An Amplitude-Homogeneous Finite-Range Count Action

The common flat-space count action has exact count-clock families and a bounded-localization theorem, but its global moments couple arbitrarily distant patterns. A different action replaces those global moments by smoothly windowed moments and averages over window anchors. With an explicit common cap, the resulting continuum equation is globally well posed, exactly independent of overall field amplitude, and additive for sufficiently separated compact supports. Its complete initial source can approximate a specified finite set of parent count-clock shells arbitrarily accurately.

These properties coexist in one changed action. The parent stationary solutions and localization theorem are not inherited. In particular, the initial approximation proved below supplies neither an exact new stationary branch nor a particle lifetime.

The coordinates are ordinary physical coordinates in shared flat $\mathbb R^{10}$, used as the working geometry. Depth classifies a pattern's dimensional involvement and extracted widths. It does not name a separate physical space. The field below is a classical complex field, equivalently two real quadratures of one field. Spatial integrals and spectral coefficients supply mathematical descriptions, not physical motion in a Hilbert space, a Born law, or mediator particles. An internal field profile remains distinct from an electron's deterministic six-dimensional trajectory and an observer's time-averaged registration.

Windowed Moments and the Parent Constitutive Function

Write $D_i=-i\partial_i$, and reserve $D=10$ for the physical spatial dimension. Let $J_\epsilon$ be convolution by a nonnegative radial smooth kernel $j_\epsilon$, supported in $B(0,\epsilon)$, with integral one. For example, normalize $\exp[-1/(1-|x|^2/\epsilon^2)]$ inside that ball and set it to zero outside. Nonnegativity gives $\|j_\epsilon\|_1=1$, hence $\|J_\epsilon\|_{2\to2}\le1$. Radiality gives rotation symmetry and $J_\epsilon^*=J_\epsilon$. A real even signed kernel would not suffice for the contraction claim.

Take a real radial smooth window $0\le\chi\le1$, equal to one on $B(0,R_0)$, zero outside $B(0,R_1)$, where $R_1=R_0+w$ and $w>0$. Set $C_\chi=\int\chi^2$. For each anchor $x$, define

$$u=J_\epsilon\psi,\qquad \phi_x(y)=\chi(y-x)u(y),\qquad z=y-x,$$ $$n_x=\int|\phi_x|^2,\quad c_x=\frac{\int z|\phi_x|^2}{n_x},\quad p_x=\frac{\operatorname{Re}\int\phi_x^*D\phi_x}{n_x},$$ $$M_x=\frac{\int zz^T|\phi_x|^2}{n_x}-c_xc_x^T,\qquad (T_x)_{ij}=\frac{\operatorname{Re}\int(D_i\phi_x)^*D_j\phi_x}{n_x} -(p_x)_i(p_x)_j. \tag{1}$$

Ratios are evaluated only when $n_x>0$. Every windowed field is smooth and compactly supported even for $\psi\in L^2$. A nonzero window has positive definite $M_x,T_x$, with $T_x\succeq M_x^{-1}/4$.

For any such moments suppress the anchor label and define

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

Here $R$ is a positive width matrix, not the window radius. Choose common constants $0<b_1<b_0$, $\mu,\Lambda,\lambda,c_{\rm s}>0$, and

$$t(s)=\frac{s-b_0}{b_1-b_0},\quad g(s)=\frac{t(s)^2}{t(s)^2+[t(s)-1]^2},\quad V(s)=\lambda t(s)^2[t(s)-1]^2.$$ $$d_{\rm eff}=\operatorname{tr}g(R),\quad K_a=\operatorname{tr}[Rg(R)M]-\frac{d_{\rm eff}}2,\quad K_n=\operatorname{tr}(RM)-5-K_a,$$ $$\mathcal F(a,d)=\mu\frac{\Gamma(a+d+1)}{\Gamma(a+1)\Gamma(d+1)},\quad W=\mathcal F(K_a,d_{\rm eff})+\Lambda K_n+\operatorname{tr}V(R), \quad h=\sqrt{W^2+c_{\rm s}^2|p|^2}. \tag{2}$$

The Gram inequality implies $K_a,K_n\ge0$, $0\le d_{\rm eff}\le10$ and $W\ge\mu$. These facts keep the Gamma quotient and square root regular on attained moments. The count function, widths and coefficients are constitutive inverse inputs. They are not derived physical scales.

For a rank-$d$ ordinary spatial projector $P$, the parent action has joint Hermite shells at $B=b_1P+b_0(I-P)$, with active degree $k$ and remaining-direction degree $l$. Their exact parent clocks are $\omega_{d,k,l}=\mu\binom{k+d}{d}+\Lambda l$. At $l=0$, the shell multiplicity is $\binom{k+d-1}{d-1}$, and the cumulative count through $k$ is $\binom{k+d}{d}=S(k+1,d)$. The approximation target below is a finite union of these parent rest shells at $d=2,3,4,5,6,10$, rather than an asserted exact shell of the changed action.

An Explicit Common Cap and One Action

Let $a(t)=e^{-1/t}$ for $t>0$, and zero otherwise. One smooth step is $\theta(s)=a(2-s)/[a(2-s)+a(s-1)]$: it is one for $s\le1$, zero for $s\ge2$, and lies between them. Define

$$\eta=\theta\!\left(\frac{\operatorname{tr}M}{L_M}\right) \theta\!\left(\frac{\operatorname{tr}M^{-1}}{L_{M^{-1}}}\right) \theta\!\left(\frac{\operatorname{tr}T}{L_T}\right) \theta\!\left(\frac{\operatorname{tr}T^{-1}}{L_{T^{-1}}}\right) \theta\!\left(\frac{|p|^2}{L_p}\right),\qquad f=\eta(h-\mu). \tag{3}$$

All five bounds are common. For the dimensionless toy widths $b_1=1,b_0=4$, the explicit choice

$$(L_M,L_{M^{-1}},L_T,L_{T^{-1}},L_p)=(1000,100,4000,30,1) \tag{4}$$

strictly contains every target joint shell with narrow degree zero and active degree $k\le94$, including arbitrary complex superpositions and spatial rotations. To see this, for the occupation Gram matrix $Q_{ij}=\langle a_i^\dagger a_j\rangle$, fixed total degree gives $0\preceq\operatorname{Re}Q\preceq kI$ and vanishing $\langle a_i a_j\rangle$. If $B=b_1P+b_0(I-P)$, then

$$M=B^{-1/2}(\operatorname{Re}Q+I/2)B^{-1/2},\qquad T=B^{1/2}(\operatorname{Re}Q+I/2)B^{1/2},\qquad p=0.$$

Thus $I/(2b_0)\preceq M\preceq(k+1/2)I/b_1$ and $b_1I/2\preceq T\preceq b_0(k+1/2)I$. In (4), the four trace ratios are bounded by $189/200,4/5,189/200,2/3$, respectively. The finite union of normalized target shells and rotation parameters is compact, so a common graph neighborhood also lies where $\eta=1$. This concerns shapes; their physical amplitudes remain arbitrary. The cap does not preserve the unlimited ladder or arbitrary boosts. It introduces no depth-five factor.

Extend $f$ by zero at singular-matrix limits, where the inverse-trace factors already switch it off. Only an open smooth extension near the compact set of attained moments is needed; arbitrary unphysical matrix pairs need not obey the parent inequalities. At a zero window the entire integrand below is defined as zero, before any division or matrix inversion.

The changed action is

$$H[\psi]=\mu\|\psi\|_2^2+C_\chi^{-1}\int n_xf(M_x,T_x,p_x)\,dx, \qquad S=\int dt\left[\frac i2\int(\psi^*\dot\psi-\dot\psi^*\psi)-H\right]. \tag{5}$$

Fubini gives $\int n_xdx=C_\chi\|J_\epsilon\psi\|_2^2$, hence $\mu N\le H\le\mu N+\|f\|_\infty\|J_\epsilon\psi\|_2^2$, where $N=\|\psi\|_2^2$. There is no norm floor. For any complex constant $a$, $H[a\psi]=|a|^2H[\psi]$; the equation below obeys $G[a\psi]=aG[\psi]$. Scaling a solution changes neither its frequency nor its normalized shape dynamics.

Complete Variation

Write $df=\operatorname{tr}(C_f\,dM+E_f\,dT)+v_f\cdot dp$. These derivatives include the cap and the complete response of $R$. For an explicit implementation of the latter, solve $MR Y_A+Y_A RM=A$ for symmetric $A$. Then

$$\operatorname{tr}(A\,dR)=\operatorname{tr}(Y_A\,dT-RY_AR\,dM).$$

Put $s_g(s)=sg(s)$. Its matrix Fréchet derivative has divided difference $[s_g(r_i)-s_g(r_j)]/(r_i-r_j)$, with diagonal value $s_g'(r_i)$. Set

$$Y_a=Y_{\mathcal L_{s_g}(R)[M]-g'(R)/2},\quad Y_d=Y_{g'(R)},\quad Y_V=Y_{V'(R)},$$ $$E=\frac\Lambda2R^{-1}+(\mathcal F_a-\Lambda)Y_a+ \mathcal F_dY_d+Y_V, \quad C=\frac\Lambda2R+(\mathcal F_a-\Lambda)[s_g(R)-RY_aR] -R(\mathcal F_dY_d+Y_V)R.$$

Here $dW=\operatorname{tr}(C\,dM+E\,dT)$, so $h_M=(W/h)C$, $h_T=(W/h)E$, $h_p=c_{\rm s}^2p/h$. Finally $df=\eta\,dh+(h-\mu)d\eta$, which specifies every derivative in the capped action. Off-diagonal divided differences remain even when the scalar derivatives vanish at individual wells.

With the physical centroid $X_x=x+c_x$, define

$$\mathcal L_x=(y-X_x)^TC_f(y-X_x)+(D-p_x)^TE_f(D-p_x) -\operatorname{tr}(C_fM_x+E_fT_x)+v_f\cdot(D-p_x),$$ $$g_x[u]=\chi_x[f+\mathcal L_x]\phi_x,\qquad i\dot\psi=G[\psi]=\mu\psi+C_\chi^{-1}J_\epsilon^* \int g_x[J_\epsilon\psi]\,dx, \quad dH[v]=2\operatorname{Re}\langle v,G\rangle. \tag{6}$$

Every derivative acts on $\phi_x=\chi_xu$, including derivatives of the window. In particular, $f\phi_x$ is the variation of the new anchor weight $n_x$. No occupied-point weight $|u(x)|^2$ occurs.

The unrestricted ratio variation can also be checked directly. Let $A=\int zz^T|\phi|^2$, $B_{ij}=\operatorname{Re}\int(D_i\phi)^*D_j\phi$, $C_0=\int z|\phi|^2$, $P_0=\operatorname{Re}\int\phi^*D\phi$. For $F_x=n_xf$, its raw partial derivatives are

$$(F_x)_A=C_f,\quad (F_x)_B=E_f,\quad (F_x)_{C_0}=-2C_fc_x,\quad (F_x)_{P_0}=v_f-2E_fp_x,$$ $$(F_x)_n=f+c_x^TC_fc_x+p_x^TE_fp_x -\operatorname{tr}(C_fM_x+E_fT_x)-v_f\cdot p_x. \tag{7}$$

Differentiating these quadratic raw moments gives (6). Empty windows have $g_x=0$.

Why the Continuum Source Is Globally Lipschitz

Treat complex $L^2$ as a real normed space for differentiation. Define the linear local jet map

$$\mathcal A_x\psi=(\phi_x,z_1\phi_x,\ldots,z_{10}\phi_x, D_1\phi_x,\ldots,D_{10}\phi_x).$$

Each raw coordinate $q_j$, other than $n$, has the form $\operatorname{Re}\langle\mathcal A_x\psi,E_j\mathcal A_x\psi\rangle$ for a fixed bounded self-adjoint operator. Let $e_j=\|E_j\|$, $r_j=q_j/n$, and regard (3) as $\widetilde f(r)$. Wherever this function or its first two derivatives are nonzero,

$$\frac{\|\mathcal A_x\psi\|^2}{n} =1+\operatorname{tr}(A/n)+\operatorname{tr}(B/n) \le\kappa:=1+R_1^2+2L_T+2L_p. \tag{8}$$

Indeed the raw derivative trace is $\operatorname{tr}T+|p|^2$. The position mean lies inside the window, and the cap bounds all centered moments, inverse moments and momentum. The raw ratios therefore range over a compact set, say $|r_j|\le\rho_j$. On this set choose finite suprema $a_0,a_j,a_{jl}$ for $|\widetilde f-r\cdot\widetilde f_r|$, $|\widetilde f_j|$, $|\widetilde f_{jl}|$. These are actual suprema of the specified smooth constitutive function, not assumed norm bounds.

For $n>0$, direct differentiation of the perspective $F_x=n\widetilde f(q/n)$ gives

$$\begin{aligned} D^2F_x[v,w]={}&(\widetilde f-r\cdot\widetilde f_r)D^2n[v,w] +\sum_j\widetilde f_jD^2q_j[v,w]\\ &+\frac1n\sum_{jl}\widetilde f_{jl} (Dq_j[v]-r_jDn[v])(Dq_l[w]-r_lDn[w]). \end{aligned} \tag{9}$$

This identity includes signed means and off-diagonal covariances. The quadratic forms and (8) imply

$$|D^2q_j[v,w]|\le2e_j\|\mathcal A_xv\|\|\mathcal A_xw\|, \quad |Dq_j[v]-r_jDn[v]| \le2(e_j\sqrt\kappa+\rho_j)\sqrt n\|\mathcal A_xv\|.$$

The two factors $\sqrt n$ cancel the apparent singularity in (9):

$$|D^2F_x[v,w]|\le C_*\|\mathcal A_xv\|\|\mathcal A_xw\|, \quad C_*=2a_0+2\sum_ja_je_j+ 4\sum_{jl}a_{jl}(e_j\sqrt\kappa+\rho_j)(e_l\sqrt\kappa+\rho_l). \tag{10}$$

At $n=0$, the whole function $\phi_x$ is zero, so its weak derivatives and position multiples vanish too: $\mathcal A_x\psi=0$. Since $|F_x|\le\|f\|_\infty n$, its derivative there is zero. Along an affine line of fields a window is either identically zero or vanishes at at most one parameter value. Integrating (10) along that line extends the Lipschitz gradient through vacuum. No vacuum Hessian is assumed.

Put $Z_\chi=\int|z|^2\chi^2$, $K_\chi=\int|\nabla\chi|^2$. The cross term in the window derivative integrates to zero because $\int\chi\nabla\chi=0$. Consequently

$$\int\|\mathcal A_xv\|^2dx =(C_\chi+Z_\chi+K_\chi)\|J_\epsilon v\|_2^2 +C_\chi\|\nabla J_\epsilon v\|_2^2 \le B_{\chi,\epsilon}\|v\|_2^2,$$ $$B_{\chi,\epsilon}=C_\chi+Z_\chi+K_\chi +C_\chi\sum_i\|\partial_i j_\epsilon\|_1^2. \tag{11}$$

Cauchy–Schwarz in the anchor integral of (10) justifies the variation and gives

$$H\in C^{1,1}(L^2;\mathbb R),\qquad \operatorname{Lip}(G)\le L:=\mu+ \frac{C_*B_{\chi,\epsilon}}{2C_\chi}. \tag{12}$$

This is a global bound independent of field amplitude, including arbitrarily high spatial frequencies at fixed smoothing. Ordinary Picard iteration and the global Lipschitz bound give a unique solution for every real time. Phase invariance yields norm conservation; the real chain rule yields energy conservation. Translation and rotation symmetry hold, with the usual domain qualifications for momentum and angular momentum observables.

The constants may grow with $R_0$ and $\epsilon^{-1}$; no unsmoothed uniform theorem follows. A nonquadratic degree-two homogeneous energy generally cannot be $C^2$ at vacuum: its putative quadratic Taylor form would have to equal $H[\psi]$ for every $\psi$, by scaling. Thus $C^{1,1}$, rather than $C^2$, is the relevant regularity.

Exact Separation of Compact Supports

For compact original supports $S_1,S_2$, assume

$$\operatorname{dist}(S_1,S_2)>2R_1+2\epsilon. \tag{13}$$

Smoothing enlarges each support by at most $\epsilon$; no radius-$R_1$ window can then meet both. At every anchor one windowed field is zero, and both its energy and complete source vanish. Therefore

$$H[\psi_1+\psi_2]=H[\psi_1]+H[\psi_2],\qquad G[\psi_1+\psi_2]=G[\psi_1]+G[\psi_2]. \tag{14}$$

The same argument covers finite collections with pairwise gaps (13), arbitrary amplitudes, phases, orientations and different isolated clocks. Empty midpoint anchors matter in this action; a weaker gap derived for an occupied-point measure does not transfer. At a fixed output point the source depends only on input within distance $2R_1+2\epsilon$. This finite interaction range does not prove finite propagation speed, preservation of compact support, or Lorentz covariance. The coefficients now vary in space, so the parent's global quadratic moment closure and bounded-localization theorem do not apply.

A Strong Complete Initial-Equation Estimate

First let smooth $u$ be supported in $B(X,r)$, with $R_0>r$, and let its global moments lie where $\eta=1$. Denote the complete parent source by $G_{\rm FB}$, and set $g_0=G_{\rm FB}[u]-\mu u$. Every anchor in $B(X,R_0-r)$ sees the whole packet, hence has identical centered moments and $g_x=g_0$. Its physical centroid is $X$, even though the raw mean $c_x=X-x$ depends on the anchor. Outside $B(X,R_1+r)$ the window is empty.

With $V_D$ the unit-ball volume, define

$$V_{\rm full}=V_D(R_0-r)^D,\qquad \beta_D=\frac{(R_1+r)^D-(R_0-r)^D}{R_0^D},\qquad V_DR_0^D\le C_\chi\le V_DR_1^D. \tag{15}$$

Fubini gives the exact mass-weighted partial-anchor fraction $C_\chi^{-1}\|u\|_2^{-2}\int_{\rm partial}n_xdx =1-V_{\rm full}/C_\chi\le\beta_D$. It bounds the energy defect by $\|u\|_2^2\beta_D\sup_{\rm partial}|f_x-f_{\rm global}|$. The full equation additionally requires derivative control.

Let $F_0,C_M,C_T,P_V$ bound $|f|,\|C_f\|,\|E_f\|,|v_f|$ on the cap, and let $Q_0$ bound $|\operatorname{tr}(C_fM+E_fT)|$. Put $p_*=\sqrt{2L_p}$, $k_1=\sup|\nabla\chi|$, $k_2=\sup\|\nabla^2\chi\|_F$, and $U_j=\|\nabla^ju\|_2$, using Frobenius norm for second derivatives. Every nonempty window's centroid is a weighted average of points in $B(X,r)$, so $|y-X_x|\le2r$ on its field and derivative support. Expanding (6), without dropping window derivatives, gives

$$\begin{aligned} \|g_x[u]\|_2\le B_u:={}& [F_0+4r^2C_M+Q_0+C_Tp_*^2+P_Vp_*]U_0\\ &+[2C_Tp_*+P_V](U_1+k_1U_0)\\ &+\sqrt D\,C_T(U_2+2k_1U_1+k_2U_0). \end{aligned} \tag{16}$$

The $\sqrt D$ bounds the matrix Frobenius norm by its operator norm. For fixed transition width $w$ and $R_0$ bounded below, $k_1,k_2$ can be uniform. Thus $B_u$ is independent of a growing plateau radius. No grazing-window norm appears in a denominator.

The complete anchor integral is the full-anchor term $V_{\rm full}g_0/C_\chi$ plus the partial-anchor integral. Minkowski's inequality, (15), and (16) give

$$\left\|C_\chi^{-1}\int g_x[u]dx-g_0\right\|_2 \le\beta_D(\|g_0\|_2+B_u). \tag{17}$$

Here both the missing full coefficient and the partial-anchor volume divided by $C_\chi$ are at most $\beta_D$. For fixed $r,w$, $\beta_D=D(w+2r)/R_0+O(R_0^{-2})$. All physical branches use $D=10$ in this volume; substituting their active depth would change the common physical action. For illustration, at $r=18,w=2$, $\beta_{10}$ is approximately $82.95645,0.385093,0.0380359$ for $R_0=36,1000,10000$. These conservative volume bounds alone are not relative source errors.

Restore smoothing with $u=J_\epsilon\psi$ and $r\le r_\psi+\epsilon$. The outer $J_\epsilon^*$ is a contraction, so (17) yields

$$\|G[\psi]-\{\mu\psi+J_\epsilon^*[G_{\rm FB}[u]-\mu u]\}\|_2 \le\beta_{10}(\|G_{\rm FB}[u]-\mu u\|_2+B_u). \tag{18}$$

For a parent rest profile $v$ with $G_{\rm FB}[v]=\omega v$, take a smooth compact truncation $v_a$. On the weighted graph space $\mathcal X=\{v\in H^2:|y-X|^2v\in L^2\}$, put $d_{\rm cut}=\|v_a-v\|_{\mathcal X}$. Smooth dependence of the parent source near the compact target family gives a finite local constant $L_{\rm FB}$. Retain the exact smoothing defect

$$E_{\rm sm}=\left\|J_\epsilon^* [G_{\rm FB}[J_\epsilon v_a]-\mu J_\epsilon v_a] -[G_{\rm FB}[v_a]-\mu v_a]\right\|_2.$$

It tends to zero as $\epsilon\to0$ for fixed smooth truncation, by graph continuity and convergence of the approximate identity. Combining all three errors proves the strong initial estimate

$$\boxed{\begin{aligned} \|G[v_a]-\omega v_a\|_2\le{}& \beta_{10}\big(\|G_{\rm FB}[J_\epsilon v_a]-\mu J_\epsilon v_a\|_2 +B_{J_\epsilon v_a}\big)\\ &+E_{\rm sm}+(L_{\rm FB}+|\omega|)d_{\rm cut}. \end{aligned}} \tag{19}$$

Choose truncation, then smoothing, then a common large plateau for any specified finite-target accuracy. Compactness extends the choice uniformly over the entire finite union of normalized joint shells and rotations. Every term scales linearly with arbitrary physical amplitude; the relative error requires no amplitude calibration. Parent Gaussian profiles have noncompact support, so the truncation defect cannot be omitted. This finite-family result is not uniform over the infinite ladder or unbounded boosts.

Finite-Time Spatial Control for the Actual Regularized Flow

For every fixed window and smoothing length, the same global solution also propagates finite spatial moments. This is a bound on spatial extent, distinct from retaining a particular Hermite shape or clock over a prescribed period.

Freeze the state only when taking a spatial operator commutator, and write the complete equation as $G=\mu\psi+JA_\psi J\psi$. Expanding the windows gives the self-adjoint differential expression

$$A_\psi=\sum_{ij}D_i a_{ij}(y)D_j+ \frac12[b(y)\cdot D+D\cdot b(y)]+V(y),$$ $$a=C_\chi^{-1}\int\chi_x^2 E_f\,dx,\qquad b=C_\chi^{-1}\int\chi_x^2(v_f-2E_fp_x)\,dx,$$ $$V=C_\chi^{-1}\int\left\{\chi_x^2 [f-\operatorname{tr}(C_fM_x+E_fT_x)+(y-X_x)^TC_f(y-X_x) +p_x^TE_fp_x-v_f\cdot p_x] -\chi_x\operatorname{tr}(E_f\nabla^2\chi_x)\right\}dx.$$

The last term retains the window stress. All coefficients have uniform state-independent bounds at fixed window: where the derivatives are active, the cap bounds $M_x,T_x,p_x$, and $|y-X_x|\le2R_1$ on the window. Thus choose finite bounds $A_*,B_*,V_*$ for the entries of $a,b,V$. In particular $V_*$ may grow as $R_1^2$.

Put $F_0=J$, $F_i=D_iJ$, and

$$JA_\psi J=\sum_{a,b=0}^{10}F_a H_{ab}(y)F_b, \quad H_{00}=V,\quad H_{0i}=H_{i0}=b_i/2,\quad H_{ij}=a_{ij}.$$

Let $h_{ab}$ bound the corresponding multiplier. If $k_a$ is the compact smooth convolution kernel of $F_a$, define

$$f_a=\|k_a\|_1,\qquad \ell_a=\left(\sum_{j=1}^{10}\|z_jk_a(z)\|_1^2\right)^{1/2}, \quad C_{R,\epsilon}=\sum_{ab}h_{ab}(\ell_af_b+f_a\ell_b).$$

Multipliers commute with position, while $[y_j,F_a]$ has kernel $z_jk_a(z)$. The product rule therefore proves

$$\|[y,JA_\psi J]\psi\|_2\le C_{R,\epsilon}\|\psi\|_2, \qquad \|(y-y_0)\psi(t)\|_2\le \|(y-y_0)\psi(0)\|_2+C_{R,\epsilon}\sqrt N\,|t|.$$

Bounded Lipschitz truncations of position justify the estimate for the global flow, followed by monotone convergence. Repeated position commutators propagate each finite spatial weight in the same way. For example, the fraction of squared field norm outside radius $r$ is at most

$$\frac{1}{r^2}\left[ \frac{\|(y-y_0)\psi(0)\|_2}{\sqrt N}+C_{R,\epsilon}|t| \right]^2.$$

These are classical field-integral bounds. Their constants remain dependent on the window and smoothing. They establish finite-time spatial control, but do not supply a radius-uniform approximation theorem, a measured lifetime, or orbital stability of a particular count profile.

Persistence Is a Separate Quantitative Question

For $\gamma(t)=e^{-i\omega t}v_a$, phase covariance makes its complete residual norm constant, $\delta=\|G[v_a]-\omega v_a\|_2$. Global Lipschitz flow and Grönwall's inequality give, for equal initial data,

$$\|\psi(t)-\gamma(t)\|_2 \le\delta\frac{e^{L_{\rm rot}|t|}-1}{L_{\rm rot}}, \qquad L_{\rm rot}=L+|\omega| \tag{20}$$

as one permissible bound. The constants in (12) can grow while (19) shrinks as the plateau expands. Arbitrarily small initial residual therefore does not certify a fixed period of persistence or an observed lifetime. The spatial bounds above control finite-time extent at fixed parameters; they do not establish modal persistence. Exact stationary branches, physical spin and charge, orbit coupling, occupied-state selection and physical scale calibration remain open for this action. The proven result is the coexistence of amplitude homogeneity, a globally regular continuum equation, finite-range reciprocity and strong finite-family initial approximation.

The construction checker and independent review checker record finite algebraic and numerical controls for the complete variation, shell cap, perspective Hessian, separation gap and volume estimates. Those controls accompany the continuum derivations above; they do not replace them or establish a particle lifetime.

The standalone construction and review checkers have been executed with 103 and 20 passing checks respectively, retaining their own source hashes.