Local Log-Curvature Patterns, Exact Motion and Admissible Domains

One local action on ordinary flat $\mathbb R^{10}$ supports exact localized Gaussian patterns and motion of their centers in all ten directions. Its positive shape penalty has a simple fourth-order source and an explicit nonnegative linear spectrum about every two-root Gaussian branch. The construction also exposes an essential domain issue: smooth positive zero-penalty profiles are Gaussians, but positive weak profiles can change width across an interface at zero penalty. Neither the Gaussian classification nor the linear spectrum establishes nonlinear persistence.

This is an inverse support-and-motion construction. Two widths, a clock, a speed and a penalty coefficient enter as constitutive inputs. The global-moment count action has exact excited simplex clocks and its own bounded-localization theorem; those statements do not transfer to this action. The Gaussian-window action has global $L^2$ evolution and asymptotic clustering under its smoothing and cap hypotheses; its theorem does not transfer either. The present local action has exact Gaussian motion, while its arbitrary-data nonlinear flow remains unresolved.

The scalar complex field describes two quadratures of an internal classical field structure. Its density is squared field amplitude, not a position-probability law. It does not replace an electron's deterministic six-dimensional trajectory or identify an internal shape with detector registration. Depth denotes a broad-width multiplicity of the pattern in shared ordinary space. It is not a separate physical sector, an extra field-component dimension, compactification or a mediator. The results below use depths $2,3,4,5,6,10$ under the same law and do not select those depths from the other allowed multiplicities.

The local action and its full variation

For $\rho=|\psi|^2>0$, let $q=\log\rho$, $S=D^2q$, and choose $0<b_1<b_0$, $\mu,c,\gamma_2>0$. Write $\Delta=b_0-b_1$ and define ordinary matrix products

$$F(S)=(S+2b_1I)(S+2b_0I),\qquad V(S)=\gamma_2\operatorname{tr}[F(S)^2],\qquad E_2[\rho]=\int\rho V(D^2\log\rho)\,dx. \tag{1}$$

Here curvature means derivatives of log density; the physical ambient metric remains flat. The symmetric matrix $F(S)$ has $\operatorname{tr}F^2=\sum_{ij}F_{ij}^2\ge0$. With $\psi=\sqrt\rho e^{i\theta}$, take

$$h(p)=\sqrt{\mu^2+c^2|p|^2},\qquad H=\int\rho h(\nabla\theta)\,dx+E_2[\rho],$$ $$\mathcal A=\int dt\left[-\int\rho\theta_t\,dx-H\right]. \tag{2}$$

For a symmetric Hessian variation $T$, cyclicity of trace gives

$$DV(S)[T]=\operatorname{tr}(A T),\qquad A=2\gamma_2F(S)[2S+2(b_1+b_0)I].$$

This formula includes noncommuting variations $T$. Since $\delta q=\delta\rho/\rho$, two integrations by parts yield

$$\delta E_2=\int\left[V\delta\rho+\rho A_{ij} \partial_i\partial_j(\delta\rho/\rho)\right]dx, \qquad W_2=\frac{\delta E_2}{\delta\rho} =V+\frac1\rho\partial_i\partial_j(\rho A_{ij}). \tag{3}$$

Indices are summed as ordered pairs over all ten spatial coordinates. The positive double-divergence sign, its inner density and the outer division by density are required. The source is fourth spatial order. The full equations are

$$\rho_t+\nabla\cdot(\rho v)=0,\qquad \theta_t=-h(\nabla\theta)-W_2,\qquad v=\frac{c^2\nabla\theta}{h(\nabla\theta)}. \tag{4}$$

Equivalently, on the positive-density domain,

$$i\psi_t=\left[h+W_2-\frac{i}{2\rho}\nabla\cdot(\rho v)\right]\psi. \tag{5}$$

The imaginary divergence translates the amplitude and cannot be dropped. These formulas require the stated integrations by parts and assert no extension through arbitrary nodes.

Multiplying $\rho$ by a positive constant leaves $S$ unchanged. Thus $H$ is degree one in $\rho$, its density source is degree zero, and $\int\rho W_2=E_2$ when boundary terms vanish. Translation, rotation and uniform phase shift are symmetries. The normalization $N=\int\rho$ is arbitrary and uncalibrated. With dimensionless phase, $b_i$ have units $\mathrm{length}^{-2}$, $\mu$ frequency, $c$ speed, and $\gamma_2$ frequency times $\mathrm{length}^8$.

Exact Gaussian support and complete motion

For any rank-$d$ orthogonal projector $P$, set $B=b_1P+b_0(I-P)$ and

$$\rho_0(x)=\frac{N\sqrt{\det B}}{\pi^5} e^{-(x-X_0)^TB(x-X_0)}. \tag{6}$$

Here $S=-2B$, so $F=V=A=0$, and the entire source $W_2$ vanishes. For any $p\in\mathbb R^{10}$,

$$\Omega=\sqrt{\mu^2+c^2|p|^2},\qquad v=\frac{c^2p}{\Omega},\qquad \psi(x,t)=\sqrt{\rho_0(x-vt)}e^{ip\cdot x-i\Omega t} \tag{7}$$

solves the full local equations. On every compact spacetime support the base amplitude has a positive minimum, so all sufficiently small smooth complex-field perturbations are admissible and polar variations exhaust them. This proves unrestricted compact-field variational stationarity, beyond a Gaussian ansatz.

The complete conserved quantities of (7) are

$$E=N\Omega,\qquad \mathcal P=Np,\qquad E^2-c^2|\mathcal P|^2=N^2\mu^2,\qquad M_{\rm rest}=\frac{N\mu}{c^2}. \tag{8}$$

Writing $\Gamma_v=(1-|v|^2/c^2)^{-1/2}$, the momentum response is the positive tensor

$$\frac{\partial\mathcal P_i}{\partial v_j} =\frac{N\mu}{c^2}\left(\Gamma_v\delta_{ij} +\Gamma_v^3v_iv_j/c^2\right).$$

All center directions remain available at every depth. Both Gaussian widths are finite; this is a localized alternative to exact uniform extension through unused infinite directions. The current law in (2) is an inverse choice, so (8) is the complete inertia of that choice, not a derivation of the choice from a different internal-motion law. The profiles do not Lorentz contract. The dispersion relation and pointwise bound $|v|<c$ establish neither Lorentz covariance nor finite propagation for the higher-derivative coupled equation.

Since $h\ge\mu$ and $V\ge0$, $H\ge\mu N$. Equality at fixed positive norm requires constant phase and zero shape penalty. Every rest Gaussian attains this global lower bound. All have clock $\mu$; there is no excited count ladder here. The proof assumes $\mu>0$ and does not identify the depth-two pattern with a photon.

Smooth classification and its weak-interface limitation

Suppose $q\in C^4(\mathbb R^{10})$, $\rho=e^q>0$, and $0<\int\rho<\infty$. If $E_2=0$, continuity and positivity imply $F(D^2q)=0$ everywhere. The Hessian eigenvalues are $-2b_1,-2b_0$. Their integer multiplicities are continuous and hence constant on connected space. Therefore the scalar Laplacian $\Delta_xq=\operatorname{tr}D^2q$ is constant. Each bounded Hessian component is harmonic, and the Euclidean bounded-harmonic Liouville theorem makes it constant. Integrating and completing the square yields precisely (6). The theorem used is Axler, Bourdon and Ramey, Harmonic Function Theory, Theorem 2.1; it concerns the ordinary Euclidean Laplacian.

The smoothness qualification is essential. Choose every $b_j\in\{b_1,b_0\}$, and put

$$q(x)=C-\sum_{j>1}b_jx_j^2- \begin{cases} b_1x_1^2,&x_1<0,\\ b_0x_1^2,&x_1>0. \end{cases} \tag{9}$$

This positive finite-norm density has $q\in C^1\cap W^{2,\infty}_{\rm loc}$. Its weak Hessian is a bounded root-valued step, with no delta term because its first derivative is continuous. Thus $F=V=A=0$ almost everywhere and $E_2=0$, although it is not a single smooth Gaussian. Dominated compact-field variation gives zero first shape variation: on a compact support its amplitude stays bounded below and perturbed Hessians stay bounded. Hence $e^{q/2-i\mu t}$, and its constant-gradient translates, are explicit variational weak histories with (8). This does not supply a weak flow for arbitrary data.

Smoothly convolving the one-dimensional part of (9) across a slab of width $O(\varepsilon)$ gives Hessians between the two roots, with penalty bounded by $\gamma_2\Delta^4$ inside the slab and zero outside. Normalizing back to the same $N$ does not change the Hessian. The energy is $O(\varepsilon)$, and the amplitudes converge in $L^2$ to the interface profile, which has positive distance from the closed fixed-norm smooth two-width Gaussian family. Thus restricting evaluation to smooth inputs does not restore an energy-only compactness theorem near that family.

There is also a smooth distant-splitting obstruction. For two equal-density Gaussians centered at $\pm Re_1$ in a broad direction, with total norm $N$,

$$\rho_R=\frac{N\sqrt{\det B}}{\pi^5} e^{-x^TBx-b_1R^2}\cosh(2b_1Rx_1).$$

Its broad Hessian entry is $-2b_1+z$, where $z=4b_1^2R^2\operatorname{sech}^2(2b_1Rx_1)$, and $\operatorname{tr}F^2=z^2(z+2\Delta)^2$. Direct integration gives

$$E_2[\rho_R]\le\gamma_2N\sqrt\pi e^{-b_1R^2} \left[40b_1^{15/2}R^7+48\Delta b_1^{11/2}R^5 +16\Delta^2b_1^{7/2}R^3\right]\longrightarrow0. \tag{10}$$

Its centered variance in that direction is $R^2+1/(2b_1)$. Near-minimal energy therefore does not bound variance or force proximity to one Gaussian. Neither obstruction contradicts the smooth exact-zero theorem or the local linear gap.

Admissible tangents and the corrected full linear spectrum

A sufficient strong domain has smooth $q,\theta$, Gaussian decay $q\le C-a|x|^2$, and polynomial growth of the required derivatives. Four derivatives of $q$ suffice for the shape source. This is a sufficient class, not a proved invariant nonlinear-flow domain or an open $L^2$ neighborhood.

At a rest Gaussian $u=\sqrt{\rho_0}$, polynomial Hermite tangents $u(f+ig)$ can be realized by positive paths

$$\psi_\varepsilon=u\exp(\varepsilon f-\varepsilon^2f^2+i\varepsilon g). \tag{11}$$

The exponent obeys $\varepsilon f-\varepsilon^2f^2\le1/4$; derivatives of the energy through second parameter order are dominated by the Gaussian times a fixed polynomial on compact parameter intervals. The Hessian of $H-\mu N$ is independent of this path's curvature because its first variation vanishes. Straight real polynomial paths can instead create nodes and leave the energy domain.

In principal coordinates let $D_i=\partial_i$, $D_i^*=-\partial_i+2b_ix_i$, with weighted inner product $\int\rho_0fg$. A real tangent $a=uf$ gives

$$\delta S=2D^2f,\qquad (\delta F)_{ij}=4(b_1+b_0-b_i-b_j)\partial_{ij}f.$$

Cross-block entries vanish. The coefficient of second order in the rotating-frame energy is

$$Q=16\gamma_2\Delta^2\int\rho_0 \left(|D^2_{\rm broad}f|^2+|D^2_{\rm narrow}f|^2\right) +\frac{c^2}{2\mu}\int\rho_0|\nabla g|^2. \tag{12}$$

All tensor pairs are ordered. In the complete Gaussian Hermite basis let $k,l$ denote total degrees in the broad and narrow blocks. The canonical real equations are $a_t=L_-z$, $z_t=-L_+a$, for $z=ug$, with eigenvalues

$$\boxed{\lambda_+=64\gamma_2\Delta^2 [b_1^2k(k-1)+b_0^2l(l-1)],\qquad \lambda_-=\frac{c^2}{\mu}(b_1k+b_0l).} \tag{13}$$

For example $b_1=1,b_0=4$ gives $\lambda_+=576\gamma_2[k(k-1)+16l(l-1)]$ and $\lambda_-=(c^2/\mu)(k+4l)$. The factors follow from $\sum_{ij}D_j^*D_i^*D_iD_j=4b^2k(k-1)$ on one equal-width block. No extra factor of two belongs in the canonical equations.

The real zero pairs are $(0,0),(1,0),(0,1),(1,1)$, with only $l=0$ available at depth ten. They represent amplitude, ten translations and unequal-width orientation tangents. Their real kernel dimensions are $1+10+d(10-d)$: respectively $27,32,35,36,35,11$ at the maintained depths. Nonconstant zero pairs have $\lambda_->0$ and size-two Jordan blocks, so conjugate phase perturbations can give linear drift. The constant imaginary mode is uniform phase.

Every other pair has positive squared shape-disturbance frequency $\lambda_+\lambda_-$, with sharp common real gap $\lambda_+\ge128\gamma_2\Delta^2b_1^2$, attained at $(2,0)$. These are disturbance frequencies about clock $\mu$, not particle mass assignments. Depth five maximizes an orientation dimension; it has no special dynamical coefficient.

After projecting out the symmetry pairs, define

$$\|(a,z)\|_E^2=\sum_\alpha [\lambda_+(\alpha)|a_\alpha|^2+\lambda_-(\alpha)|z_\alpha|^2]. \tag{14}$$

Multiplying each pair by the square roots of its two weights reduces the flow to a rotation of frequency $\sqrt{\lambda_+\lambda_-}$. The direct sum defines a strongly continuous isometric group in this energy space; finite-prefix approximation controls its square-summable tail. Its generator domain is the subspace with $\sum_\alpha\lambda_+\lambda_-[\lambda_+|a_\alpha|^2+\lambda_-|z_\alpha|^2]<\infty$. The real and imaginary form domains are oscillator powers one and one-half, respectively. This is a complete linear theorem. Imaginary eigenvalues alone do not imply a uniform ordinary $L^2$ bound because $\lambda_+/\lambda_-$ grows with degree.

The shape potential is not globally convex: at $S=-(b_0+b_1)I$, $D^2V(S)[T,T]=-4\gamma_2\Delta^2\operatorname{tr}(T^2)$. The root Gaussian has a good highest-order symbol, but global positivity of that symbol cannot be inferred from nonnegative energy. A nonlinear Cauchy theorem, coercivity modulo symmetries and nonlinear persistence remain open.

Additional stationary Gaussian clocks

The zero-penalty roots are not the complete stationary Gaussian family. For any positive definite constant $B$, the unrestricted source is $$W_2=V(-2B)-2\operatorname{tr}(BA) +4(x-X)^TBAB(x-X).$$ It is constant precisely when $A=0$. Consequently each eigenwidth can be $b_1$, $b_*=(b_0+b_1)/2$, or $b_0$. If $j$ directions have the midpoint width, the complete rest clock is $$\omega=\mu+j\gamma_2(b_0-b_1)^4,\qquad j=0,\ldots,10.$$ These are eleven clocks from 66 multiplicity triples, with arbitrary amplitude and orientation. They solve the full field equation, but are not the simplex mass family. Every midpoint direction has the negative fixed-norm width curvature $-16N\gamma_2(b_0-b_1)^2$. The optional third-log term vanishes along Gaussian width changes and cannot remove that negative direction. An energy saddle is not a measured lifetime.

For these nonzero-shape-energy branches, the original current law gives $\Omega=\sqrt{\mu^2+c^2p^2}+\omega-\mu$ and inertia $N\mu/c^2$. It therefore misses the shape contribution to the rest clock in its translation inertia. The distinct frequency-following action, with $E=\mu N+E_2+E_3$, $w=\delta E/\delta\rho$, and $$H_{\rm FL}=E+\int\rho\left[\sqrt{w^2+c^2|\nabla\theta|^2}-|w|\right],$$ gives all these branches exact motion with $\Omega=\sqrt{\omega^2+c^2p^2}$, $v=c^2p/\Omega$, energy $N\Omega$ and rest inertia $N\omega/c^2$. The full extra density derivative is $(Dw)^*[\rho(w/\Omega-1)]$; it vanishes at the branch because homogeneity implies $(Dw)^*\rho=0$. Locality and positivity make every compact smooth field variation admissible at a Gaussian. This changes the action, retains the negative width direction and supplies no nonlinear flow theorem.

Local momentum balance

For the fourth-order shape density $e=\rho V(D^2q)$, define the full shape stress by $$\frac{K}{\rho}=(W_2-V)I+(D^2q)A -\nabla q\otimes(A\nabla q+\operatorname{div}A).$$ Direct differentiation gives $\partial_jK_{ij}=\rho\partial_iW_2$. Combining this identity with (4), the momentum density $m=\rho\nabla\theta$ obeys $$\partial_t m_i+\partial_j(m_i v_j+K_{ij})=0.$$ This displays the same-field stress in reciprocal momentum exchange. The canonical higher-gradient stress need not be symmetric pointwise; an angular-momentum improvement and the optional third-gradient stress require their respective derivations. The identity is a conservation law under the stated regularity, not an existence theorem.

The optional third-log interface repair

A changed action adds

$$E_3=\gamma_3\int\rho|D^3\log\rho|^2dx,\qquad \gamma_3>0,$$ $$W_3=\gamma_3|D^3q|^2 -\frac{2\gamma_3}{\rho}\partial_i\partial_j\partial_k (\rho\partial_i\partial_j\partial_kq). \tag{15}$$

The source becomes $W_2+W_3$, of sixth spatial order. Its coefficient has units frequency times $\mathrm{length}^6$, distinct from $\gamma_2$. Every Gaussian has $D^3q=0$, so its added energy and first variation vanish; (7)–(8) survive. The Hessian jump in (9) has a distributional third derivative and is not admissible for this positive weighted-square term. Thus (15) removes that particular zero-energy interface mechanism. It is not a proof of nonlinear regularity, a global compactness theorem or persistence for the changed action. The pair results below permit $\gamma_3\ge0$ and retain vanishing excess at large separation even with this term. The fourth-order linear theorem above was established for $\gamma_3=0$; adding a source requires its own evolution/domain analysis.

Coherent pair energies: fixed norm and fixed momentum

Take two identical equally oriented Gaussian amplitudes at $\pm Rv$, where $|v|=1$, and relative phase $\phi\in(-\pi,\pi)$. This is an equal-shape control, not a restriction on unlike particles' clocks or independently evolving orientations. Put

$$w=Bv,\quad\beta=v^TBv,\quad t=w\cdot x,\quad z=2Rt, \quad C=\cos\phi,\quad D_z=\cosh z+C,\quad f=\log D_z,$$ $$\psi=Ae^{-x^TBx/2-R^2\beta/2} [e^{Rw\cdot x}+e^{i\phi}e^{-Rw\cdot x}]. \tag{16}$$

Adjust $A$ to keep the complete $N$ fixed. Define

$$D=1+Ce^{-\beta R^2},\quad g_\beta(t)=\frac{e^{-t^2/\beta}}{\sqrt{\pi\beta}},\quad L(t)=\frac{e^{-\beta R^2}D_z}{D},\quad a=[(b_1+b_0)I-2B]w,$$ $$M_0=w\cdot w,\quad M_1=w\cdot a,\quad M_2=a\cdot a, \quad K=M_0M_2+M_1^2,\quad Z=4R^2f''.$$

The Gaussian marginal $g_\beta$ is an integration identity, and $\int g_\beta L=1$. Exact rank-one differentiation gives

$$S=-2B+Zww^T,\quad f''=\frac{1+C\cosh z}{D_z^2},\quad f'''=\frac{\sinh z(C^2-2-C\cosh z)}{D_z^3},\quad \nabla\theta=-\frac{R\sin\phi}{D_z}w,$$ $$U_2=2Z^2K+4Z^3M_0^2M_1+Z^4M_0^4,\quad U_3=64R^6(f''')^2M_0^3,\quad U_c=\sqrt{\mu^2+c^2R^2\sin^2\phi\,M_0/D_z^2}-\mu.$$

The complete energy is

$$H=\mu N+N\int g_\beta L(U_c+\gamma_2U_2+\gamma_3U_3)dt. \tag{17}$$

The possibly negative expanded cubic term does not negate $U_2=\operatorname{tr}F^2\ge0$. No isolated energy has been discarded. At fixed nonnodal phase, the small-$R$ contributions are

$$\frac{E_c}{N}=\frac{c^2M_0}{2\mu}\tan^2(\phi/2)R^2+O(R^4),\quad \frac{E_2}{N}=\frac{32\gamma_2K}{(1+C)^2}R^4+O(R^6),$$ $$\frac{E_3}{N}=\frac{128\gamma_3\beta M_0^3(C-2)^2}{(1+C)^4}R^8+O(R^{10}). \tag{18}$$

At $\phi=0$, current energy vanishes identically. At coalescence $H=\mu N$; every finite $R>0$ has strictly positive shape excess. These limits are not uniform as $\phi\to\pi$.

For large $R$, let $J_4(C)=\int(\cosh z+C)(f'')^4dz>0$ and $J_{33}(C)=\int(\cosh z+C)(f''')^2dz$. Dominated expansion gives

$$\frac{E_2}{N}=\frac{128\gamma_2M_0^4J_4}{\sqrt{\pi\beta}} R^7e^{-\beta R^2}[1+O(R^{-2})],\quad \frac{E_3}{N}=\frac{32\gamma_3M_0^3J_{33}}{\sqrt{\pi\beta}} R^5e^{-\beta R^2}[1+O(R^{-2})],$$ $$0\le\frac{E_c}{N}\le \frac{cR|\sin\phi|\sqrt{M_0}e^{-\beta R^2}}{D}. \tag{19}$$

Thus both endpoints have zero excess, with a positive barrier between. The generalized separation response $-\partial_RH$ is inward near coalescence and outward at large separation, where it is $(H-\mu N)[2\beta R-7/R+O(R^{-3})]$. No stable binary minimum, encounter solution or acceleration follows from this restricted energy curve.

For $U=U_c+\gamma_2U_2+\gamma_3U_3$, the complete fixed-$N$ derivative is

$$\partial_RH=N\int g_\beta L [\partial_RU+U\partial_R\log L]dt, \quad \partial_R\log L=-2R\beta+\frac{2t\sinh z}{D_z} +\frac{2R\beta Ce^{-\beta R^2}}{D}. \tag{20}$$

The last term is the normalization correction. For full separation $\ell=2R$, its conjugate radial force is $-\partial_RH/2$; center components are reciprocal in a collective-coordinate interpretation. Anisotropy permits transverse responses, and accelerations require a kinetic/symplectic reduction or the actual PDE.

At nonzero phase this family also has

$$\mathcal P=-\frac{NR\sin\phi\,w e^{-\beta R^2}}{D}. \tag{21}$$

Consequently (20) is not a fixed-total-momentum derivative. The leading current term in (18) partly represents overall coherent motion and must not all be called a rest interaction potential. Adding the uniform gradient $p_0=R\sin\phi\,w e^{-\beta R^2}/D$ sets total momentum to zero without changing density or shape energy. It changes current energy, and its derivative must vary $p_0$ as well. For that different zero-momentum preparation,

$$\nabla\theta+p_0=\frac{R^3\sin\phi\,w(2t^2-\beta)}{(1+C)^2}+O(R^5),\qquad \frac{E_{c,0\mathcal P}}N= \frac{c^2M_0\beta^2\sin^2\phi}{\mu(1+C)^4}R^6+O(R^8). \tag{22}$$

The positive $R^4$ shape term then leads at every fixed nonnodal phase. At large $R$, changing the current by $p_0$ costs at most $c|p_0|$ per unit norm, so the leading shape asymptotic survives. The two limiting response signs therefore also hold in this zero-momentum comparison; the tabulated intermediate finite-phase responses in the source report concern the original preparation.

With $\eta=|Pv|^2$, the invariants reduce to

$$\beta=b_1\eta+b_0(1-\eta),\quad M_0=b_1^2\eta+b_0^2(1-\eta),\quad M_1=\Delta[b_1^2\eta-b_0^2(1-\eta)],\quad M_2=\Delta^2M_0. \tag{23}$$

Transverse determinants cancel at fixed $N$. There is no $d$ in (23): all depths below ten admit $0\le\eta\le1$, and depth ten has only $\eta=1$. Common rotation of $B,v$ preserves the energy and rotates its spatial derivative. Thus this equal-shape family has directional dependence but no depth-five suppression. Independently oriented unlike shapes and full physical polarization remain untested. Phase and angular energy derivatives are not locking rates or solved rotational accelerations; the source examples even change angular-response sign. Unlike physical particles must begin with their own isolated clocks if a later encounter calculation is undertaken.

For a dimensionless illustration with $b_1=1,b_0=4$, $\mu=c=\gamma_2=\gamma_3=N=1$, $\phi=\pi/3$ and $\eta=\cos^2\alpha$, $\alpha=\pi/4$, the existing calculation gives:

These are fixed-norm trial responses in arbitrary common units, not measured forces or rates. Their angular sign change is a concrete reason to calculate orientation evolution rather than impose universal alignment.

Nodes, finite components and admissibility

A scalar simple nodal hypersurface has $\rho\sim Cs^2$, hence $D^2q\sim-2/s^2$ in its normal direction. Its second-log energy density behaves as $16\gamma_2C|s|^{-6}$; its third-log density is $O(|s|^{-4})$. Both diverge. In particular the coherent pair at $\phi=\pi$ has infinite shape energy for every finite $R>0$, and its zero-separation field cannot be normalized to positive $N$.

This scalar obstruction does not exclude all finite-component excited profiles. For a finite-component classical field $\Psi$, explicitly extend the shape terms by $\rho=\sum_A|\Psi_A|^2$. Their component sources, where defined, are $W_j\Psi_A$. A scalar polar current law does not thereby supply complete vector polarization dynamics.

Let $y_i=\sqrt{b_1}x_i$ in the broad directions, and multiply the following real polynomials by a common ten-dimensional two-width Gaussian amplitude. For degree $k>0$, use

$$H_k(y_1),\quad H_k(y_2),\quad y_1H_{k-1}(y_2),\quad y_iH_{k-1}(y_1)\ (i=2,\ldots,d). \tag{24}$$

Every component has total oscillator degree $k$. Consecutive physicists' Hermite polynomials have no common zero, as follows from $H_{j+1}=2yH_j-2jH_{j-1}$, $H_0=1$; their derivative is $H'_k=2kH_{k-1}$. These identities are tabulated in NIST DLMF §18.9.

For even $k>0$, only $H_k(y_1),H_k(y_2),y_2H_{k-1}(y_1)$ are needed. They have no simultaneous zero, and their squared sum is positive, coercive in $(y_1,y_2)$, and bounded below globally. Both shape energies, both $L^2$ shape sources and unrestricted compact smooth first variations are regular. Degree zero is the constant Gaussian control.

For odd $k$, (24) vanishes exactly at $y=0$, with Jacobian rank $d$. Its nodal manifold has codimension $d$, extending through the complementary $10-d$ spatial coordinates. Setting $h=H_{k-1}(0)$, $a=H'_k(0)=2kh$, one can normalize the first two entries by $\sqrt2a$, the cross entries for $y_1,y_2$ by $\sqrt2h$, and the remaining entries by $h$. Their squared sum is then $r^2+O(r^4)$, $r=|y|$. There are at most $d+2$ real components, at most twelve real or six complex components across the maintained depths. Components add no physical spatial directions. These profiles are explicit admissibility controls, not stationary solutions of the changed action and not identifications of spin or charge.

For a controlled nondegenerate nodal expansion of codimension $c_\perp$, $\rho=r^{2m}a(\omega,z)[1+O(r)]$, with smooth positive angular coefficient and controlled differentiated remainders, the sharp energy tests are

$$E_3<\infty\ \Longleftrightarrow\ c_\perp+2m>6,\qquad E_2<\infty\ \Longleftrightarrow\ c_\perp+2m>8. \tag{25}$$

Equality gives logarithmic divergence. Radial derivatives and the positive tensor norms prevent angular cancellation of the leading divergence. The relevant volume is transverse, not automatically ten-dimensional.

Energy, first variation and $L^2$ source have different requirements. Sufficient absolute compact-variation conditions are $c_\perp+m>6$ for $E_3$ and $c_\perp+m>8$ for $E_2$. Sufficient $L^2$-source conditions are $c_\perp+2m>12$ and $>16$, respectively. For isotropic simple zeros, the explicit nonzero leading sources are

$$W_3=-\frac{16\gamma_3(2c_\perp^3-15c_\perp^2+31c_\perp-22)}{r^6},\quad W_2=-\frac{16\gamma_2(4c_\perp^2-37c_\perp+72)}{r^8}+O(r^{-6}). \tag{26}$$

Thus their $L^2$ component sources require $c_\perp>10$ and $>14$, respectively; no isotropic simple zero in this ambient space meets either condition. The sufficient compact-path differentiability conditions for simple vector zeros are $c_\perp>5$ and $>7$. They follow by controlling nearby nodal graphs and uniformly integrable tube contributions and assert no global Fréchet or flow theorem.

At codimension five the third-log energy can be finite but non-differentiable. Lifting an isotropic simple zero into an unused real component gives $\rho_\varepsilon=r^2+\varepsilon^2$. With a compact cutoff equal to one near the zero,

$$E_3(\varepsilon)-E_3(0) =-210\pi\gamma_3|S^4|\,|\varepsilon|+o(|\varepsilon|) \tag{27}$$

in unit local-density coordinates. For the normalized odd Hermite controls, the coefficient acquires a positive Gaussian/tangential cutoff factor and the cusp persists. The cutoff transition contributes only $O(\varepsilon^2)$. Although the pointwise first density variation along this lift is zero, the integrated energy has no two-sided directional derivative. Finite energy alone cannot justify the formal source there.

Using the maintained label $k=n-1$, the explicit controls give:

“Regular” here means finite shape energies, $L^2$ shape sources and unrestricted compact smooth variations for these explicit positive-density controls. Eight assignments are regular, two have finite but cusp-singular $E_3$, and four have divergent $E_3$. Depth five is a threshold of this penalty's nodal regularity, not a derived force suppression; its even control is fully regular. Positive density does not make the excited shape source zero. Their stationary equations and frequencies under the changed action remain to be solved.

Reproducibility and unresolved physical questions

The existing independent Hessian audit reports 137 exact identities and 994 finite spectral pairs, together with the analytic infinite-degree proof and a rational matrix check. The pair calculation reports 75 checks, including 45 analytic-versus-differenced separation derivatives, and nonrigorous quadrature corroboration of the analytic reduction. The nodal calculation reports 185 exact checks, including all fourteen shell identities and the exact cusp integral. Portable companion checks reproduce the accepted calculations with the executed source hashes and original provenance kept distinct.

The established coexistence is local exact Gaussian support, arbitrary normalization, all-direction positive complete inertia, a projected linear energy group, and explicit phase/direction-sensitive trial pair energies. Smoothness and nodal domains constrain those statements. The physical constitutive origin, selected depths, excited count clocks, nonlinear well-posedness and persistence, reciprocal encounter evolution, spin, charge, full polarization and deterministic orbit coupling remain unresolved. These are separate research questions; the present local action supplies no Born rule, mediator mechanism or observed particle identification.

Portable verification: curvature, pair response, nodal domains, and additional stationary branches. The pair companion retains 21 exact and 54 numerical comparisons; its quadrature is not a rigorous enclosure.