Responding Spatial Count Patterns and Their Persistence

Current interpretation. This article preserves an 18 October inverse construction. Its supplied count-valued spectral clocks and separately excited response modes are mathematical comparison models. They are not a realization of the current proposal of C portions sharing one fundamental frequency, whose additive average registration rate is C times that fundamental. Persistence and response conclusions below apply only to the action and domains actually written here; they do not transfer to a different action. The current particle-geometry account develops that distinction and the newer constructions.

An explicit conservative action can join a count clock, four independently changing spatial profiles, responding anisotropic support and a full spatial response of the same field. Its isolated circular families have a nonlinear persistence basin at fixed total phase charge, including the response contribution. Two patterns in this same action exchange charge and exert actual spatial forces. A controlled finite encounter theorem survives, while an exact coincident-source solution shows why positive energy and total charge alone do not ensure that each retained displacement remains nonzero forever.

Two related phase-neutral actions instead conserve each pattern's phase charge. One adds a positive direct overlap term and is repulsive where overlap decreases with separation. The other uses a common saturated source and has an attractive static cross contribution. Each has a precise sufficient two-core persistence budget. They change the temporal phase behavior as well as the force law.

This advances the previous persistence and response construction, which established spatial persistence and responding modes in separate actions. Here the spatial profiles, supporting geometry and response are varied in one action before proving its energy identity. The fixed-template pair calculations supply useful spectral comparisons, but their kinetic alternatives and coefficients are not silently transferred to this action.

All spatial directions are ordinary directions of one shared physical space, taken here as $\mathbb R^{10}$, without compactification. Components and coefficients describe $\Psi_\infty$; they are not additional spatial dimensions, mediator particles or position probabilities. The internal classical fields below remain distinct from the electron's literal external trajectory in its surrounding six-dimensional space. A finite normal envelope is an input to these examples, not a derivation of exact layer confinement.

Spatial carrier, clock and support

For $d\in\{2,3,4,5,6,10\}$, let $W_J$ be the complete real Hermite shell of degree $J$ in the $d$ active spatial directions, with a ground Gaussian in each remaining direction. Work at a specified finite cutoff,

$$\mathcal H_d=\bigoplus_{J=d}^{L}W_J,\qquad M_d=\binom{N_P}{d},\qquad M_d|_{W_J}=c_J I,\quad c_J=\binom Jd. \tag{1}$$

In dimensionless material coordinates, $a_j=(y_j+\partial_j)/\sqrt2$, $N_P=\sum_{j\le d}a_j^*a_j$. Summing all order-$d$ derivative products gives

$$\frac1{d!}\sum_{|\alpha|=d}\frac{d!}{\alpha!} (a^\alpha)^*a^\alpha=\binom{N_P}{d}.$$

Indeed the coefficient of $t^d$ in $\prod_j(1+t)^{\beta_j}$ is $\binom Jd$ on a product Hermite state of degree $J$. At $K=n+d-1$, its eigenvalue is the desired count $S(n,d)=\binom{n+d-1}{d}$. The cumulative prefix $N_P\le n-1$ has the same dimension, but is a different space. Neither identity selects its physical occupation.

Let $f_a\in S(\mathcal H_d)$, $a=(i,j)\in\{0,1\}^2$, be four independently variable real unit profiles. With orthogonal component vectors $e_a$, define

$$p_{ij}=\frac{1+(-1)^ik}{2}\frac{(1,x^2)_j}{1+x^2},\quad v=\sum_a\sqrt{p_a}f_ae_a,\quad \|v\|=1,$$ $$r>0,\quad |k|<1,\quad x>0,\quad G_k=\frac1{4(1-k^2)},\quad G_x=\frac1{(1+x^2)^2}. \tag{2}$$

The product weights and common temporal phase are explicit restrictions of the retained field description. Define the full weighted count moments

$$m=\sum_a p_a\langle f_a,M_df_a\rangle,\quad z_c=\sum_a p_a\langle f_a,M_d^2f_a\rangle,\quad \mathcal B=m+\eta(m)\frac{z_c-m^2}{m}, \quad \Lambda=\mu_d\mathcal B(1+\gamma w),\quad\lambda=\Lambda^2. \tag{3}$$

Here $\gamma>0$ is common and $\mu_d>0$ is supplied. The variance includes the differences between component means. Two coefficient branches are retained:

$$\eta(m)=1+\frac{(d!m)^{1/d}}d\quad\text{at any chosen finite cutoff}; \qquad \eta=128\quad\text{on }L=d+99. \tag{4}$$

Use $b=1/2$ and either of the following whole current laws:

$$w=\frac{1+k^2}{2}x^2-bx+\frac{b^2}{2}, \quad(k_0,x_0)=(0,1/2),$$ $$f_{\rm cur}(k)=\frac{k^2}{8(1+k^2)},\quad w_0=w-f_{\rm cur}(k_0)-f_{\rm cur}'(k_0)(k-k_0) +\frac14(k-k_0)^2, \quad(k_0,x_0)=(3/4,8/25). \tag{5}$$

Below $w$ means the selected law, each with $w=dw=0$ at its reference. The second is a changed common constitutive law, not a label-dependent adjustment.

For material position $Y$, common length $\ell$, center $X$, and $F\in GL^+(10)$, let

$$(U_{X,F}f)(X+FY)=(\det F)^{-1/2}f(Y),\quad q_{AB}=\ell^2\sum_a p_a\int\partial_Af_a\partial_Bf_a\,d^{10}Y,$$ $$S_q=I+\chi q/a_0^2,\quad T=a_0^2(\det S_q)^{-1/10}S_q, \quad H_F=F^TF,\quad V_s=\frac\kappa8\operatorname{tr}[(T^{-1}H_F-I)^2]. \tag{6}$$

The positive constants are common. Although the product in the trace need not be symmetric, it is similar to the symmetric strain $D=T^{-1/2}H_FT^{-1/2}-I$, so $V_s=\kappa\|D\|_F^2/8\ge0$. Also $\det T=a_0^{20}$. A rotation-covariant law thus supports anisotropic configurations without a fixed preferred axis.

Write $z=r n(\theta)\in\mathbb R^2$, $n=(\cos\theta,\sin\theta)$, and $Jn=(-\sin\theta,\cos\theta)$. The retained field is $z\otimes Uv$. Its specified material connection subtracts $(\partial_tU)U^{-1}$; it yields

$$T_i=\frac{|\dot z_i|^2}{2}+T_{{\rm shape},i}+T_{{\rm geo},i}, \quad T_{\rm shape}=\frac{r^2}{2} \left(G_k\dot k^2+G_x\dot x^2+\sum_a p_a\|\dot f_a\|^2\right),$$ $$T_{\rm geo}=\frac{I_F}{2}\|\dot F\|_F^2+\frac{M_0}{2}|\dot X|^2, \qquad I_F,M_0>0. \tag{7}$$

The response below remains in laboratory coordinates with its full time derivative. This material connection and positive collective inertia are supplied kinetic premises; they are not derived boost inertia. Additive overlapping packet kinetics also use the stated component decomposition and collective coordinates, rather than following automatically from a single scalar Cartesian field norm.

For dimensionless $t=t_{\rm phys}/t_0$, $X=X_{\rm phys}/\ell_0$, let the physical action be $A_*\int L\,dt$. Energies, phase charges and forces then have units $A_*/t_0$, $A_*$, and $A_*/(t_0\ell_0)$. Amplitude and $A_*$ remain uncalibrated; choosing a common occupied action is additional input.

Two actual source maps

Both constructions use a normalized source $s$, but they describe different laws:

$$s_A=U_{X,F}v\in L^2(\mathbb R^{10};\mathbb R^4), \qquad \rho=\sum_a p_a|U_{X,F}f_a|^2,\quad s_I=\rho/\|\rho\|_2\in L^2(\mathbb R^{10};\mathbb R). \tag{8}$$

The first retains signed spatial amplitudes and component contractions. The second is a normalized field intensity; squaring removes nodal signs. Neither is a probability law. The amplitude $r$ is outside these normalized shape maps. The source and its actual spatial adjoint are

$$B_s z=g\,s\otimes z,\qquad B_s^*y=g\int s\mathbin{\lrcorner}y\,d^{10}X. \tag{9}$$

There is one common real $g$, with no separately fitted return coefficient. The response occupies another real component block of $\Psi_\infty$, of dimension two or eight according to (8). These mathematical component counts are not spatial dimensions or additional particles.

Every geometric dependence is varied. At fixed laboratory position,

$$\delta s_A=U\left[\delta v-(F^{-1}\delta X)\cdot\nabla_Yv -(F^{-1}\delta F Y)\cdot\nabla_Yv -\tfrac12\operatorname{tr}(F^{-1}\delta F)v\right],$$ $$\delta v=\sum_a\left[\frac{\delta p_a}{2\sqrt{p_a}}f_a +\sqrt{p_a}\delta f_a\right]e_a, \quad\langle f_a,\delta f_a\rangle=0,$$ $$\delta s_I=\frac{\delta\rho-s_I\langle s_I,\delta\rho\rangle}{\|\rho\|_2}, \quad \delta\rho=\sum_a\delta p_a|Uf_a|^2+2\sum_a p_a(Uf_a)\delta(Uf_a). \tag{10}$$

Consequently $\langle s,\delta s\rangle=0$ for every allowed tangent. It cancels an isolated self force, but does not cancel the pair derivative $\langle\delta s_i,s_j\rangle$.

The common coherent action and full variation

Choose the flat laboratory response operator $L_s=\Omega^2 I$, $\Omega>0$, on the entire appropriate $L^2$ space. This includes every response component complementary to the instantaneous sources. It has no spatial-gradient propagation; it is an effective spatial response law, not a local causal master equation.

For one or two patterns the action is

$$L=\sum_i\left[T_i-V_{s,i}-\frac{r_i^2}{2}(\lambda_i+C_i)\right] +\frac{\|\dot y\|^2-\Omega^2\|y\|^2}{2} +g\sum_i\langle s_i\otimes z_i,y\rangle, \quad C_i=\frac{g^2}{\Omega^2-\lambda_i},\quad\lambda_i<\Omega^2. \tag{11}$$

Only self response is compensated. No cross overlap is canceled. Crucially, $C_i$ is a function on the whole current/profile neighborhood, not a constant assigned at a target shell. The inverse pole defines a genuine domain boundary.

At fixed $y$, its full potential differential for pattern $i$ is

$$\delta V_i=r_i(\lambda_i+C_i)\delta r_i +\frac{r_i^2}{2}\left[1+\frac{g^2}{(\Omega^2-\lambda_i)^2}\right]\delta\lambda_i +\delta V_{s,i}-g\langle\delta(s_i\otimes z_i),y\rangle. \tag{12}$$

For the clock derivatives,

$$\mathcal B_m=1+\eta'\frac{z_c-m^2}{m}-\eta-\eta\frac{z_c}{m^2}, \quad\mathcal B_{z_c}=\eta/m, \quad\eta'=\frac{(d!m)^{1/d}}{d^2m}$$

on the variable branch, and $\eta'=0$ on the fixed branch. Thus

$$\delta\lambda=2\Lambda\mu_d \left[(1+\gamma w)(\mathcal B_m\delta m+\mathcal B_{z_c}\delta z_c) +\gamma\mathcal B\delta w\right]. \tag{13}$$

Both $\delta m$ and $\delta z_c$ include all $\delta p_a$ and profile variations. With $Z_T=T^{-1}$ and $t_q=(\det S_q)^{-1/10}$, the support differential is

$$A_H=\frac\kappa4(Z_TH_FZ_T-Z_T),\quad A_T=\frac\kappa4(Z_TH_FZ_T-Z_TH_FZ_TH_FZ_T),$$ $$C_q=\chi t_q\left[A_T-\frac{A_T:S_q}{10}S_q^{-1}\right],\quad \delta V_s=A_H:\delta H_F+C_q:\delta q, \quad\delta H_F=\delta F^TF+F^T\delta F. \tag{14}$$

For general retained coordinates $q^A$, including the profiles in local sphere charts, write (7) as $H_{AB}\dot q^A\dot q^B/2$. Their complete equations and the response equation are

$$H_{AB}\ddot q^B+\Gamma_{A,BC}\dot q^B\dot q^C +\partial_A\sum_i\left[V_{s,i}+r_i^2(\lambda_i+C_i)/2\right] -g\sum_i\langle\partial_A(s_i\otimes z_i),y\rangle=0,$$ $$\ddot y+\Omega^2 y=g\sum_i s_i\otimes z_i. \tag{15}$$

Here $\Gamma$ is the Christoffel symbol of the first kind of the displayed kinetic metric. This compact form retains the current loads $-r^2\sum_a p_{a,s}\|\dot f_a\|^2/2$, profile momentum derivatives $(r^2p_a)'\dot f_a$, all frame rotations, and every radial/source derivative. It does not freeze the supports or profiles during an encounter.

Complete energy, charge and isolated circular states

The conserved energy and simultaneous temporal-rotation charge are

$$E=\sum_i[T_i+V_{s,i}+r_i^2(\lambda_i+C_i)/2] +\frac{\|\dot y\|^2+\Omega^2\|y\|^2}{2} -g\sum_i\langle s_i\otimes z_i,y\rangle,$$ $$Q=\sum_i(Jz_i)\cdot\dot z_i+\langle Jy,\dot y\rangle. \tag{16}$$

The response charge is essential. It is phase action, not a derived electric charge.

For one shell $K$, put

$$c=\binom Kd,\quad h=\mu_dc,\quad\Delta=\Omega^2-h^2>0, \quad a=g/\Delta,\quad Z=1+a^2.$$

Choose any four $f_a\in S(W_K)$, either current reference, $F^TF=T(q)$, and a constant center. Then, for $\sigma=\pm1$,

$$z=r_0n(\sigma ht+\theta_0),\quad y=a\,s\otimes z, \quad Q=\sigma hZr_0^2,\quad E=h^2Zr_0^2=h|Q|. \tag{17}$$

All equations hold: $\delta\lambda=\delta V_s=0$ on the constrained reference, and each remaining source variation is proportional to $\langle\delta s,s\rangle$. Radial balance reads $h^2=\lambda+C-ga$, with $C=ga$, and the response satisfies $a(\Omega^2-h^2)=g$. At fixed $A=|Q|$, $r_0^2=A/(hZ)$. Response changes the radius at fixed phase action, not the supplied clock. Both temporal signs have positive equal energy.

The family contains four full same-shell spheres and all 45 material rotations, with dimension $45+4[\binom{K+d-1}{d-1}-1]$ before phase and center coordinates. It is much larger than the rotation orbit of one shape.

An exact nonlinear persistence basin

Let $P_J$ project onto $W_J$, and define

$$t_J=\sum_a p_a\|P_Jf_a\|^2,\quad D_J=(c_J-c)/c, \quad a_c=\sum_Jt_JD_J,\quad b_2=\sum_Jt_JD_J^2.$$

Then $m=c(1+a_c)$, $z_c-m^2=c^2(b_2-a_c^2)$. If $s_{\rm leak}=\sum_{J\ne K}t_J$, Cauchy–Schwarz gives $a_c^2\le s_{\rm leak}b_2$ and $s_{\rm leak}\le b_2/\delta_c^2$. Use

$$\delta_c=d/K\ (K>d),\qquad\delta_c=d\ (K=d<L),$$ $$(\eta_U,R_g)= \begin{cases}(1+K/d,1/d),&\text{variable law},\\ (128,128-K/d),&\text{fixed law on its hundred-shell domain},\end{cases}$$ $$\alpha=\min(1/2,R_g/(16\eta_U)),\quad s_b=R_g/(8\eta_U),\quad \beta=\min(\alpha^2,\delta_c^2s_b). \tag{18}$$

If $L=K=d$, there is no spectral leakage and its barrier is omitted. Pascal's identity gives $c/(c-c_{K-1})=K/d$, while the binomial product bounds the variable $\eta(c)$ between $(K+1)/d$ and $1+K/d$. Thus $D_J+\eta(c)D_J^2\ge R_gD_J^2$. On $|a_c|\le1/2$, $H(a_c)=\eta(c(1+a_c))/(1+a_c)$ obeys $0<H\le2\eta_U$, $|H'|\le4\eta_U$. Therefore throughout $b_2\le\beta$,

$$\frac{\mathcal B}{c}-1=a_c+H(a_c)(b_2-a_c^2) \ge[R_g-4\eta_U|a_c|-2\eta_Us_{\rm leak}]b_2 \ge R_gb_2/2. \tag{19}$$

This controls the full nonlinear variance, including changes of weights; it is not merely a Hessian sign.

For either current law, take

$$|k-k_0|<\rho_k=(1-|k_0|)/2,\quad |x-1/[2(1+k^2)]|<1/8. \tag{20}$$

Completing its square gives $w\ge v_x^2/2+C_k(k-k_0)^2$, with $v_x=x-1/[2(1+k^2)]$, $C_k=1/16$ or $7/32$. The corresponding bounds are $p_a\ge1/260$ or $1/1040$, and boundary costs $\overline W=1/128$ or $7/2048$. In this basin,

$$\Lambda/h-1\ge R_gb_2/2+\gamma w\ge0. \tag{21}$$

Fix total $Q=\sigma A$, $A>0$. Direct completion of (16), without eliminating any response mode, yields

$$\begin{aligned} e:=E-hA={}&T_{\rm shape}+T_{\rm geo} +\tfrac12|\dot z-\sigma hJz|^2 +\tfrac12\|\dot y-\sigma hJy\|^2+V_s\\ &+\tfrac\Delta2\|y-a\,s\otimes z\|^2 +\frac{r^2(\lambda-h^2)}2 \left[1+\frac{g^2}{(\Omega^2-\lambda)\Delta}\right]. \end{aligned} \tag{22}$$

All terms are nonnegative on this basin below the pole. In particular, all response components outside the instantaneous source are retained.

A common explicit certificate for either source uses $\lambda_{\rm cap}=(\Omega^2+h^2)/2$, $E_0=hA$. Begin strictly inside (18),(20), with $r>0$, $F\in GL^+(10)$, $\lambda<\lambda_{\rm cap}$, finite retained velocities, and $y,\dot y\in L^2$. If

$$0\le e<\min\left\{ \frac{E_0}{4},\frac{A\Delta}{16h}, \frac{E_0R_g\beta}{16Z},\frac{E_0\gamma\overline W}{8Z}, \frac\kappa{32},\frac{E_0}{16Z}(\lambda_{\rm cap}/h^2-1) \right\}, \tag{23}$$

the full isolated evolution stays in the basin for every real time.

To see why, put $N=r^2+\|y\|^2$. Charge and the rotating kinetic squares give

$$|A-hN|\le\sqrt{2eN},\quad A/(2h)\le N\le2A/h,\quad N\le2Zr^2+4e/\Delta,$$ $$A/(8hZ)\le r^2\le2A/h. \tag{24}$$

Using (21) in (22) now gives the lower barrier

$$e\ge T_{\rm shape}+T_{\rm geo}+T_{\rm rot}+V_s +\tfrac\Delta2\|y-a\,s\otimes z\|^2 +\frac{E_0}{8Z}(R_gb_2/2+\gamma w), \tag{25}$$

where $T_{\rm rot}$ is the sum of both rotating kinetic squares. A first spectral, current or cap exit costs its corresponding term in (23). The material threshold keeps $\|D\|_F<1/2$, so $T/2<F^TF<3T/2$. These exits contradict conserved energy.

At each finite band, the profile spheres, nonzero weights, bounded nonsingular $F$, lower radius bound and pole margin bound the kinetic metrics and all source derivatives. Density normalization stays positive uniformly on the compact internal sets modulo translations. The response is a bounded linear oscillator forced by these smooth finite coordinates. Its Duhamel formula and the finite equations give locally Lipschitz evolution on the finite-coordinate/$L^2\times L^2$ phase space. The bounds prevent finite-time escape. Translation covariance makes the estimates independent of the center location; bounded center velocity allows drift without finite-time blowup. No compactness of an infinite-dimensional $L^2$ ball is assumed.

As $e\to0$, (24) and (22) imply $r^2\to A/(hZ)$, response convergence to its dressed source, and vanishing rotating velocity errors. Equation (25), shell projection and support following give proximity to the entire stationary family. Same-shell shapes, orientation and accumulated phase can drift. This is Lyapunov persistence modulo that family, without damping or a transverse gap along its neutral directions.

There is a separate valid large-gap certificate: with $\gamma=1$, both current tubes have $w<1$, and $\mathcal B/c<1+\alpha+2\eta_U\beta<3$, hence $\Lambda<6h$. Taking $\Omega=16\max_i h_i$ places every declared tube below $\Omega/2$. One can then omit the cap term and use the stricter radial thresholds $E_0/8$, $A\Delta/(32h)$, retaining the other three barriers. For arbitrary $\gamma$, one must recompute the whole-tube clock bound or use (23). It is not an arbitrary-gap theorem without a cap.

What spatial localization is controlled

The count moment controls the retained physical core. Concavity gives

$$\overline J=\sum_Jt_JJ \le d-1+(d!c)^{1/d}(1+\sqrt\beta)^{1/d} \le d-1+3K/2.$$

The harmonic oscillator identity then bounds the summed position and gradient integrals by $L_*=2d+3K+8$. Setting $S_*=1+\chi L_*/a_0^2$, the strain bound gives

$$\frac{\int |Z_{\rm phys}-X|^2|\Psi_{\rm ret}|^2\,d^{10}Z_{\rm phys}} {\|\Psi_{\rm ret}\|^2} \le\frac32\ell^2a_0^2S_*L_*, \quad \frac{\|\nabla\Psi_{\rm ret}\|^2}{\|\Psi_{\rm ret}\|^2} \le\frac{2S_*L_*}{a_0^2\ell^2}. \tag{26}$$

Here $Z_{\rm phys}=X+\ell Fy$ when $y$ is dimensionless material position. Physical position uses $F$, and the gradient uses $F^{-T}$. These are field-norm estimates, not probabilities.

The response obeys $\|y-a\,s\otimes z\|^2\le2e/\Delta$. Its dominant source is localized: for $s_A$ by (26), and for $s_I$ by the normalized finite-band Gaussian-polynomial density and compact internal bounds. The remainder is small in global $L^2$. A small arbitrary remainder may nevertheless have infinite second spatial moment or be centered arbitrarily far away. Consequently there is no uniform full-response second-moment theorem. An infinite Hermite-band evolution theorem, uniform higher-derivative envelope and local causal propagation also do not follow.

Exact coherent pair balances and finite encounters

For two patterns, only their simultaneous temporal rotation is generally a symmetry. The retained charges obey

$$Q_{{\rm ret},i}=r_i^2\dot\theta_i,\quad \dot Q_{{\rm ret},i}=g\langle s_i\otimes Jz_i,y\rangle, \quad\dot Q_y=-\sum_i\dot Q_{{\rm ret},i}. \tag{27}$$

There is no unique division of the overlapping response charge into two individual dressed charges. The isolated theorem therefore cannot be applied separately to each core during a coherent encounter.

The exact center force and translation momentum are

$$M_0\ddot X_i=g\langle\partial_{X_i}s_i\otimes z_i,y\rangle, \quad P=M_0\sum_i\dot X_i-\int\dot y\cdot\nabla y\,d^{10}X. \tag{28}$$

Where this momentum integral exists, differentiation and integration by parts give $\dot P=0$. The full $L^2$ existence result alone does not guarantee gradient momentum for every datum. Common rotations similarly conserve center, deformation and response angular momentum. Instantaneous center forces need not be opposite: the response supplies the missing momentum balance.

For an instantaneous sum of isolated dressed preparations $y=\sum_jg(s_j\otimes z_j)/\Delta_j$, $\Delta_j=\Omega^2-h_j^2$, put $\rho_{12}=\langle s_1,s_2\rangle$. Then

$$F_{X_i}=\frac{g^2(z_i\cdot z_j)}{\Delta_j}\partial_{X_i}\rho_{12}, \quad \dot Q_{{\rm ret},i}=\frac{g^2\rho_{12}}{\Delta_j}(Jz_i)\cdot z_j, \quad j\ne i. \tag{29}$$

Both signs $z_i=r_in(\sigma_i h_it+\phi_i)$ and their actual frequency difference are retained. Different clocks are not synchronized. Equation (29) is an initial-data evaluation, not a pair trajectory with all nonzero shape forces discarded.

At frozen geometry the full response pencil, with squared frequency $\zeta$, is

$$\mathcal F_{ii}(\zeta)=h_i^2+g^2/\Delta_i-\zeta-g^2/(\Omega^2-\zeta), \quad\mathcal F_{12}=-g^2\rho_{12}/(\Omega^2-\zeta),$$ $$-\mathcal F'(\zeta)=I+\frac{g^2\operatorname{Gram}(s_1,s_2)}{(\Omega^2-\zeta)^2}. \tag{30}$$

This is the complete dynamic norm. Its full normal modes include upper response branches. Frozen-mode or justified adiabatic forces use $-I\partial_X\nu$ at fixed complete modal action; they are not an automatic theorem for the moving family with shape zero directions. Equal-clock low symmetric and antisymmetric modes can attract or repel, respectively, when the relevant overlap decreases. Detuned dispersive shifts also survive a zero first beat average. None is a universal force sign for arbitrary populations.

For the co-rotating equal-clock leading normal form,

$$H_{\rm res}=h(I_1+I_2)-J_{\rm ex}(a_1^*a_2+a_2^*a_1), \quad J_{\rm ex}=\frac{g^2\rho_{12}}{2h\Delta Z},\quad I_i=|a_i|^2,$$ $$\dot I_1=-2J_{\rm ex}\operatorname{Im}(a_1^*a_2),\quad \dot I_2=-\dot I_1,\quad \frac d{dt}\operatorname{Re}(a_1^*a_2)=0. \tag{31}$$

The last invariant holds for every real time-dependent $J_{\rm ex}(t)$, even with unequal actions. Initial quadrature can therefore keep zero leading force while charge exchanges. Detuning changes that derivative to $-(h_1-h_2)\operatorname{Im}(a_1^*a_2)$. At constant coupling the detuned two-mode maximum transfer fraction is $4J_{\rm ex}^2/[(h_1-h_2)^2+4J_{\rm ex}^2]$. For fixed nonzero detuning, integration by parts bounds a first-order transfer amplitude by $(2\sup|J_{\rm ex}|+\int|\dot J_{\rm ex}|dt)/|h_1-h_2|$. These are frozen or leading-order controls, not full nonlinear long-time results; opposite temporal signs need their actual real equations.

Two direct finite-time comparison theorems require no individual invariant. In a specified regular tube of the full finite-coordinate/$L^2$ phase space, let $L_V$ bound the vector-field derivative. Comparison with an uncoupled $g=0$ trajectory, at initial distance $b_0$, gives

$$\operatorname{dist}(t)\le e^{L_V|t|}b_0 +C|g|\frac{e^{L_V|t|}-1}{L_V}. \tag{32}$$

Here $C$ bounds $\|V_g-V_0\|/|g|$ in that tube. The right side must be less than half the reference distance to every spectral, current, support, radius and pole boundary. Initially dressed data incur an $O(g)$ initial distance from zero response.

Alternatively, use two independently dressed circles with fixed reference geometries and their summed response as the comparison curve. Its response equation and self equations hold exactly; its remaining residual is entirely cross force, of order $g^2$. If $D_V$ bounds its residual norm divided by $g^2$, and $K_V$ bounds the vector-field derivative on a radius-$a_T$ tube, then

$$\operatorname{dist}(t)\le e^{K_V|t|}\epsilon_0 +g^2D_V\frac{e^{K_V|t|}-1}{K_V}<a_T. \tag{33}$$

At zero derivative bound the fractions become $|t|$. The constants include source derivatives, all inverse kinetic metrics and the response domain; small scalar overlap alone does not bound them. Gronwall's inequality prevents the first exit while these conditions hold. For every fixed finite $T$, a sufficiently small common $g$ and initial error provide a nonempty encounter interval, including actual center acceleration and profile/support changes. These are computable supremum definitions, not calculated observed durations or estimates uniform in $T$.

There is also an exact charge-loss budget. While both clock tubes hold, let $\lambda_{\min}=\min_i h_i^2$ and require $2g^2/\Omega^2\le\lambda_{\min}/2$. Completing the full pair potential gives

$$E\ge T_{\rm all}+\frac{\lambda_{\min}}4(r_1^2+r_2^2) +\frac{\Omega^2}{2}\left\|y-\frac g{\Omega^2}\sum_i s_i\otimes z_i\right\|^2.$$

Thus

$$C_Q=2|g|\sqrt{E/\lambda_{\min}} \left[\frac{\sqrt{2E}}\Omega+\frac{4|g|\sqrt{E/\lambda_{\min}}}{\Omega^2}\right], \quad |Q_{{\rm ret},i}(t)-Q_{{\rm ret},i}(0)|\le C_Q|t|. \tag{34}$$

For $|t|<\min_i|Q_{{\rm ret},i}(0)|/(2C_Q)$, each retains its sign and at least half its initial magnitude, and $r_i\ge |Q_{{\rm ret},i}(0)|/(2\sqrt{2E})$. Combine this interval with (32) or (33), which supplies the required clock tubes. Incoming response energy is included in $E$; the bound can be conservative, especially for very different clocks.

An exact invariant transfer solution, and its chart boundary

Choose equal depths and shells, identical four reference profiles, current reference, center, deformation and source-component identification. Then $s_1=s_2=s$. Set the geometric velocities to zero and $y=s\otimes Y(t)$. For every shape/current/support/center tangent, $\langle\delta s_i,s\rangle=0$, while $\delta\lambda_i=\delta V_{s,i}=0$. All spatial forces vanish for arbitrary temporal $z_i,Y$, and the kinetic loads also vanish with their geometric velocities. This is an invariant subsystem of both unchanged coherent source actions, not a generic frozen-force approximation.

Writing $k_h=h^2+g^2/\Delta$, its equations are

$$\ddot z_1+k_hz_1=gY,\quad\ddot z_2+k_hz_2=gY, \quad\ddot Y+\Omega^2Y=g(z_1+z_2). \tag{35}$$

The dark coordinate $z_-=(z_1-z_2)/\sqrt2$ has frequency $\omega_a=\sqrt{k_h}$. The low bright coordinate $z_+=(z_1+z_2)/\sqrt2$ has

$$\omega_s^2=\frac{k_h+\Omega^2-\sqrt{(\Omega^2-k_h)^2+8g^2}}2. \tag{36}$$

Assume $g\ne0$ and $k_h\Omega^2>2g^2$, ensuring $0<\omega_s<\omega_a$. With amplitude $R>0$,

$$z_+=Rn(\omega_st+\phi_s),\quad z_-=Rn(\omega_at+\phi_a), \quad Y=b_sRn(\omega_st+\phi_s), \quad b_s=\frac{\sqrt2g}{\Omega^2-\omega_s^2},\quad Z_s=1+b_s^2. \tag{37}$$

The upper bright mode and all response complements remain in the action but are unexcited in this solution. The complete conserved quantities are

$$E=R^2(\omega_s^2Z_s+\omega_a^2),\quad Q=R^2(\omega_sZ_s+\omega_a). \tag{38}$$

There is no single $hQ$ relation for this two-frequency state. Reversing both temporal signs reverses $Q$ and preserves $E$.

For $\delta=(\omega_a-\omega_s)t+\phi_a-\phi_s$,

$$r_{1,2}^2=R^2(1\pm\cos\delta),\quad Q_{{\rm ret},1,2}=\frac{R^2(\omega_s+\omega_a)}2(1\pm\cos\delta). \tag{39}$$

Start at $\delta=\pi/2$. Both squared radii lie between $R^2/2$ and $3R^2/2$ for $|t|\le\pi/[6(\omega_a-\omega_s)]$. The first forward zero $z_1=0$ occurs at $t_* =\pi/[2(\omega_a-\omega_s)]$. However,

$$|\dot z_1(t_*)|^2=\frac{R^2(\omega_a-\omega_s)^2}{2}>0, \qquad T_{{\rm ret},1}(t_*)=\frac{R^2(\omega_a-\omega_s)^2}{4}. \tag{40}$$

Vanishing retained displacement and phase charge do not mean vanishing kinetic field data. The regular $r>0$ phase/profile chart reaches its boundary. The restricted Cartesian oscillator solution continues smoothly, but this does not prove unique generic continuation of the full degenerate profile chart through $r=0$. It is not annihilation, an observed decay or a separated-particle interaction. As $g\to0$, these preparations approach independently dressed circles and the beat time diverges as $O(g^{-2})$. The example excludes an unrestricted permanent-nonzero-displacement theorem based only on total charge and positive energy; it leaves more restrictive separated-core persistence questions open.

Two phase-neutral replacements with separate charges

Both replacements use the positive normalized density $s_I$, a phase-neutral scalar response $u$, and the same retained spatial kinetic energy and base potential $V_{\rm base}=\sum_i[r_i^2\Lambda_i^2/2+V_{s,i}]$. For source functions $J_i$, write the common template explicitly as

$$L_\xi=\sum_iT_i-V_{\rm base} +\frac{\|\dot u\|^2-\Omega^2\|u\|^2}{2} +\langle J_1+J_2,u\rangle -\frac{\|J_1\|^2+\|J_2\|^2}{2\Omega^2} -\xi\frac{\langle J_1,J_2\rangle}{\Omega^2}. \tag{41}$$

For one pattern omit absent-pattern terms. The repulsive law is $\xi=2,\ J_i=gr_i^2s_{I,i}$. The attractive saturated law is $\xi=0,\ J_i=g f(r_i)s_{I,i}$, with one common new scale $r_*>0$, $f(r)=r^2/(r_*^2+r^2)$. These are different actions, not two approximations to (11).

Each $\theta_i$ is independently cyclic, so $Q_i=r_i^2\dot\theta_i$ is separately conserved. The scalar response transforms trivially under these temporal rotations; its phase charge is zero by representation, not by omission. Its full equation is $\ddot u+\Omega^2u=J_1+J_2$. The complete energy is

$$E=E_{\rm base}+\frac{\|\dot u\|^2}{2} +\frac{\Omega^2}{2}\left\|u-\frac{J_1+J_2}{\Omega^2}\right\|^2 +(\xi-1)\frac{\langle J_1,J_2\rangle}{\Omega^2}, \tag{42}$$

where $E_{\rm base}=\sum_i[T_i+r_i^2\Lambda_i^2/2+V_{s,i}]$. Response energy and spatial momentum remain present.

At fixed response the exact interaction force covector for any variation of pattern $i$ is

$$\langle\delta J_i,u\rangle -\frac{\langle\delta J_i,J_i+\xi J_j\rangle}{\Omega^2}. \tag{43}$$

For the repulsive law, $\delta J_i=g[2r_is_i\delta r_i+r_i^2\delta s_i]$. For the attractive law,

$$\delta J_i=g[f'(r_i)s_i\delta r_i+f(r_i)\delta s_i],\quad f'(r)=\frac{2rr_*^2}{(r_*^2+r^2)^2}. \tag{44}$$

The self radial derivatives in (43) are nonzero and must be included. Equations (10),(13),(14) still give every source/profile/current/support derivative. At an isolated static response $u=J_i/\Omega^2$, all self derivatives cancel, including radial ones. The original circular family has $r_i^2=|Q_i|/h_i$, $\dot\theta_i=\operatorname{sign}(Q_i)h_i$, $E_i=h_i|Q_i|$. Its static response has nonzero norm but no reference kinetic energy. It does not carry the coherent action's rotating factor $Z$.

At the instantaneous static pair response, (43) reduces to $(1-\xi)\langle\delta J_i,J_j\rangle/\Omega^2$. Thus the repulsive source gives $-\partial_{X_i}\langle J_1,J_2\rangle/\Omega^2$, while the attractive source gives the opposite sign. These names refer to intervals where the actual positive density overlap decreases with separation; nodal profiles can make that overlap nonmonotone. During motion the full response equation applies. Static elimination is not imposed.

For fixed nonzero $Q_i$, set $A_i=|Q_i|$, $E_{i0}=h_iA_i$. Inside each parent basin the exact nonnegative excess is

$$P_i=T_{{\rm red},i} +\frac12(A_i/r_i-r_i\Lambda_i)^2 +A_i(\Lambda_i-h_i)+V_{s,i},$$ $$B_i=\min\{E_{i0}R_{g,i}\beta_i/2, E_{i0}\gamma\overline W_i,\kappa/32\}. \tag{45}$$

Here $T_{\rm red}$ includes radial, shape, deformation and center kinetic energy, and the absent spectral barrier is again omitted in the one-shell case.

For the repulsive law, $\langle J_1,J_2\rangle\ge0$, so

$$e_{\rm rep}=E-E_{10}-E_{20} =P_1+P_2+\frac{\|\dot u\|^2}{2} +\frac{\Omega^2}{2}\left\|u-\frac{J_1+J_2}{\Omega^2}\right\|^2 +\frac{\langle J_1,J_2\rangle}{\Omega^2}. \tag{46}$$

Strict initial parent tubes and $0\le e_{\rm rep}<\min(B_1,B_2)$ prevent either first exit. Fixed nonzero charges and radial squares bound both radii; the finite-band metrics and bounded-response continuation yield all-time two-core persistence. Reference cores with static summed response become admissible at sufficiently large separation because their Gaussian density overlap tends to zero, even at fixed common $g$. Finite contact must pay its actual positive overlap energy.

For the attractive saturated law, $0\le f<1$, $0\le\rho_{12}\le1$, hence $0\le\langle J_1,J_2\rangle\le g^2$. Put $b_g=g^2/\Omega^2$. Then

$$e_+=E-E_{10}-E_{20}+b_g =P_1+P_2+\frac{\|\dot u\|^2}{2} +\frac{\Omega^2}{2}\left\|u-\frac{J_1+J_2}{\Omega^2}\right\|^2 +b_g-\frac{\langle J_1,J_2\rangle}{\Omega^2}. \tag{47}$$

Strict parent tubes and $0\le e_+<\min(B_1,B_2)$ give the same sufficient persistence result. Saturation supplies the uniform negative-energy allowance; an unsaturated attractive quartic is not covered by this argument. With reference cores, zero center velocities and static summed response,

$$e_+=b_g[1-f(r_1)f(r_2)\rho_{12}]. \tag{48}$$

One common coupling with $b_g<\min(B_1,B_2)$ therefore supplies admissible data at every separation, and an open neighborhood by continuity. For a finite collection of specified nonzero charges, one may use the minimum over all fourteen positive barriers without per-particle retuning. At fixed nonzero $g$, however, separating the cores makes $e_+\to b_g$, not zero. Arbitrarily precise proximity does not follow from separation alone; it requires a smaller common coupling or a demonstrably smaller actual shifted excess.

Both replacements permit either temporal sign and unequal clocks because of their separate symmetries. Both lose temporal phase sensitivity and coherent phase-charge exchange. Their all-time two-core theorems cannot be transplanted back to (11). Response complements remain controlled in norm, with the same localization qualification as before.

The gradient-response conditional branch

For a fixed positive operator $L_s$, such as $\Omega^2-c_s^2\Delta_X$, replace the self term in (11) by

$$C=\langle b,(L_s-\lambda)^{-1}b\rangle,\quad b=gs, \quad R_\lambda=(L_s-\lambda)^{-1}.$$

Its variation is

$$\delta C=2\langle\delta b,R_\lambda b\rangle +\langle b,R_\lambda^2b\rangle\delta\lambda, \tag{49}$$

with an additional $-\langle b,R_\lambda(\delta L_s)R_\lambda b\rangle$ if the operator varies. The isolated circular response is $y=rR_{h^2}b\otimes n$, with complete norm $Z=1+\|R_{h^2}b\|^2$. The source-derivative terms cancel the reference self force. The rotating identity replaces the response square by $\|(L_s-h^2)^{1/2}(y-rR_{h^2}b\otimes n)\|^2/2$, and its final bracket by $1+\langle b,R_\lambda R_{h^2}b\rangle$. These are stationary and form-domain identities below the gap.

For the gradient operator the form domain is $H^1$, operator domain $H^2$. Its filtered overlap is an actual Fourier integral with denominator $\Omega^2+c_s^2|p|^2-\zeta$, not a flat overlap divided by a constant. Equal target clocks do not guarantee equal self functions or exchange parity for different source shapes. A weak-cross normalized reduction using $C_{12}(h^2)/\sqrt{Z_1Z_2}$ needs derivative control as well; smallness of that one value alone is insufficient. Keep the full pencil and dynamic cross norm if cancellation supplies the small value.

The accepted all-time evolution theorem in this note is the flat-response theorem. The gradient branch additionally needs wave regularity and uniform source derivatives in its energy norm. The identities do not establish that theorem, causal completion of the whole effective action, or continuation through a response pole. No pending regularity result is assumed here.

All-depth controls and their evidential limits

The actual supplied labels are:

In archived energy-equivalent MeV units, with $\hbar=1$ for the numerical prescription,

$$m_e=0.51099895,\quad \mu_2=m_e\sqrt{2890},\quad\mu_3=m_e\sqrt{32\sqrt7}, \quad\mu_4=\frac{m_e}{15}\sqrt{48/\sqrt7},$$ $$\mu_6=\mu_{10}=m_e/18564,\quad \mu_5=10^{-6}\sqrt{\frac{7.49\times10^{-5}}{11628^2-2002^2}}, \quad h_i=\mu_{d_i}c_i. \tag{50}$$

These are dated supplied scales, not current empirical lookups or derived force constants. The neutrino unit is a separate solar-splitting calibration. No finite-chain correction belonging to another action is inserted. The photon zero and conditional bottom geometric mean are outside this positive-count chart.

Let $H=\max_i h_i$, $h_{\min}=\min_i h_i$. Two separately executed common witnesses use $\Omega=2H$ with the cap theorem, or $\Omega=16H$ with the whole-tube large-gap theorem, each with $g=\Omega h_{\min}/10$. Both give $2g^2/\Omega^2=h_{\min}^2/50$, sufficient for global pair-energy positivity and the charge budget in the tubes. A chosen finite-encounter estimate may require reducing this one common $g$ further. These small couplings demonstrate existence, not measured interaction strengths. Equal-unit arithmetic controls are separate from the actual clock comparison.

The signed-source control uses Hermite indices $(n,1,\ldots,1,0,\ldots,0)$, with every active direction excited and its actual support $q=\operatorname{diag}(n+1/2,3/2\ [d-1],1/2\ [10-d])$, $F=\sqrt{T(q)}$. At depths two through six, swapping one active odd direction with a normal direction through $\pi/2$ gives exact zero overlap by parity, even with unequal widths. At depth ten, rotating two equal-width $h_1$ factors gives overlap $\cos(2\varphi)$, zero at $\varphi=\pi/4$. Aligned centered different-depth members also have an unmatched odd direction. Separation or rotation can remove that cancellation. Along a single degree-$K$ excited direction, the translated amplitude overlap is

$$\rho_K(R)=e^{-R^2/4}L_K(R^2/2),\qquad \rho_2(R)=e^{-R^2/4}(1-R^2+R^4/8). \tag{51}$$

Here $R$ is material separation; the physical gradient includes $F^{-T}$. Signed nodes permit both signs and actual torques; one zero overlap does not prove all derivatives vanish.

The density-source controls instead distribute $K$ as evenly as possible over the $d$ active Hermite indices $\beta_j$, use normal ground factors and the actual stationary support. With $a_0=\chi=1$, $F_j^2=(\beta_j+3/2)/[\prod_k(\beta_k+3/2)]^{1/10}$. All 105 unordered including-identical overlaps at separation $\ell_0/2$ along the first active axis were independently recomputed, agreeing within $5.7232\times10^{-14}$. This is a finite set of actual supported geometries, not all possible orientations or encounters. The simple equal-width density controls are

$$\langle s_{h_1},s_{h_0}\rangle=1/\sqrt3,\quad D(R)=e^{-R^2/2}(1-2R^2/3+R^4/3). \tag{52}$$

An active/normal swap gives $1/3$ at equal widths. Density overlap is positive, but need not be monotone because of nodes. Force signs use its actual derivative.

The 91 distinct-label detuning controls are clock diagnostics, not 91 nonzero spatial couplings: a signed source integral can vanish. The 105 density overlaps likewise do not establish interaction rates. Depth five follows the same laws and can share parity zeros or positive density overlap with other depths. No uniquely protected depth-five axis, weak-interaction rarity or photon polarization law follows. The construction supplies compatibility at all maintained depths, including six and ten, rather than new empirical dimension selection.

Reproducibility and remaining physical inputs

The accepted original calculations contain 1,567 checks for the signed-source construction and 366 for the density-source construction. A separate 530-check comparison rederived the shared identities, exact transfer and boundary velocity, both neutral completions, all fourteen clock rows, 91 diagnostic detunings and 105 spatial overlaps. A further independently implemented 152-check backcheck verified the new transfer, cap and neutral-budget results. Each was replayed unchanged; these are separate executed scopes, not a sum of independent physical confirmations. No full nonlinear encounter time integration or observed lifetime was computed.

A new standalone scoped checker and its results verify selected displayed identities with exact rational arithmetic, check the exact invariant oscillator solution and nonzero boundary velocity numerically, and check the fourteen count labels. It uses only Python's standard library and no private research inputs. It does not rerun the original 1,567, 366, 530 or 152 checks, the 105 spatial quadratures, or a nonlinear persistence simulation. The provenance record identifies its exact executed bytes and distinguishes the original evidence scopes.

The supplied clock units and occupied labels, finite band, normalized source representation, four-profile/common-phase restriction, material connection, support and variance laws, inverse compensation and collective center inertia remain inverse inputs. The new neutral laws additionally supply either a direct cross term or a saturation scale. The results establish useful same-action stationarity, finite-band persistence, full-response accounting and conditional pair mechanisms. Physical boost inertia, fundamental field content, layer confinement, electric charge, spin, measured coupling strengths and observed persistence still require derivation. No photon identification, completed mass operator or empirical preference for particular depths is claimed.