\alpha t\Delta}\psi_0 -i\int_0^t e^{i\alpha(t-s)\Delta}F(\psi(s))\,ds. \tag{27}$$

The integral is a Bochner integral in $L^2$. No contraction or uniqueness argument is used. The trap is removed entirely: it is a compactness device, not a physical wall or extra field in the final action. Finite-energy superpositions of unlike-frequency profiles and nodal initial fields are admitted, but their evolution has not been proved unique or quantitatively resolved into scattering histories.

Complete spatial Hessian and an all-degree identity

This section concerns the constrained quadratic form of $E-\omega N$ at (11). Derive its variations first for bounded smooth relative-density disturbances or valid physical amplitude paths, then close the form in $L^2(\rho_*dx)$. Unbounded Hermite polynomials are not asserted to define positive two-sided paths $\rho_*(1+\varepsilon f)$. Their mode formulas below describe the closed form. This does not assert a Fréchet $C^2$ energy or nonlinear flow theorem.

Diagonalize $C_*$, and write

$$r_i=1-2\sigma^2C_i>0,\quad B_i=C_i/r_i,\qquad T_\sigma f=\frac{G_\sigma*(\rho_* f)}u.$$

Completing the convolution square gives conditional mean $r_ix_i$ and normalized-coordinate correlation $\sqrt{r_i}$. Thus a normalized physical relative-density Hermite mode of multi-index $\mathbf n$ filters to the corresponding normalized $u$-weighted mode multiplied by $\prod_i r_i^{n_i/2}$. Put $t_{\mathbf n}=\prod_i r_i^{n_i}$ and $M_{\mathbf n}=\sum_iB_in_i$.

For the local filtered energy $\int u\Phi(-D^2\log u/2)$, direct second variation along $u(1+\varepsilon f)$ gives

$$\delta C=-\tfrac12D^2f,\qquad \delta^2 C=\tfrac12D^2(f^2),$$ $$\delta^2J_\Phi =\int u\left[\Phi_C:(\nabla f\otimes\nabla f) +\frac14\Phi_{CC}[D^2f,D^2f]\right]. \tag{28}$$

The explicit density-factor cross term cancels the $fD^2f$ part; the gradient term remains because $\Phi_C\ne0$. Spectral differentiation of (8), including divided differences off diagonal, gives

$$(\Phi_C)_i=-\frac\alpha{2r_i^2},\qquad \Phi_{CC}[H,H]=\sum_{ij}q_{ij}H_{ij}^2, \quad q_{ij}=-\frac{\alpha\sigma^2(r_i+r_j)}{r_i^2r_j^2}. \tag{29}$$

On the diagonal $q_{ii}=-2\alpha\sigma^2/r_i^3$; both ordered off-diagonal entries are included. Normalized Hermite derivative norms are $2C_in_i$ for $\partial_i$, and $4C_iC_jn_i(n_j-\delta_{ij})$ for $\partial_{ij}$, times the mode norm. The physical amplitude kinetic density Hessian is $(\alpha/2)\int\rho_*|\nabla f|^2$, giving $\alpha M_{\mathbf n}$. Together with (28)–(29), it yields the diagonal residual

$$L_{\mathrm{res}}=\alpha M_{\mathbf n} -\alpha t_{\mathbf n}\sum_i\frac{B_i}{r_i}n_i -\alpha\sigma^2t_{\mathbf n}\sum_{ij}B_iB_j \left(\frac1{r_i}+\frac1{r_j}\right)n_i(n_j-\delta_{ij}). \tag{30}$$

For total degree $n=\sum_i n_i$, make a repeated list of $n_i$ copies of $h_i=2\sigma^2B_i>0$. Let $S$ be its sum, $\Pi=\prod_{\ell=1}^n(1+h_\ell)$, and $e_j$ its elementary symmetric polynomials. Then exactly

$$\boxed{L_{\mathrm{res}}= \frac\alpha{2\sigma^2\Pi} \left[3e_3+S\sum_{j=3}^n e_j\right].} \tag{31}$$

To verify it at every degree, the numerator before simplification is

$$S\Pi-\sum_\ell h_\ell(1+h_\ell)(1+S-h_\ell) =S\Pi-S-S^2-Sp_2+p_3,$$

where $p_j=\sum_\ell h_\ell^j$. Substitute $p_2=S^2-2e_2$, $p_3=S^3-3Se_2+3e_3$, and $\Pi=1+S+e_2+\sum_{j\ge3}e_j$ to obtain (31). Thus the residual vanishes at degrees zero, one and two and is strictly positive at every degree at least three. Empty sums are zero.

The count term has the following complete matrix Hessian. For symmetric $H$, write $u_P=\operatorname{tr}H_{PP}$, $u_Q=\operatorname{tr}H_{QQ}$, and $s=a-b$. Then

$$T[H,H]=A(u_P-u_Q)^2+Z_cu_Q^2 +R_0\left[\left\|H_{PP}-\frac{u_P}{d}I\right\|^2 +\frac{u_Q^2}{d}+\|H_{QQ}\|^2\right],$$ $$A=\frac{2\Lambda\pi^2b^2}{d^2a^4},\qquad Z_c=\frac{8\Lambda\pi^2}{(a-b)^2},\qquad R_0=2\eta(a-b)^2. \tag{32}$$

Indeed $\delta a=(u_P-u_Q)/d$, $\delta D=2u_Q/s$, $\delta K=-b(u_P-u_Q)/(da^2)$, and $\delta R_{PP}=s(H_{PP}-\delta aI)$, $\delta R_{QQ}=-sH_{QQ}$, $\delta R_{PQ}=0$. The sine penalties and $2\eta\|\delta R\|^2$ give (32). It is positive on all nonzero block-diagonal variations and zero on the cross blocks. At $d=10$, $u_Q=0$, so both trace and traceless variations remain controlled.

The complete count density contribution is

$$\frac14\int u\,T[D^2(T_\sigma f),D^2(T_\sigma f)]\,dx\ge0. \tag{33}$$

Filtering acts on mixed-width Hermite blocks by positive diagonal congruence, not one scalar multiplier. It is nonzero at every degree. Combining (31) and (33), the full density form is nonnegative, with kernel

$$f=c+v\cdot x+x_P^TM_{PQ}x_Q, \qquad \dim\ker L=11+d(10-d), \tag{34}$$

or 11 at depth ten. These are amplitude, translation and orientation tangents. Every other form direction has positive quadratic energy. The phase Hessian is $2\alpha\int\rho_*|\nabla\theta|^2\ge0$. There is no exceptional depth-five suppression in these expressions.

At high degree, the residual is $\alpha M_{\mathbf n}$ plus terms bounded by a fixed polynomial in $n$ times $r_{\max}^n$, with $r_{\max}<1$; the filtered count operator has the same exponential suppression. Consequently the closed weighted density operator is a bounded perturbation of the anisotropic Gaussian number operator, has compact resolvent, and grows linearly with degree. Polynomial cutoffs converge in the form norm, justifying the Hermite description of the closed form. This function-space statement does not add physical dimensions. The result supports nonnegative linear shape-frequency squares; symmetry drift and nonlinear orbital persistence require separate analysis.

Translation, inertia and physical limits

The final action is invariant under phase, translation and common physical rotations. Its density-dependent interaction gives the ordinary Galilean boost

$$\psi_v(x,t)=\exp\left[\frac{i v\cdot x}{2\alpha} -\frac{i|v|^2t}{4\alpha}\right]\psi(x-vt,t).$$

For a rest Gaussian,

$$E_v=N\omega+\frac{N|v|^2}{4\alpha},\qquad P_v=\frac{Nv}{2\alpha},\qquad M_{\mathrm{inertial}}=\frac{N}{2\alpha}. \tag{35}$$

The complete translational inertia is positive but independent of $\omega$ at fixed norm. This action therefore does not derive physical inertia proportional to the selected count frequency. The alternative frequency-following action has its own domain and evolution questions; its motion law cannot be transferred to (9).

Gaussian smoothing has infinite spatial range. No finite propagation, finite-distance quantitative source-clustering rate, calibrated reciprocal pair scattering or observed force law follows from the continuity theorem. Occupation selection, the derivation of depth scales, observed spin and charge, general uniqueness, nonlinear branch stability and a special depth-five interaction suppression remain open. The exact stationary branches, source estimates, existence construction and all-degree quadratic identities retain their stated mathematical scope.

Reproducible finite checks

The standalone exact checker uses Python's standard library and rational arithmetic. It checks all 960 cutoff margins and stationary source/energy closures, the fourteen maintained count controls, compensation divided differences, and active/complement disturbance splits through degree twelve. All 30,484 assertions pass. The all-degree result is the algebraic proof (31), rather than extrapolation from this finite range. The checker records the SHA-256 of the code actually executed separately from its input provenance.

An independent exact-matrix calculation gives a positive-definite diagonal matrix of the fourteen third-degree active-block residuals. A rigorous 128-bit enclosure for the cutoff margin is

$$1-(4/5)^{1/4} =0.054258390996824186698303880112785\ \pm\ 2.06\times10^{-35}.$$

These checks verify finite algebra and a cutoff margin. They are not a numerical evolution, a uniqueness test or an empirical identification of the Gaussian patterns with observed particles.

Exact Energy Conservation and Conditional Uniqueness

The following argument uses the same action, without a current correction. It upgrades (22) to equality and strong H1 continuity. Its separate Gaussian-envelope uniqueness class is a sufficient assumption, with no claim of propagation for general data.

Action and hypotheses

Let G be the centered normalized Gaussian of covariance sigma² I, with sigma>0. Put rho=|psi|², u=G*rho, C=-D²log(u)/2, J=integral u Z(C), and E=alpha||grad psi||²+J, alpha>0. The real smooth matrix function Z and all derivatives needed below are bounded. Set

W=Z-(1/(2u)) partial_i partial_j(u Z_Cij),
H=G*W, F=psi H.

Use the source theorem proved above:

integral u |W|^p <= K_p N, N=||psi||², for every finite p.

Suppose psi belongs to C(I;L2) and L_infinity,loc(I;H1), is weakly continuous in H1, and solves i psi_t=-alpha Delta psi+psi H in distributions. Its L2 source is continuous. The norm N is constant and its local mass equation is

rho_t=-div j, j=2alpha Im(conjugate(psi) grad psi).

The zero-norm solution is trivial; assume N>0. All statements below apply on any finite time interval with the stated H1 bound.

1. Shifted Gaussian score estimate

For a multi-index beta and a bounded displacement h define

L_(beta,h)(z)=D^beta G(z+h)/G(z).

This is a polynomial times an exponential of a linear function of z. Its every finite Gaussian moment is finite, uniformly for h in a compact set. Conditional Cauchy–Schwarz, followed by Cauchy–Schwarz in u dx and conditional Jensen, gives

integral rho(y) |D^beta H(y+h)|² dy
  <= integral u(x) |W(x)|² E_x |L_(beta,h)(x-y)|² dx
  <= (integral u |W|⁴)^(1/2)
     (N integral G |L_(beta,h)|⁴)^(1/2)
  <= C_(beta,h0) N, |h|<=h0.                         (1)

Here E_x is integration in the normalized classical convolution measure G(x-y)rho(y)dy/u(x). No derivative or moment of rho enters (1). The same proof with larger exponents gives analogous bounds in every finite Lp(rho dy).

Consequently

||psi partial_j H||_2 <= C sqrt(N),
||psi [H(.+h e_j)-H]/h||_2 <= C sqrt(N), |h|<=1.     (2)

Moreover the second expression converges strongly in L2 to psi partial_j H as h->0. On a fixed compact set this follows from smoothness of H. On its complement, the uniform Lp(rho) bounds with p>2 and tightness of rho give uniform integrability. This argument works at each time, and (2) supplies domination for time integration.

2. Exact kinetic balance without differentiating F in H1

Fix nonzero h and put q_h=[psi(.+h e_j)-psi]/h. It is an H1 weak solution of

i (q_h)_t=-alpha Delta q_h+H(.+h e_j)q_h
                    +psi [H(.+h e_j)-H]/h.

Every displayed potential product belongs to L2: H(.+h)psi(.+h) is the translated source, and H(.+h)psi is covered by (1). The standard H1/H(-1) norm identity therefore applies, and the real potential cancels:

||q_h(t)||²-||q_h(s)||²
  =2 integral_s^t Im integral conjugate(q_h) psi delta_h H dx dr.   (3)

For every r, q_h(r)->partial_j psi(r) strongly in L2, and ||q_h(r)||_2<=||partial_j psi(r)||_2. Equation (2) and dominated convergence pass (3) to the limit. Summing j gives the absolutely continuous identity

alpha||grad psi(t)||²-alpha||grad psi(s)||²
  =- integral_s^t integral grad H . j dx dr.           (4)

This avoids the invalid inference that F maps H1 to H1. In particular, the unbounded real multiplication term H grad psi never needs its own L2 estimate.

3. Exact potential-energy balance

Define v=j/rho where rho>0 and zero otherwise. Since j vanishes at rho=0,

integral rho |v|² <=4alpha²||grad psi||².              (5)

Gaussian convolution of the continuity equation gives u_t=-sum_j (partial_j G)*j_j. Define R_beta=(D^beta u)/u and

S_beta=(D^beta u_t)/u
       =-sum_j E_x[(D^(beta+e_j)G/G)(x-y) v_j(y)].

For every fixed polynomial P of the normalized scores R_beta and each Gaussian score H_gamma=D^gamma G/G,

integral u |P(R) E_x(H_gamma v_j)|
  <= (integral u |P(R)|² E_x|H_gamma|²)^(1/2)
     (integral u E_x|v_j|²)^(1/2)
  <= C sqrt(N) ||v||_L2(rho).                        (6)

The first factor is finite by the source theorem's arbitrary score-moment estimates and conditional Jensen; the second is exactly the physical rho-weighted norm. Thus no weighted spatial moment is hidden in (6).

Write a=u_t/u=S_0. Then

partial_i partial_j a
  =S_ij-S_i R_j-S_j R_i-a R_ij+2a R_i R_j,

and pointwise differentiation gives

partial_t [u Z(C)] =u[a Z-(1/2) Z_C:D²a].             (7)

Every term in (7) is bounded in L1(dx), uniformly on finite time intervals, by C sqrt(N)||grad psi||. Local time differentiation is justified by the smoothed continuity equation. Fubini and the integral fundamental theorem, using (6), therefore make J absolutely continuous with derivative the integral in (7).

Spatial integration by parts twice gives

J_t=integral W u_t.

This is a genuine integration by parts: compact cutoffs first give the identity; all cutoff-error terms are integrable by (6), including derivatives of Z_C, and their tails vanish. The remaining convolution calculation is also absolutely integrable by (6):

integral W(x) u_t(x) dx
  =-sum_j integral W(x) partial_j G(x-y) j_j(y) dx dy
  =integral grad H(y) . j(y) dy.                    (8)

The last sign follows from differentiating G(x-y) with respect to y. Equations (4) and (8) cancel exactly:

E(psi(t))=E(psi(s)) for all s,t in I.                (9)

Equation (4) makes the H1 seminorm continuous; weak H1 continuity plus continuity of its norm proves psi in C(I;H1). For the global constructed solutions, the coercive energy estimate now uses equality rather than only lower semicontinuity.

4. Concrete Gaussian-envelope uniqueness class

A useful sufficient condition can be stated without any lower bound on the physical density itself. Let q_t(x) be a positive Gaussian density on R10. Its covariance eigenvalues stay between positive finite bounds on each finite time interval. Suppose two solutions psi and phi obey, with uniform finite M and positive m,

|psi|², |phi|² <= M q_t,
G*|psi|², G*|phi|² >= m (G*q_t)                     (10)

pointwise on that interval. The same q_t is used for both solutions. Its center can move. Then the two solutions with the same initial data agree throughout the interval.

Here is the bounded-response estimate underlying that assertion. Interpolate densities rho_s=(1-s)|psi|²+s|phi|², so (10) persists. This density interpolation avoids zeros from interference along a field interpolation. Write U=G*q_t. All normalized scores of rho_s obey polynomial bounds in the centered spatial coordinate, because

|D^beta(G*rho_s)| <= M (|D^beta G|*q_t), G*rho_s>=m U.

Linearizing the explicit score-polynomial formula for W therefore gives

|delta W(z)| <= P(z) sum_(|beta|<=4)
                        |D^beta G*delta rho(z)|/U(z),           (11)

where P is a fixed polynomial bound; its constants are uniform in s and time. Also

|delta rho(y)| <= (|psi(y)|+|phi(y)|)|psi(y)-phi(y)|
                <=2 sqrt(M q_t(y)) |psi(y)-phi(y)|.

After multiplying G*delta W by phi, (11) is dominated by integral operators whose kernels are a polynomial times

K(x,y)=sqrt(q_t(x)q_t(y))
          integral G(z-x) G(z-y)/U(z) dz.            (12)

Derivatives of G add only polynomials. Each such kernel is Hilbert–Schmidt. To see this directly, diagonalize the Gaussian covariance. In one eigen-direction with variance tau², write s=sigma² and a=tau². The exponent in (12), after integrating z, is

-A(x²+y²)+Bxy,
A=(2a²+2as+s²)/[4as(2a+s)],
B=(a+s)/[s(2a+s)].

Its quadratic form is strictly positive because

2A-B=s/[2a(2a+s)]>0,
2A+B=(2a+s)/(2as)>0.

For clarity, the z integral contributes (a+s)(x+y)^2/[2s(2a+s)] to the exponent, which gives these coefficients directly.

Thus polynomial multiples remain square integrable. The response bound is

||phi (H_phi-H_psi)||_2 <= C ||phi-psi||_2.           (13)

For d=psi-phi, write F(psi)-F(phi)=H_psi d+phi(H_psi-H_phi). The real H_psi term cancels in the L2 difference identity. Equation (13) and Gronwall give uniqueness. The individual H_psi phi products are integrable under (10) and the polynomial score bounds, so this cancellation is justified.

Condition (10) is a sufficient comparison-class assumption, not a proved invariant neighborhood for arbitrary H1 data. It includes each exact target Gaussian history and permits physical nodes whenever the convolved density still satisfies its lower envelope. General H1 uniqueness, or propagation of (10), remains open. An unbounded L2 Frechet derivative of F at a Gaussian is compatible with this result: the unbounded real multiplication part cancels in the difference norm.

The bounded-current completion retains the count branches while adding frequency-dependent inertia and proving its own energy balance. Separated-pattern and orientation results establish energy and source clustering and delimit the energetic meaning of the positive Gaussian Hessian.