A Local Inverse Action with Exact Gaussian Count Clocks

A constitutive energy depending on local spatial derivatives can support a whole family of exact positive localized patterns with the supplied simplex frequencies. Its full variation, rather than a Gaussian ansatz variation, gives the stationary equation. The same energy has a nonnegative spatial Hessian in every smooth compact density direction. A frequency-following current term then supplies exact moving histories and positive complete inertia in all ten spatial directions.

The construction is inverse: the count function, width sequence, sector frequency units and positive restoring penalties are chosen. What follows is their coexistence in one local field equation. The raw infinite family has a restricted derivative domain. A second, globally smooth constitutive potential repairs that domain for a declared finite band and preserves all fourteen maintained frequency controls. These are distinct versions of the action, with different domain and large-width behavior.

The field is classical on ordinary shared $\mathbb R^{10}$. Density is squared field amplitude, not a position probability. Depth classifies broad directions of a spatial pattern; it is not a separate physical space. No mediator, compact dimension or particle-exchange mechanism is introduced. An internal profile, its center, and an electron's deterministic trajectory in six ordinary spatial dimensions remain distinct. The Gaussian construction does not itself derive that electron trajectory, observed spin or charge.

Local curvature descriptors and a common potential

Let $\psi=\sqrt\rho e^{i\theta}$, with smooth $\rho>0$, and define

$$q=\log\rho,\qquad B=-\frac12D^2q,\qquad C=B-bI, \qquad t=\operatorname{tr}C,\quad z=\operatorname{tr}C^2,$$ $$D=\frac{t^2}{z},\qquad a=b+\frac zt,\qquad K=\frac ba-2, \qquad b>0. \tag{1}$$

The raw potential is defined on the open matrix domain $t<0,z>0,a>0$. Here $a<b$, so $K>-1$, and $0<D\le10$ by Cauchy–Schwarz. Equality $D=10$ at scalar $C$ does not make this matrix domain nonopen. The scalars in (1) describe ordinary spatial derivatives and add no physical dimensions.

For an orthogonal rank-$d$ projector $P$, put $Q=I-P$. At $B=aP+bQ$, $0<a<b$, equation (1) recovers $D=d$ and the same $a$. The proposed branch widths $a=b/(k+2)$, $k\in\mathbb Z_{\ge0}$, give $K=k$.

Choose common positive $\Lambda,\eta$, and set

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

The constants carry the units needed for $V$ to be a frequency; $b,a$ have inverse-squared-length units and $K,D$ are dimensionless. Since $g$ is nondecreasing, $g(K)>-1$ and $g(D)>0$, every Gamma argument is positive. Thus (2) is smooth and positive on its domain.

One may take $m(v)=\mu>0$ for a common-unit comparison. To retain given positive units $\mu_d$ at the maintained depths $\mathcal D=\{2,3,4,5,6,10\}$, a single explicit choice is

$$L_d(v)=\prod_{\substack{e\in\mathcal D\\e\ne d}}\frac{v-e}{d-e}, \qquad m(v)=\mu_{\rm ref} \exp\!\left[\sum_{d\in\mathcal D}L_d(v) \log\frac{\mu_d}{\mu_{\rm ref}}\right],\qquad\mu_{\rm ref}>0. \tag{3}$$

This gives $m(d)=\mu_d$ with dimensionless logarithms. It is an interpolation of supplied scales, not a derivation of them. It does not resolve the neutrino absolute unit. Values at other depths are constitutive choices. No per-particle amplitude or independent mass correction enters.

Complete constitutive and density derivatives

Use the matrix inner product $A:H=\operatorname{tr}(AH)$ for symmetric matrices. For an arbitrary symmetric variation $H=\delta B$, define

$$A_a=\frac{2C}{t}-\frac{zI}{t^2},\qquad A_D=\frac{2tI}{z}-\frac{2t^2C}{z^2},\qquad A_K=-\frac b{a^2}A_a.$$

Then $\delta a=A_a:H$, $\delta D=A_D:H$, $\delta K=A_K:H$, and

$$\delta R=CH+HC-(a-b)H-C(A_a:H). \tag{4}$$

Write $U(K,D)$ for the first two terms of (2), $u=g(K),v=g(D)$, and $\Psi_0$ for the logarithmic Gamma derivative. The needed scalar derivatives are

$$F_1=F[\Psi_0(u+v+1)-\Psi_0(u+1)],\qquad F_2=F[\Psi_0(u+v+1)-\Psi_0(v+1)],$$ $$U_K=m(v)F_1g'(K)+\Lambda\pi\sin(2\pi K),$$ $$U_D=[m'(v)F+m(v)F_2]g'(D)+\Lambda\pi\sin(2\pi D),$$ $$V_B=U_KA_K+U_DA_D +2\eta[CR+RC-(a-b)R-\operatorname{tr}(RC)A_a]. \tag{5}$$

This includes the response of every descriptor and noncommuting matrix direction. Let

$$E[\rho]=\int_{\mathbb R^{10}}\rho V(B)\,dx.$$

For compact smooth density variations, $\delta B=-D^2(\delta\rho/\rho)/2$. Two integrations by parts give the complete stationary source

$$w[\rho]=\frac{\delta E}{\delta\rho} =V-\frac1{2\rho}\partial_i\partial_j(\rho V_{B,ij}). \tag{6}$$

In particular this is a fourth-order local source, not the value of $V$ alone. For $\sigma=\delta\rho$, $H=-D^2(\sigma/\rho)/2$ and $T=V_{BB}$, its full differential is

$$Dw[\sigma]=V_B:H +\frac{\sigma}{2\rho^2}\partial_i\partial_j(\rho V_{B,ij}) -\frac1{2\rho}\partial_i\partial_j \{\sigma V_{B,ij}+\rho(T[H])_{ij}\}. \tag{7}$$

The energy is homogeneous: $E[\lambda\rho]=\lambda E[\rho]$, since the constant shift in $q$ leaves $B$ unchanged.

Exact Gaussian clocks and the fourteen maintained controls

For arbitrary amplitude $A_0>0$, center $X$, and projector $P$, take

$$B_*=aP+bQ,\qquad a=\frac b{k+2},\qquad \rho_*(x)=A_0^2e^{-(x-X)^TB_*(x-X)}. \tag{8}$$

At this matrix $K=k,D=d,R=0$. Both $g'$ and $g''$ vanish at integers. Hence $V_B(B_*)=0$ in every symmetric direction, and (6) gives

$$\boxed{w[\rho_*]=\omega_{d,k}=m(d)\binom{k+d}{d}.} \qquad N=\frac{A_0^2\pi^5}{a^{d/2}b^{(10-d)/2}},\qquad E=N\omega_{d,k}. \tag{9}$$

All ten directions are localized. The $d$ broad directions rotate continuously with $P$, while all ten center translations are available. The same formula admits all integer $1\le d\le10$, $k\ge0$ on the raw domain; it supplies no occupation rule. In particular $k$ is a width label, not an established physical Hermite degree. With the maintained assignment notation $n_{\rm assignment}=k+1$, the supplied simplex count is $S(n_{\rm assignment},d)=\binom{k+d}{d}$.

These are exact targets in the chosen potential. They are not new measured mass predictions or an independent empirical test of the count law.

Full positive spatial Hessian

Put $s=a-b<0$, $m_\perp=10-d$. Decompose a symmetric $H$ into blocks $H_P,H_Q,H_{PQ}$, and set $u=\operatorname{tr}H_P$, $v=\operatorname{tr}H_Q$. At (8),

$$\delta a=(u-v)/d,\quad \delta D=2v/s,\quad \delta K=-\frac{b(u-v)}{da^2},$$ $$\delta R_{PP}=s(H_P-\delta a I),\qquad \delta R_{QQ}=-sH_Q,\qquad\delta R_{PQ}=0.$$

The entire $mF$ term has zero first and second derivative there. Define

$$A=2\Lambda\pi^2\left(\frac b{da^2}\right)^2, \qquad Z=\frac{8\Lambda\pi^2}{s^2},\qquad R_0=2\eta s^2.$$

The complete matrix Hessian is

$$T[H,H]=A(u-v)^2+Zv^2 +R_0\left[\left\|H_P-\frac udI\right\|^2 +\frac{v^2}{d}+\|H_Q\|^2\right]. \tag{10}$$

It is positive on every block-diagonal nonzero direction; only the cross blocks vanish. These $d(10-d)$ zero directions are precisely rotations of the projector. For $m_\perp>0$, the trace form has matrix

$$\begin{pmatrix}A&-A\\-A&A+Z+R_0/d+R_0/m_\perp\end{pmatrix}, \quad\det=A(Z+R_0/d+R_0/m_\perp)>0.$$

At $d=10$, $v=0$: $A$ controls trace and $R_0$ controls all traceless directions. Thus the vanishing derivative of $D$ at depth ten does not leave an uncontrolled width disturbance. There is no special depth-five suppression.

Since the rotation orbit is compact, (10) gives a genuine local minimum normal to that orbit. In a sufficiently small tubular matrix neighborhood, $V\ge\omega$, with equality only on the orbit. Accordingly $E-\omega N\ge0$ for densities whose local $B(x)$ stays in that neighborhood. This is a condition on derivatives, not an open $L^2$ stability theorem. For $C^4$ log densities the equality case is Gaussian: on the orbit $\Delta q$ and $\|D^2q\|^2$ are constant, so

$$0=\Delta\|D^2q\|^2 =2\|D^3q\|^2+2D^2q:D^2(\Delta q)=2\|D^3q\|^2.$$

Hence $q$ is quadratic and completing the square gives (8). No weak nonsmooth equality classification is asserted.

For $f=\delta\rho/\rho_*$, the unrestricted density Hessian is

$$\delta^2(E-\omega N)[f,f] =\frac14\int\rho_* T[D^2f,D^2f],\qquad Lf=\frac1{4\rho_*}\partial_i\partial_j (\rho_*T_{ijkl}\partial_k\partial_lf). \tag{11}$$

This is also $Dw[\rho_*f]$. Its natural Gaussian weighted form-domain kernel is exactly

$$f=c+u_P\cdot x_P+u_Q\cdot x_Q+x_P^TMx_Q. \tag{12}$$

Indeed zero form forces both pure-block Hessians to vanish; integration of those equations gives (12). These are amplitude, translation and orientation tangents, of dimension $11+d(10-d)$, or 11 at depth ten. Every nonzero compact smooth density disturbance has strictly positive form. The appendix specifies the entire non-Gaussian disturbance spectrum.

Exact motion under full compact field variation

Apply the frequency-following mobility law to this local $E$. On its smooth positive-$w$ domain, set

$$H_{\rm FL}=E+\int\rho[\sqrt{w^2+c^2|\nabla\theta|^2}-w], \qquad S=\int dt[-\int\rho\theta_t-H_{\rm FL}],\quad c>0. \tag{13}$$

The canonical equations are $\rho_t=\delta H/\delta\theta$, $\theta_t=-\delta H/\delta\rho$. With $h=\sqrt{w^2+c^2|\nabla\theta|^2}$, full variation gives

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

Homogeneity implies $E=\int\rho w$; differentiating this identity with all source dependence retained gives $(Dw)^*\rho=0$. At (8), $w=\omega$. For uniform $\nabla\theta=p$, $w/h-1$ is constant, so the entire adjoint term vanishes. Therefore

$$\psi(x,t)=e^{ip\cdot x-i\Omega t} \sqrt{\rho_0(x-X_0-vt)},\qquad \Omega=\sqrt{\omega^2+c^2|p|^2},\quad v=\frac{c^2p}{\Omega}, \tag{15}$$ $$\mathcal E=N\Omega,\quad\mathcal P=Np,\quad \mathcal E^2-c^2|\mathcal P|^2=N^2\omega^2,\quad M_{\rm rest}=\frac{N\omega}{c^2}I. \tag{16}$$

Here $\rho_0(y)=A_0^2e^{-y^TB_*y}$ is the centered profile from (8). At finite speed the inertia tensor is $(N\omega/c^2)[\gamma I+\gamma^3vv^T/c^2]$, $\gamma=(1-|v|^2/c^2)^{-1/2}$. The comoving phase rate is $\Omega-p\cdot v=\omega^2/\Omega$, distinct from the laboratory factor $\Omega$. The common current law supplies these response identities; no local Lorentz covariance or profile contraction follows. The uncalibrated norm $N$ remains in complete energy and inertia, while the selected frequency $\omega$ supplies the target mass in equivalent frequency units.

Equation (15) is stationary against every compact smooth complex spacetime variation $\zeta$. Over the finite time support of $\zeta$, its moving Gaussian has a positive minimum on the compact test region. Thus $\psi+\epsilon\zeta$ admits smooth density and phase there for sufficiently small $\epsilon$. Its local $B,w$ are unchanged outside the test support; inside, smooth dependence and the strict branch inequalities preserve the raw domain and $w>0$, uniformly over that time interval. All compact real amplitude and phase tangents arise this way. Equations (6), (7) and (14) therefore apply to the complete field variation, and integration by parts using the continuity and phase equations gives $\delta S=0$. Nothing in this argument freezes $w$ or restricts the test to Gaussian parameters.

These are global explicit histories. They are not a nonlinear neighborhood existence or persistence theorem. Generically $w$ contains fourth spatial derivatives, and the density equation of (13) contains eighth spatial derivatives. The evolution theorems of the global-moment count action belong to that different action and cannot be transferred here.

Why the infinite raw family needs a domain qualification

An everywhere-positive finite-norm field need not have admissible local descriptors. For $d=10,k=22,b=1,a=1/24$, consider the sum of amplitudes

$$\psi_R(x)=e^{-a|x-Re_1|^2/2}+e^{-a|x+Re_1|^2/2}.$$

At the midpoint $B=aI-a^2R^2e_1e_1^T$. With $R=24$,

$$t=-127/12,\qquad z=3485/288,\qquad \widehat a=1+z/t=-437/3048<0. \tag{17}$$

Each isolated Gaussian is an exact branch, but their positive sum leaves the raw action domain. Spatially local dependence therefore does not by itself establish a usable pair theory or clustering.

A common smooth continuation for 960 labels

A finite-band repair can retain every target with $1\le d\le10$, $0\le k\le95$, including all fourteen maintained controls. Their matrix rotation orbits form a compact finite union. Choose smooth scalar cutoffs $\chi_t,\chi_z,\chi_a\in[0,1]$, equal to one on the inner intervals and supported in the outer intervals below.

Set $\chi=\chi_t(t/b)\chi_z(z/b^2)\chi_a(a/b)$ where the descriptors are defined, and zero elsewhere. The cutoffs vanish on neighborhoods of every descriptor singularity. Since $z=\|B-bI\|^2$, their support is compact in matrix space and lies strictly inside the raw domain. Every one of the 960 target orbits lies strictly within the inner intervals, so $\chi=1$ on open neighborhoods of all of them.

Use a quadratic exterior reference, with common $\eta_{\rm ref}>0$:

$$U_{\rm ref}(B)=\mu_{\rm ref}+\eta_{\rm ref}\operatorname{tr}(B-bI)^2, \qquad V_{\rm ext}=(1-\chi)U_{\rm ref}+\chi V. \tag{18}$$

Outside the descriptor domain this means $V_{\rm ext}=U_{\rm ref}$. It is a globally smooth positive rotation-invariant function of every symmetric $B$. Its derivative is explicitly

$$(V_{\rm ext})_B=(1-\chi)2\eta_{\rm ref}C+\chi V_B +(V-U_{\rm ref})\chi_B,$$ $$\chi_B=\frac{\chi_t'}b\chi_z\chi_a I +\chi_t\frac{\chi_z'}{b^2}\chi_a\,2C +\chi_t\chi_z\frac{\chi_a'}b A_a. \tag{19}$$

The primes are evaluated at their displayed dimensionless arguments. Equation (6), and its full differential (7), apply with this extended potential. Every derivative agrees with (2) near every finite-band target, so its clocks, spatial Hessian, spectrum and compact-variation proof survive. At depths outside $\mathcal D$, the scales from (3) remain uncalibrated constitutive values rather than additional physical predictions.

Unlike a constant exterior reference, (18) penalizes arbitrary exterior curvature. In a region with $\chi=0$, a Gaussian of constant positive definite $B$ has constant $w$ only if $(U_{\rm ref})_B=0$: the quadratic term in (6) is $-2x^TB(U_{\rm ref})_BBx$. Thus $B=bI$ is the only wholly exterior stationary Gaussian. Its clock is $\mu_{\rm ref}$, with density Hessian

$$L_{\rm ref}f=\frac{\eta_{\rm ref}}{2\rho_*} \partial_i\partial_j(\rho_*\partial_i\partial_j f),\qquad \lambda_{\rm ref}(n)=2\eta_{\rm ref}b^2n(n-1). \tag{20}$$

Only amplitude and translations vanish. This removes the continuum of arbitrary-width stationary Gaussians produced by a flat constant exterior. It does not classify transition-zone or non-Gaussian stationary branches.

The tradeoff is loss of a uniform upper bound proportional to norm: $E_{\rm ext}=\int\rho V_{\rm ext}$ contains curvature energy. Finite positive in-phase Gaussian sums still have finite energy. Indeed their log amplitude is a finite log-sum-exp of quadratics. Its normalized positive weights have derivatives formed from affine gradient differences and fixed Hessians, so all fixed log-density derivatives grow at most polynomially. The same holds for $V_{\rm ext}(B(x))$, $w(x)$ and its fixed spatial derivatives: the quadratic reference is polynomial in $B$ and the remaining matrix modification is smooth with compact support. Gaussian upper tails consequently give finite shape energy and $\sqrt\rho\,w\in L^2$. Uniform phase gradients also give finite current energy, and compact smooth complex field variations are admissible. Thus (17) becomes an initial field with an actual compact variational source. This says nothing about arbitrary complex sums or phase-induced nodes. A constant exterior would instead make $E_{\rm ext}$ finite for every smooth positive finite-norm density, but at the cost of the extra flat-shape continuum. Neither choice supplies a pair force or scattering solution by itself.

The infinite ladder cannot be preserved by any globally continuous extension with the same widths and clocks. At fixed $d$, $k\to\infty$ gives $B_*\to bQ$, a finite matrix, while $m(d)\binom{k+d}{d}\to\infty$. Continuity at $bQ$ would contradict that divergence. A global extension must change this limit or its frequency law; the finite-band restriction in (18) states that change explicitly.

For a globally smooth current continuation, let $\omega_{\min}=\min_{1\le d\le10,\,0\le k\le95} m(d)\binom{k+d}{d}>0$, and choose one $0<\delta<\omega_{\min}$. Take smooth $\alpha(w)\in[0,1]$, zero for $w\le\delta/2$ and one for $w\ge\delta$, and define

$$f_+(w)=(1-\alpha(w))\delta/2+\alpha(w)w>0,$$ $$H_{\rm ext}=E_{\rm ext}+\int\rho[h-f_+(w)],\qquad h=\sqrt{f_+(w)^2+c^2|\nabla\theta|^2}. \tag{21}$$

Its full density derivative is

$$\frac{\delta H_{\rm ext}}{\delta\rho} =w+h-f_+(w) +(Dw)^*[\rho f_+'(w)(f_+(w)/h-1)], \tag{22}$$

and its phase derivative remains the divergence in (14), using the new $h$. At each retained branch $f_+(w)=w$, $f_+'(w)=1$ on a neighborhood, so (15)–(16) and the complete rest linearization are unchanged. The excess obeys $0\le\rho(h-f_+)\le c\rho|\nabla\theta|$; finite $H^1$ field norm suffices for this current integral. It does not by itself ensure finite curvature energy. The logarithmic shape energy still requires positive density; arbitrary nodal fields and nonlinear evolution need separate analysis.

Scope of the local construction

One common constitutive formula realizes the supplied clocks as actual positive stationary spatial fields, admits unrestricted compact variations, has a positive Hessian away from symmetry directions, and permits exact all-direction motion. The finite-band extension addresses a concrete two-Gaussian domain failure while retaining 960 common-law labels.

The simplex multiplier and sector-unit interpolation remain inputs. Gaussian width labels are not inherited Hermite shells of the different global-moment action. Observed occupation, electron orbit dynamics, spin, charge, nonlinear persistence, source clustering, finite propagation, reciprocal pair scattering and a physical depth-five interaction difference remain unresolved. The common square-root current is a chosen inertial completion. Its successful exact histories do not derive a complete fundamental action for $\Psi_\infty$.

Appendix: complete spatial disturbance spectrum

Work in $L^2(\rho_*dx)$, with coordinates diagonalizing $B_*$. Let $D_i=\partial_i$, $D_i^*=-\partial_i+2B_i x_i$, and let $N_P,N_Q$ be the ordinary Gaussian number operators from $D_i/\sqrt{2B_i}$, $D_i^*/\sqrt{2B_i}$. Let $\Delta_P,\Delta_Q$ be the partial Laplacians, and use stars for their weighted adjoints. Expanding (10) gives exactly

$$L=R_0[a^2N_P(N_P-1)+b^2N_Q(N_Q-1)] +\frac{A-R_0/d}{4}\Delta_P^*\Delta_P +\frac{A+Z+R_0/d}{4}\Delta_Q^*\Delta_Q -\frac A4(\Delta_P^*\Delta_Q+\Delta_Q^*\Delta_P). \tag{23}$$

It preserves total disturbance degree $n=N_P+N_Q$. This $n$ is unrelated to $k$ or $n_{\rm assignment}$. The Hermite expansion here describes small classical field disturbances, not physical particle positions or a Born law.

For $m_\perp>0$, fix ordinary harmonic-polynomial degrees $\ell,h$ in the two blocks and total radial excitation $J$. In the orthonormal radial basis $r=0,\ldots,J$, $t_r=J-r$, set

$$n_P=\ell+2r,\quad n_Q=h+2t_r, \quad\tau_P(r)=4r(r+\ell+d/2-1), \quad\tau_Q(t_r)=4t_r(t_r+h+m_\perp/2-1).$$

All eigenvalues in that harmonic sector are those of the finite real symmetric tridiagonal matrix

$$M_{rr}=a^2[R_0n_P(n_P-1)+(A-R_0/d)\tau_P(r)] +b^2[R_0n_Q(n_Q-1)+(A+Z+R_0/d)\tau_Q(t_r)],$$ $$M_{r,r+1}=-Aab\sqrt{\tau_P(r+1)\tau_Q(t_r)}. \tag{24}$$

Its multiplicity is $h_d(\ell)h_{m_\perp}(h)$, where for $j\ge2$

$$h_j(l)=\binom{l+j-1}{j-1}-\binom{l+j-3}{j-1},$$

with the second term zero for $l<2$. For the extra finite-band labels with a one-dimensional block, use $h_1(0)=h_1(1)=1$, $h_1(l)=0$ for $l\ge2$; only those allowed harmonic degrees enter (24). All maintained nonzero $m_\perp$ are at least four.

At depth ten there is no $Q$ block. For $n=\ell+2r$,

$$\lambda_{n,\ell}=a^2\{R_0n(n-1) +(A-R_0/10)[n(n+8)-\ell(\ell+8)]\}, \tag{25}$$

with multiplicity $h_{10}(\ell)$. Only $n=0,1$ vanish; (10) proves positivity for every $n\ge2$, including when $A-R_0/10<0$.

There is no omitted continuum in the natural Friedrichs realization. The positive trace form and traceless terms in (10) give a constant $\gamma>0$ with $T[H,H]\ge\gamma(\|H_P\|^2+\|H_Q\|^2)$. On degree $n$, equation (11) is therefore bounded below by

$$\gamma\min(a^2,b^2)\, \min_{n_P+n_Q=n}[n_P(n_P-1)+n_Q(n_Q-1)] \ge\frac{\gamma\min(a^2,b^2)}2 n(n-2),\quad n\ge3. \tag{26}$$

At depth ten the corresponding bound is $\gamma a^2n(n-1)$. Finite-dimensional norm bounds also give an $O(n^2)$ upper bound. Polynomial modes are complete in the Gaussian weighted space and smooth compact cutoffs approximate them in the form norm. Every degree block is finite and (26) tends to infinity, so the resolvent is compact and (24)–(25) exhaust the spectrum.

The frequency-following rest linearization is also explicit. Write $\rho=\rho_*(1+f)$, $\theta=-\omega t+\vartheta$, and set

$$J_0=-\rho_*^{-1}\nabla\cdot(\rho_*\nabla) =2aN_P+2bN_Q.$$

Then

$$f_t=(c^2/\omega)J_0\vartheta,\qquad\vartheta_t=-Lf. \tag{27}$$

On each nonconstant block, squared shape frequencies are $(c^2/\omega)$ times the eigenvalues of $J_0^{1/2}LJ_0^{1/2}$; in (24), $J_0$ has diagonal $2an_P+2bn_Q$. At depth ten they are $(c^2/\omega)2an\lambda_{n,\ell}$. All are nonnegative. Symmetry modes can have secular Jordan drift, and the degree-zero amplitude/phase pair is neutral. Linear spectral positivity does not prove nonlinear orbital stability, observed persistence or an empirical decay law.

The principal symbol also states a limit of the center-speed result. For $U_\xi=|\xi_P|^2$, $V_\xi=|\xi_Q|^2$, the fourth-order density symbol is

$$\sigma_L(\xi)=\frac14\{A(U_\xi-V_\xi)^2+ZV_\xi^2 +R_0[(1-1/d)U_\xi^2+(1+1/d)V_\xi^2]\}. \tag{28}$$

For $d<10$, its coefficient matrix has first entry $A+R_0(1-1/d)>0$ and determinant

$$AZ+2AR_0+R_0Z(1-1/d)+R_0^2(1-1/d^2)>0.$$

At depth ten it is $(A+9R_0/10)|\xi|^4/4$. Thus $L$ is strongly elliptic at every branch, despite the matrix rotation kernel: a nonzero rank-one Hessian cannot be purely cross-block. Freezing the coefficients of (27) gives $\omega_{\rm shape}(\xi)^2\sim(c^2/\omega)|\xi|^2\sigma_L(\xi)$. Its short-wavelength shape frequency grows as $|\xi|^3$, and its formal group speed as $|\xi|^2$. The bound $|v|<c$ for the exact centers therefore does not bound general linear shape propagation. This symbol calculation alone proves no nonlinear propagation or ill-posedness theorem.

Reproducible verification

Five standalone scripts reproduce 10,923 mathematical checks, with a separate structural cardinality assertion. These cover the complete derivatives, branch counts, Hessian blocks, finite-band continuation, independent source and symbol review, and supplied-scale interpolation. The all-degree derivation above establishes the infinite disturbance statement; finite checks corroborate it. Executed source hashes are retained in dated results, separately from historical provenance.

• local-count-pattern-checks
• local-count-spectrum-checks
• local-count-continuation-checks
• local-count-pattern-review
• local-count-scale-checks

A Smooth Spatial Carrier Completion

The inverse count action can include internal carrier compatibility without altering its fourteen supplied clocks or exact moving histories. This is a changed constitutive action, with its full extra variation retained. Let $\psi\in\mathbb R^{32}$ be the reviewed ambient real spin carrier, $\rho=|\psi|^2>0$, $e=\psi/\sqrt\rho$, and let $J$ be the fixed ambient skew clock structure. Set

$$a_i=\langle Je,\partial_i e\rangle,\qquad \mathcal T=\sum_i(|\partial_i e|^2-a_i^2)\ge0.$$

For a gapped spatial projector $P(B)=1_{(-\infty,3b/4)}(B)$, the embedded rotation group determines the Schur decomposition of the real carrier. With $g=X^\dagger X/2$, $n=\rho/2$, use the normalized scalar

$$s(P,e)=\frac{W_P(g)}{n^2},\qquad W_P=\frac{\beta_s}{4}(n^2-\operatorname{Re}\operatorname{tr}g^2) +\frac{\gamma_s}{4}\|[g,j]\|^2.$$

The coefficients are common across depths. On a fixed-rank plane manifold this is smooth and independent of the spin frame. One explicit invariant definition uses the real Haar twirl $T_P(e)$ of $ee^T$ over $\operatorname{Spin}(\operatorname{range}P)$. If $T_\alpha$ is its isotypic block and $r_\alpha$ the real dimension of one irreducible copy,

$$s=\frac{\beta_s}{4}\left(1-\sum_\alpha r_\alpha \operatorname{tr}_{\mathbb R}T_\alpha^2\right) +\frac{\gamma_s}{4}\sum_\alpha r_\alpha\|[T_\alpha,J]\|_{\mathbb R}^2.$$

The sum includes both chiral blocks where present. It reproduces the Gram normalization, including real and quaternionic trace factors, and is covariant under simultaneous spatial and carrier rotation. It introduces no physical probability interpretation or extra spatial coordinate.

The sharp projector alone is not a general pair law. Two identical positive Gaussians separated by $\pm Ru$ in an inactive direction have $B_{\rm pair}=B_*-b^2R^2\operatorname{sech}^2(bRu\cdot x)uu^T$. For $R>1/(2\sqrt b)$ this crosses the threshold, even though neither field has a node. The scalar continuation $V_{\rm ext}$ does not fix that problem.

A smooth spectral notch does. Choose a common smooth $0\le h\le1$ with $h=1$ below $9b/16$ and above $15b/16$, and $h=0$ on $[5b/8,7b/8]$. Put

$$\chi(B)=\det h(B),\qquad S_\sigma(B,e)=\chi(B)s(P(B),e)+(1-\chi(B))\sigma_0,\quad\sigma_0\ge0.$$

The first term is extended by zero wherever the projector fails: it vanishes on an open neighborhood of every threshold crossing. Thus the resulting scalar, rather than an exact variable-rank projector, is globally smooth in all symmetric $B$. When it is nonzero the selected and excluded clusters have a uniform gap. At every branch $B_*=bP/(k+2)+b(I-P)$, $k\ge0$, all derivatives agree with the original selector on an operator-norm neighborhood of radius less than $b/16$. Both choices $\sigma_0=0$ and $\sigma_0>0$ are allowed: the first turns compatibility selection off in the notch, while the second replaces it with a scalar cost. Neither preserves a definite depth or an orientation preference inside the notch. For definiteness one construction sets $s=0$ at unoccupied ranks; this is an exterior constitutive choice, not evidence excluding those ranks.

With the finite-band $V_{\rm ext}$, the complete rest action is

$$E=\int\rho[V_{\rm ext}(B)+S_\sigma(B,e)+\kappa\mathcal T/2],\quad \mathcal S=\int dt\,[\int\psi^TJ\psi_t-E],\quad\kappa>0.$$

Its kinetic normalization gives $-\int\rho\theta_t$ on a fixed phase plane, matching the scalar count action. Let $A=V_{\rm ext}+S_\sigma+\kappa\mathcal T/2$ and $Z=(V_{\rm ext})_B+(S_\sigma)_B$. Complete radial variation gives

$$w=E_\rho=A-\frac1{2\rho}\partial_i\partial_j(\rho Z_{ij}).$$

In particular $(S_\sigma)_B=(s-\sigma_0)\chi_B+\chi(DP)^*s_P$; both cutoff and plane responses matter. The unit-carrier variation contains $\rho(S_\sigma)_e$ and the full variation of $\mathcal T$. On a compatible constant polarization, the additions and their first derivatives vanish. Every retained Gaussian therefore keeps $w=\omega=\mu_d\binom{k+d}{d}$. The full mixed shape/carrier Hessian is nonnegative, since the added energies are nonnegative and vanish there. Old individual scalar eigenvalues need not survive the new mixed terms.

Use this entire $w$ in the current completion $H=E+\int\rho(\sqrt{w^2+c^2|a|^2}-w)$, or its already stated positive mobility continuation. Homogeneity gives $(D_\rho w)^*\rho=0$ and $(D_e w)^*\rho=E_e$. On the compatible branch $E_e=0$, so both adjoint source contributions cancel for a uniform phase gradient. The exact histories retain $\Omega=\sqrt{\omega^2+c^2|p|^2}$, $v=c^2p/\Omega$, energy $N\Omega$, momentum $Np$ and complete rest inertia $N\omega/c^2$. This cancellation does not require the mixed derivative $D_ew$ to vanish pointwise.

The continuation admits the positive gap-crossing pairs with finite local energy and compact variational source. It does not yet solve their dynamics. Positive density remains essential because $D^2\log\rho$ and the normalized carrier derivatives are undefined at nodes. The nonlinear evolution theorem for the different fixed-decomposition entropy action cannot be transferred. The selector alone extends through the accumulation of all $k\ge0$ branches; the full infinite count potential still cannot extend continuously because its prescribed clocks diverge at a finite limiting curvature matrix. Finite-band and singular infinite-ladder versions remain distinct.

The two transition constructions have 119 exact checks with fourteen independent positive-matrix certificates and 57 exact checks, respectively. The construction is a common inverse action with admissible encounters, not a forward derivation of the count law, observed occupation, charge or a unique depth-five response.