Gaussian Window Patterns with Global Classical Field Evolution

A Gaussian-window continuation of the flat-space count action combines exact localized stationary profiles with a globally regular classical field equation and asymptotic separation of distant patterns. The additional common centroid cap is essential to the regularity estimate. Call this action GC-R1. It differs from both the global-moment parent and the compact-window finite-range action; their stronger stationary, localization or finite-range properties do not transfer automatically.

The physical coordinates are ordinary coordinates in shared flat $\mathbb R^{10}$. Depth classifies a pattern's extracted spatial widths, not a separate physical sector. The complex scalar notation represents two classical field quadratures. Squared amplitude, mode coefficients and spatial integrals are not a Born position law. The internal profile remains distinct from the deterministic center trajectory of an electron in six ordinary spatial dimensions and from time-averaged detector registration. Interactions below arise from one field's action, without mediator particles.

Common Gaussian Windows and Constitutive Law

Set $D=10$, $\alpha=R_w^{-2}$, and choose $R_w,\sigma>0$. Define

$$j_\sigma(z)=(2\pi\sigma^2)^{-D/2}e^{-|z|^2/(2\sigma^2)},\quad J_\sigma\psi=j_\sigma*\psi,\quad \chi_x(y)=e^{-\alpha|y-x|^2/2},\quad u=J_\sigma\psi,\quad\phi_x=\chi_xu.$$

For $z=y-x$ and $D_y=-i\nabla_y$, the nonzero window's moments are

$$n_x=\int|\phi_x|^2,\quad c_x=n_x^{-1}\int z|\phi_x|^2,\quad X_x=x+c_x,\quad p_x=n_x^{-1}\operatorname{Re}\int\phi_x^*D_y\phi_x,$$ $$M_x=n_x^{-1}\int zz^T|\phi_x|^2-c_xc_x^T,\qquad (T_x)_{ij}=n_x^{-1}\operatorname{Re}\int(D_i\phi_x)^*D_j\phi_x-(p_x)_i(p_x)_j. \tag{1}$$

Gaussian smoothing is injective on $L^2$, because its Fourier multiplier is strictly positive. Thus nonzero input gives $n_x>0$ at every anchor. Gaussian multiplication and Gaussian derivative convolution provide all moments in (1). The spatial Gram inequality gives $M_x,T_x>0$ and $T_x\succeq M_x^{-1}/4$.

Suppress the anchor label. The positive matrix $R$ is defined by

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

Choose common constants $0<b_1<b_0$ and $\mu,\Lambda,\lambda,c_{\rm s}>0$. Define

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

The attained Gram moments obey $K_a,K_n\ge0$, $0\le d_{\rm eff}\le10$, and $W\ge\mu$. The Gamma quotient and material scales are inverse constitutive inputs, not a derivation of physical sector scales.

Let $a(t)=e^{-1/t}$ for $t>0$, zero otherwise, and $\theta(s)=a(2-s)/[a(2-s)+a(s-1)]$. It equals one for $s\le1$ and zero for $s\ge2$. With common positive bounds, take

$$\eta_0=\theta\!\left(\frac{\operatorname{tr}M}{L_M}\right) \theta\!\left(\frac{\operatorname{tr}M^{-1}}{L_{M^{-1}}}\right) \theta\!\left(\frac{\operatorname{tr}T}{L_T}\right) \theta\!\left(\frac{\operatorname{tr}T^{-1}}{L_{T^{-1}}}\right) \theta\!\left(\frac{|p|^2}{L_p}\right),$$ $$\eta(c,M,T,p)=\eta_0(M,T,p)\theta(|c|^2/L_c),\quad f=\eta(h-\mu),\quad C_\chi=\int\chi^2=(\pi R_w^2)^5,$$ $$H_{\rm GC}[\psi]=\mu\|\psi\|_2^2+C_\chi^{-1}\int n_xf_x\,dx,\qquad S=\int dt\left[\frac i2\int(\psi^*\dot\psi-\dot\psi^*\psi)-H_{\rm GC}\right]. \tag{3}$$

The integrand is zero at the zero field. The inverse-trace caps switch it off before singular covariance boundaries; differentiation requires only a smooth neighborhood of the compact attained moment set. The centroid cap controls the anchor-relative centroid $c_x$. Translating anchor and field together leaves it unchanged, preserving translation symmetry. All cap arguments are rotation scalars; phase symmetry also survives. There is no norm floor, and

$$H_{\rm GC}[a\psi]=|a|^2H_{\rm GC}[\psi],\qquad G_{\rm GC}[a\psi]=aG_{\rm GC}[\psi].$$

Thus no physical amplitude is fitted. Fubini and Gaussian contraction give

$$\int n_xdx=C_\chi\|J_\sigma\psi\|_2^2,\qquad \mu N\le H_{\rm GC}\le\mu N+\|f\|_\infty\|J_\sigma\psi\|_2^2 \le(\mu+\|f\|_\infty)N, \quad N=\|\psi\|_2^2. \tag{4}$$

Complete Centroid-Cap Variation

Write the complete differential, holding the other displayed moment arguments fixed, as

$$df=w_f\cdot dc+\operatorname{tr}(C_f\,dM+E_f\,dT)+v_f\cdot dp.$$

In particular $w_f=(h-\mu)\partial_c\eta$. Every coefficient includes the cap response and the response of $R$. An explicit computation of the latter uses the symmetric Sylvester solution

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

For $s_g(s)=sg(s)$, its matrix derivative $\mathcal L_{s_g}(R)$ uses divided differences, including off-diagonal entries. Put

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

Then $dW=\operatorname{tr}(C\,dM+E\,dT)$, $h_M=(W/h)C$, $h_T=(W/h)E$, $h_p=c_{\rm s}^2p/h$, and $df=\eta\,dh+(h-\mu)d\eta$. These formulas specify every derivative.

With $\xi=y-X_x$, the complete source is

$$\mathcal L_x=\xi^TC_f\xi+(D_y-p_x)^TE_f(D_y-p_x) -\operatorname{tr}(C_fM_x+E_fT_x)+v_f\cdot(D_y-p_x)+w_f\cdot\xi,$$ $$g_x[u]=\chi_x[f+\mathcal L_x]\phi_x,\qquad G_{\rm GC}[\psi]=\mu\psi+C_\chi^{-1}J_\sigma^*\int g_x[J_\sigma\psi]dx, \quad i\dot\psi=G_{\rm GC}[\psi],\quad dH[v]=2\operatorname{Re}\langle v,G_{\rm GC}\rangle. \tag{5}$$

Derivatives act on the entire windowed field. Omitting $w_f\cdot\xi$ would lose the centroid cap's response. To check (5), use raw moments $A=\int zz^T|\phi|^2$, $B_{ij}=\operatorname{Re}\int(D_i\phi)^*D_j\phi$, $C_0=\int z|\phi|^2$, $P_0=\operatorname{Re}\int\phi^*D\phi$. For $F_x=nf$,

$$(F_x)_A=C_f,\quad(F_x)_B=E_f,\quad (F_x)_{C_0}=w_f-2C_fc,\quad(F_x)_{P_0}=v_f-2E_fp,$$ $$(F_x)_n=f+c^TC_fc+p^TE_fp-\operatorname{tr}(C_fM+E_fT)-v_f\cdot p-w_f\cdot c. \tag{6}$$

Expanding the raw quadratic variations gives (5). On general $L^2$ inputs the source is interpreted through the bounded weak derivative proved below; a pointwise anchor integral is unnecessary.

Exact Count-One Ground Patterns at All Maintained Depths

Let $P$ be any ordinary spatial orthogonal projector of rank $d\in\{2,3,4,5,6,10\}$, and $Q=I-P$. Choose the same constants satisfying

$$\alpha<\min(b_1,b_0),\qquad \sigma^2\max(b_1-\alpha,b_0-\alpha)<1.$$

Set

$$B'=b_1P+b_0Q,\quad B=B'-\alpha I,\quad B_{\rm in}=B(I-\sigma^2B)^{-1},\quad \psi(y)=a\exp[-(y-z_0)^TB_{\rm in}(y-z_0)/2]. \tag{7}$$

Here $a$ is any complex amplitude. Gaussian convolution yields

$$u(y)=a\det(I+\sigma^2B_{\rm in})^{-1/2} \exp[-(y-z_0)^TB(y-z_0)/2].$$

Every window has the same inverse width $B'$, with

$$X_x=(B')^{-1}(Bz_0+\alpha x),\quad M=(B')^{-1}/2,\quad T=B'/2,\quad p=0, \quad K_a=K_n=0,\quad d_{\rm eff}=d,\quad U=0,\quad h=\mu. \tag{8}$$

Choose the common moment cap one on these moments. The centroid cap varies across anchors and need not equal one. To verify the unrestricted parent variation, write $H_d=\sum_{j=1}^d1/j$. At (8),

$$\mathcal F_a(0,d)=\mu H_d,\quad\mathcal F_d(0,d)=0,\quad Y_d=Y_V=0,\quad Y_a=P(B')^{-1}/2,$$ $$E_{\rm par}=(\mu H_dP+\Lambda Q)(B')^{-1}/2,\qquad C_{\rm par}=(\mu H_dP+\Lambda Q)B'/2.$$

The actual adjoint Sylvester input is $P(B')^{-1}/2$: $M$ has no off-block entries in a basis adapted to $P$. This evaluates the complete divided-difference formula at the ground state; it does not delete off-diagonal terms generally. The identities

$$C_{\rm par}=B'E_{\rm par}B',\qquad \operatorname{tr}(E_{\rm par}T+C_{\rm par}M)=\operatorname{tr}(E_{\rm par}B')$$

annihilate the complete normalized quadratic source on each Gaussian window. Equivalently it is $[\mu H_dA_P+\Lambda A_Q]\phi_x=0$, with each block number operator $A_P=\tfrac12[D_y^TP(B')^{-1}D_y+\xi^TPB'\xi-d]$.

In (5), $f=w_f=v_f=0$ and $C_f,E_f$ are $\eta_x$ times these parent derivatives. Every derivative of the cap multiplies $h-\mu=0$. Therefore the entire nonlinear source vanishes, including transition anchors:

$$G_{\rm GC}[\psi]=\mu\psi,\qquad \psi(t)=e^{-i\mu t}\psi(0),\qquad H_{\rm GC}=\mu N. \tag{9}$$

This is an unrestricted field solution for every amplitude, translation and orientation at all six depths. It attains the global fixed-norm energy minimum by (4). The cap can create other minimizers, so this is neither coercivity nor orbital stability. The branch realizes count one, $\binom{d}{d}=1$, at every depth; it does not realize the maintained excited particle frequencies or select one depth. There is no depth-five-specific property here.

Global Continuum Regularity and Evolution

Define the bounded window analysis jet

$$\mathcal A_x\psi=(\phi_x,z_1\phi_x,\ldots,z_D\phi_x,D_1\phi_x,\ldots,D_D\phi_x).$$

Each raw coordinate $q_j$, other than $n$, is a bounded self-adjoint quadratic form of this jet, with form norm $e_j$. Let $r_j=q_j/n$, and write $f=\widetilde f(r)$. On the support of $f$ or its first two derivatives, the common caps imply

$$\frac{\|\mathcal A_x\psi\|^2}{n} =1+\operatorname{tr}M+|c|^2+\operatorname{tr}T+|p|^2 \le\kappa:=1+2L_M+2L_c+2L_T+2L_p. \tag{10}$$

The inverse covariance caps keep this attained ratio set in a compact regular region. Choose $\rho_j\ge|r_j|$ there, and finite constitutive derivative suprema

$$a_0\ge|\widetilde f-r\cdot\widetilde f_r|,\quad a_j\ge|\widetilde f_j|,\quad a_{jl}\ge|\widetilde f_{jl}|.$$

These are suprema of the specified law on its compact regular set, not numerical sharp constants. Direct perspective differentiation gives

$$\begin{aligned} D^2F_x[v,w]={}&(\widetilde f-r\cdot\widetilde f_r)D^2n[v,w] +\sum_j\widetilde f_jD^2q_j[v,w]\\ &+n^{-1}\sum_{jl}\widetilde f_{jl} (Dq_j[v]-r_jDn[v])(Dq_l[w]-r_lDn[w]). \end{aligned}$$

Quadratic-form Cauchy–Schwarz and (10) cancel the apparent inverse norm:

$$|D^2F_x[v,w]|\le C_*\|\mathcal A_xv\|\|\mathcal A_xw\|,$$ $$C_*=2a_0+2\sum_ja_je_j+ 4\sum_{jl}a_{jl}(e_j\sqrt\kappa+\rho_j)(e_l\sqrt\kappa+\rho_l). \tag{11}$$

At an empty window its field, weak derivatives and position multiples vanish. The bound $|F_x|\le\|f\|_\infty n$ gives derivative zero there. Integration of (11) along affine field lines extends the Lipschitz derivative through vacuum. Gaussian injectivity means an entire window can be empty only for zero input.

Gaussian integration and $\int\chi\nabla\chi=0$ give

$$\int\|\mathcal A_xv\|^2dx =(C_\chi+Z_\chi+K_\chi)\|J_\sigma v\|^2+C_\chi\|\nabla J_\sigma v\|^2 \le B_{\rm jet}\|v\|^2,$$ $$Z_\chi/C_\chi=5R_w^2,\quad K_\chi/C_\chi=5/R_w^2,\quad \|\partial_i j_\sigma\|_1=\sqrt{2/\pi}/\sigma, \quad B_{\rm jet}/C_\chi=1+5R_w^2+5/R_w^2+20/(\pi\sigma^2). \tag{12}$$

Because $DF_x(0)=0$, (11) implies $|DF_x(\psi)[v]|\le C_*\|\mathcal A_x\psi\|\|\mathcal A_xv\|$. This is anchor-integrable by (12), and controls the integrated Taylor remainder. Riesz representation defines the unique $L^2$ gradient, agreeing with (5) weakly on smooth tests. Hence

$$H_{\rm GC}\in C^{1,1}(L^2;\mathbb R),\qquad \operatorname{Lip}(G_{\rm GC})\le L_G:=\mu+\frac{C_*B_{\rm jet}}{2C_\chi}. \tag{13}$$

Here complex $L^2$ is only the real normed function space used to describe a classical spatial field. Picard iteration gives a unique $C^1(\mathbb R;L^2)$ solution for every $L^2$ initial field, with continuous dependence. Phase symmetry and the real chain rule conserve $N,H$. Momentum and angular momentum as finite observables retain their usual generator-domain requirements. This theorem does not assert Schwartz propagation for arbitrary data or localization of neighboring fields. Its constant is not uniform as $\sigma\to0$, or when the finite caps are removed. Lipschitzness of $G$ does not itself assert Fréchet differentiability of $G$.

Complete Energy and Source Clustering

Let $\tau_L\psi_2(y)=\psi_2(y-L)$ and $e_j(x)=\|\mathcal A_x\psi_j\|$ for arbitrary fixed $L^2$ fields. Then $e_j\in L^2$ of anchor space. Integrating a mixed second derivative of $F_x$, using (11), gives

$$|H_{\rm GC}[\psi_1+\tau_L\psi_2]-H_{\rm GC}[\psi_1]-H_{\rm GC}[\psi_2]| \le2\mu|\langle\psi_1,\tau_L\psi_2\rangle| +\frac{C_*}{C_\chi}\int e_1(x)e_2(x-L)dx\longrightarrow0. \tag{14}$$

Both correlations tend to zero by compact-support approximation and Cauchy–Schwarz. For the complete source, the window derivative difference is bounded by either packet as perturbation:

$$|DF_x(a+b)[v]-DF_x(a)[v]-DF_x(b)[v]| \le2C_*\min(e_1(x),e_2(x-L))\|\mathcal A_xv\|.$$

The complex-gradient factor two and (12) therefore yield

$$\|G_{\rm GC}[\psi_1+\tau_L\psi_2]-G_{\rm GC}[\psi_1]-\tau_LG_{\rm GC}[\psi_2]\| \le\frac{C_*\sqrt{B_{\rm jet}}}{C_\chi} \|\min(e_1,e_2(\cdot-L))\|_2\longrightarrow0. \tag{15}$$

To justify the last limit, put all but small $L^2$ tails of the two anchor functions in compact sets. Once those sets separate, at least one factor is in its small tail at every anchor. Arbitrary phases, orientations and amplitudes are allowed. Gaussian tails give asymptotic clustering, not finite interaction range or finite propagation speed.

For equal-width Gaussian profiles (7), let their center separation be $L$, with norms $N_j$ and smoothed norms $U_j$. Set

$$S=B(B+\alpha I)^{-1},\quad K=\alpha S,\quad A_0=1+\tfrac12\operatorname{tr}(B+\alpha I)^{-1}+\tfrac12\operatorname{tr}(B+\alpha I).$$

The exact jet norm is $e_j(x)^2=n_j(x)[A_0+|S(x-z_j)|^2]$, where $n_j(x)=\text{constant}_j e^{-(x-z_j)^TK(x-z_j)}$. Completing the anchor square and using $S^2K^{-1}=S/\alpha$ gives

$$C_\chi^{-1}\int e_1(x)e_2(x)dx \le\sqrt{U_1U_2}\,e^{-L^TKL/4} [A_0+\operatorname{tr}S/(2\alpha)+|SL|^2/4],$$ $$|\langle\psi_1,\psi_2\rangle| \le\sqrt{N_1N_2}\,e^{-L^TB_{\rm in}L/4}. \tag{16}$$

In (16) the two functions already include their distinct centers, so no further translation is inserted. These are directional Gaussian-times-polynomial bounds for (14), not a force sign or an interaction acceptance law. Unequal widths and orientations retain (14)–(15).

Finite-Time Dynamical Clustering and Scope

Let $\psi_j(t)$ be the isolated solutions, and $\Psi_L(t)$ the solution initialized by their translated sum. For fixed finite $T$, define

$$R_L(T)=\sup_{|t|\le T}\|G_{\rm GC}[\psi_1(t)+\tau_L\psi_2(t)] -G_{\rm GC}[\psi_1(t)]-\tau_LG_{\rm GC}[\psi_2(t)]\|.$$

Each continuous isolated trajectory has compact image on $[-T,T]$. The residual map is Lipschitz in its two field arguments, uniformly in $L$, with bound $2L_G(\|\delta a\|+\|\delta b\|)$. A finite net of these compact images promotes (15) to $R_L(T)\to0$. Variation of constants and Grönwall give

$$\sup_{|t|\le T}\|\Psi_L(t)-\psi_1(t)-\tau_L\psi_2(t)\| \le R_L(T)\frac{e^{L_GT}-1}{L_G}\longrightarrow0. \tag{17}$$

This is a fixed-finite-time limit, not scattering or a bound uniform as $T\to\infty$. General isolated patterns retain their own histories; no common frequency is imposed. For the exact ground branch their common clock $\mu$ follows from (9), and phase covariance makes the residual norm constant in time.

The proven coexistence is exact localized count-one solutions, arbitrary amplitude, global classical $L^2$ flow, and complete energy/source/dynamical clustering. Excited clocks, coercive stability, empirical spin and charge, deterministic orbit coupling and occupied-state selection remain open. Translation symmetry is insufficient to establish an exact moving Gaussian family of GC-R1. The shape-frequency mobility analysis distinguishes that obstruction from exact motion under changed phase actions.

The construction checker and independent review checker supply finite controls for the complete variation, smoothing inversion and ground-source cancellation. Their executed source calculations reported 63 exact construction checks, six off-ground numerical centroid-chain-rule comparisons and six rotated Gaussian cancellations. These finite diagnostics accompany the proofs; they do not supply a continuum theorem or an observed particle identification by themselves.