A Common Flat-Space Action with Exact Count Clocks and Bounded Localization

One explicit action can support the maintained depth classifications as different patterns in the same ordinary ten-dimensional space. It has exact orthogonal mode families with the target simplex clocks, positive complete energy, center motion in every spatial direction, and a theorem of persistent bounded localization for an open set around every maintained branch.

This is an inverse realization of the frequency proposal. Its constitutive function includes the simplex law; its width scales and positive penalties are chosen. The result establishes that these requirements can coexist in one equation. It does not derive the constitutive law, identify the observed occupied states, or reproduce interactions between separated particles.

One Field on Ordinary Spatial Coordinates

Let $\psi(x,t)$ be a complex classical field on $\mathbb R^{10}$. Two real quadratures give the same notation. Define ordinary spatial integrals

$$N=\int|\psi|^2,\quad X=N^{-1}\int x|\psi|^2,\quad p=N^{-1}\operatorname{Im}\int\psi^*\nabla\psi, \quad \xi=x-X,\quad q=-i\nabla-p,$$ $$M_{ij}=N^{-1}\int\xi_i\xi_j|\psi|^2, \qquad T_{ij}=N^{-1}\operatorname{Re}\int(q_i\psi)^*q_j\psi.$$

Assume nonzero norm, finite second moments and the regularity needed below. These integrals describe an internal field profile. They do not assign a position probability, and they do not replace an electron's deterministic six-dimensional center trajectory.

Both matrices are positive definite. Integration by parts and the Gram inequality for the functions $\xi_i\psi$ and their derivatives give $T\succeq M^{-1}/4$. Let $R$ be the unique positive matrix satisfying

$$RMR=T,\qquad R=M^{-1/2}(M^{1/2}TM^{1/2})^{1/2}M^{-1/2}.$$

The symbol $R$ here is an inverse-squared-length matrix, not a radius. It measures oscillator widths extracted from the field itself.

Choose two positive widths $b_1<b_0$, and define for $s>0$

$$t(s)=\frac{s-b_0}{b_1-b_0},\qquad g(s)=\frac{t(s)^2}{t(s)^2+[t(s)-1]^2},\qquad V(s)=\lambda t(s)^2[t(s)-1]^2.$$

Thus $g(b_0)=0$, $g(b_1)=1$, and both $g'$ and $V'$ vanish at the wells. Matrix functions use their ordinary spectral definitions. Set

$$D=\operatorname{tr}g(R),\quad U=\operatorname{tr}V(R),\quad K=\operatorname{tr}(RM)-5,$$ $$K_a=\operatorname{tr}[Rg(R)M]-\frac D2,\qquad K_n=K-K_a.$$

The Gram inequality implies $R^{1/2}MR^{1/2}\succeq I/2$. Consequently $K_a,K_n\ge0$ and $0\le D\le10$, including for noncommuting $M$ and $R$. There is no discontinuous matrix rank in the action: $D$ is a smooth real variable away from the domain boundary.

The Action and Its Complete Variation

For positive common constants $\mu,\Lambda,\lambda,c$, take

$$F(a,D)=\mu\frac{\Gamma(a+D+1)}{\Gamma(a+1)\Gamma(D+1)},\qquad W=F(K_a,D)+\Lambda K_n+U,$$ $$h=\sqrt{W^2+c^2|p|^2},\quad H=Nh,\qquad S=\int dt\left[\frac i2\int(\psi^*\dot\psi-\dot\psi^*\psi)-H\right]. \tag{1}$$

The Gamma quotient is smooth and positive on the entire attained domain, with $F\ge\mu$. Thus $H\ge\mu N>0$. Overall field multiplication scales $H$ by the squared amplitude but leaves every normalized shape variable unchanged. No amplitude is chosen to fit a particle mass.

The full equation follows by varying every moment. Write $dW=\operatorname{tr}(C\,dM+E\,dT)$. Then

$$i\dot\psi=\left\{h+\frac Wh \left[q^TEq+\xi^TC\xi-\operatorname{tr}(ET+CM)\right] +\frac{c^2p}{h}\cdot q\right\}\psi. \tag{2}$$

The matrices $C,E$ include the response of $R$; they are not frozen coefficients. An explicit way to compute them avoids differentiating eigenvectors. For any symmetric $A$, solve the Sylvester equation

$$MR Y_A+Y_A RM=A.$$

Differentiating $RMR=T$ gives $\operatorname{tr}(A\,dR)=\operatorname{tr}(Y_A\,dT-RY_AR\,dM)$. Let $f(s)=sg(s)$, and let $\mathcal L_f(R)[M]$ be its matrix Fréchet derivative, with divided difference $[f(r_i)-f(r_j)]/(r_i-r_j)$ and diagonal value $f'(r_i)$. Define

$$Y_a=Y_{\mathcal L_f(R)[M]-g'(R)/2},\qquad Y_D=Y_{g'(R)},\qquad Y_U=Y_{V'(R)}.$$

With derivatives of $F$ evaluated at $(K_a,D)$,

$$E=\frac\Lambda2R^{-1}+(F_a-\Lambda)Y_a+F_DY_D+Y_U,$$ $$C=\frac\Lambda2R+(F_a-\Lambda)[f(R)-RY_aR] -R(F_DY_D+Y_U)R. \tag{3}$$

Equations (2)–(3) allow unrestricted changes in norm, centroid, phase gradient, widths and orientation. In particular, off-diagonal divided differences must remain even when $g'$ vanishes at each well.

Exact Orthogonal Mode Families

For any ordinary spatial orthogonal projector $P$ of rank $d$, put $B=b_1P+b_0(I-P)$. This describes a solution family; $P$ is not an external field supplied to (1). At fixed $B$, define the ordinary Hermite number operator

$$A_B=\frac12(q^TB^{-1}q+\xi^TB\xi-10)=A_P+A_{I-P}.$$

Take any complex superposition in a joint shell $A_Pu=ku$, $A_{I-P}u=lu$. Orthogonality between different total Hermite degrees implies $\langle a_i a_j\rangle=0$, hence $T=BMB$ and $R=B$. Joint degree also removes cross-block correlations. The unrestricted variations in (3) consequently reduce to

$$N\frac{\delta K_a}{\delta\psi^*}=(A_P-k)\psi, \qquad N\frac{\delta K_n}{\delta\psi^*}=(A_{I-P}-l)\psi.$$

They vanish pointwise on the entire joint shell. So do the normalized $D,U$ sources. The resulting exact stationary frequency is

$$\boxed{\omega_{d,k,l}=\mu\binom{k+d}{d}+\Lambda l.} \tag{4}$$

For the remaining directions in their ground state, $l=0$, the orthogonal shell dimension is $\binom{k+d-1}{d-1}$, and the cumulative dimension through $k$ is $\binom{k+d}{d}$. With $n=k+1$, this is the maintained count $S(n,d)$. Each fixed shell is self-consistent; a superposition across different shells is not asserted to be stationary.

All ten coordinates remain spatially localized. Depth counts one class of extracted oscillator widths. This is a finite-norm localized alternative to exact uniform extension through unused infinite directions; it does not assert that those directions are absent or compactified. Depths two, three, four, five, six and ten follow the same equation. No five-specific interaction suppression has been derived.

A common $\Lambda=64\,512\,240\mu$ puts every constructed branch with $l\ge1$ above the largest of the fourteen maintained toy-scale targets. This is a finite-band statement about the constructed families, not a classification of every stationary solution. The entire arbitrarily high ladder cannot exclude transverse excitation with a finite penalty.

Center Motion and Normalization

For any normalized joint-shell profile $u$, any $N>0$ and any $p\in\mathbb R^{10}$, the exact moving solution is

$$\psi(x,t)=\sqrt N\,e^{ip\cdot x-i\Omega t} u(x-X_0-vt),\qquad \Omega=\sqrt{\omega^2+c^2|p|^2},\quad v=\frac{c^2p}{\Omega}.$$

Complete energy and momentum obey $\mathcal E=N\Omega$, $\mathcal P=Np$, and $\mathcal E^2=N^2\omega^2+c^2|\mathcal P|^2$. Rest inertia is $N\omega/c^2$ in every direction. The uncalibrated common phase-action normalization $N$ stays explicit. This inverse center law does not by itself prove a local Lorentz-covariant field theory; these laboratory profiles do not contract.

A rest-hyperplane realization supplies a complete changed-variable action with Lorentz-contracted physical scalar histories and the same proper-time dynamics. It retains the norm factor, angular center shift and nonlocal scope explicitly.

Exact Evolution and Persistent Bounded Localization

Equation (2) is a spatial quadratic operator with coefficients determined by the complete field. This permits exact evolution, without a Gaussian ansatz. Define

$$Z_{ij}=N^{-1}\operatorname{Re}\int\xi_i\psi^*q_j\psi,\qquad \Sigma=\begin{pmatrix}M&Z\\Z^T&T\end{pmatrix}.$$

The conserved quantities are $N,p,h,W$. In internal time $\tau=Wt/h$, the exact covariance equation is

$$\Sigma'=A\Sigma+\Sigma A^T,\qquad A=\begin{pmatrix}0&2E\\-2C&0\end{pmatrix},\qquad \Sigma(\tau)=S(\tau)\Sigma(0)S(\tau)^T, \tag{5}$$

where $S'=AS$, $S(0)=I$, and $S$ is symplectic. This matrix describes physical position/derivative moments. It is not an additional physical space. Its quadratic spatial propagator acts exactly on any initial Schwartz function; the standard mathematical construction is the metaplectic propagator.

Writing $s=\operatorname{tr}(ET+CM)$, the reconstruction includes the scalar phase $\theta(\tau)=\int_0^\tau[W-s(\sigma)]d\sigma$, the translation $X(t)=X_0+c^2pt/h$, and constant modulation. These terms make the reconstructed function solve the full nonlinear equation, not just its moment projection.

An explicit sufficient trapping condition proves global existence and bounded localization. In inverse-length-squared units take $b_1=1$, $b_0=4$. The dimensionless potential $V/\lambda$ has values $49/1296,1/16,49/1296$ at the interval boundaries $1/2,5/2,9/2$. If all initial $R$ eigenvalues lie in the two intervening intervals and

$$W(0)<\lambda\frac{49}{1296}, \tag{6}$$

energy conservation prevents any eigenvalue from crossing a boundary. Both the positive lower width bound and the number of eigenvalues in each interval persist.

The rest of the proof controls every field direction:

1. If $D<1/2$, then $g(R)\preceq D I$ and $K_n\ge K/2$, so $\Lambda K_n$ bounds $K$.
2. If $D\ge1/2$, then $F(K_a,D)\ge F(K_a,1/2)$, which tends to infinity with $K_a$. Together with $\Lambda K_n$, it again bounds $K$.
3. Bounded $R,R^{-1},K$ bound $M,T,Z$. The classical Gram matrix $\Sigma+iJ_0/2\succeq0$ forces $\Sigma$ to be strictly positive. Its determinant is conserved by (5), so it stays uniformly away from singularity. The smooth matrix equation therefore extends for all time.
4. Equation (5) bounds $S$ and its symplectic inverse. Conjugating position and derivative operators by the quadratic propagator then bounds every centered weighted derivative norm. Sobolev embedding supplies uniform centered Schwartz seminorm bounds for each fixed initial Schwartz field.

Let $W_{\max}=64\,512\,240\mu$. The single choice $\lambda=32W_{\max}$ gives a barrier $(98/81)W_{\max}>W_{\max}$. Every maintained branch consequently has an open neighborhood satisfying (6). Thus persistence is proved for nearby unrestricted smooth fields, not only for the exact profiles.

This is bounded localization, not convergence or orbital stability of a particular Hermite mode. Shape and orientation can change within the bounded family; some linear symmetry modes have secular drift. The theorem concerns the canonical action (1), not additional trajectories of a different second-order kinetic completion.

Conserved Shape Information

The exact quadratic closure also limits the dynamics. With $L=J_0\Sigma$, equation (5) implies $L'=[L,A^T]$. Its characteristic polynomial is constant, including all positive symplectic covariance eigenvalues $\nu_i$. On any complex fixed-total-degree Hermite shell of degree $m$, these are the eigenvalues of $\langle a_i^\dagger a_j\rangle+I/2$, so $\sum_i\nu_i-5=m$.

The isolated action consequently cannot connect exact single-profile Hermite shells of different total degree. This does not exclude every packet transition with radiation or another pattern, because the invariant concerns the complete field. It does explain why the localization theorem does not establish occupied-state selection or decay. Changing only the scalar function of global second moments preserves the same restriction; spatially varying or nonquadratic sources require a new evolution analysis.

Physical Scope and Reproducibility

One fixed smooth positive function $\mu(D)$ can replace the common $\mu$, preserving (4) with the specified values $\mu(d)$. The localization proof uses its positive minimum on $[0,10]$, with bounds recomputed. This realizes a declared scale prescription; it does not derive those scales or make their interpolation independent evidence.

The important constructive result is coexistence: one smooth flat-space action has genuine count-clock shell families, finite total energy, all-direction center motion and persistent bounded localization. Its global moments still couple distant patterns. A cluster-separating interaction, local propagation, spin, observed charge, physical sector scales and occupied-state selection require further work.

The branch checker and evolution checker retain the complete noncommuting variation, arbitrary complex shell controls, fourteen exact count inputs, covariance evolution, mode-response matrices and rational barrier bounds. The branch calculation passes 301 checks; the evolution calculation passes 158, and an independent review passes 40 exact checks. Their finite tests accompany the proofs above; no long particle simulation is used to infer persistence.

A Finite-Range Continuation

The windowed count action supplies a changed, amplitude-homogeneous continuum law with global evolution, exact separated-support additivity, finite-time spatial bounds and controlled initial approximation to the finite target shell family. Its locality properties and the exact parent count profiles belong to different actions: exact stationary branches and the parent bounded-localization theorem do not transfer automatically.