Nonlinear Count Persistence and Multiple-Frequency Spatial Response

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.

Two classes of effective actions extend the constructive count clocks. The first now has a nonlinear persistence theorem for its entire finite spatial dynamics, including current, independent profiles and responding material geometry. The second accommodates several different supplied clocks in one positive linear action, despite overlap of their actual spatial sources. Its alternative kinetic and response laws have different physical content. These results do not combine into an unwritten master action.

The spatial dimensions are ordinary directions in the shared physical space, taken here as $\mathbb R^{10}$. Real field components, profile coefficients and response coordinates describe $\Psi_\infty$; they are neither extra spatial dimensions nor mediator particles. Field-norm localization is distinct from a position-probability law. In particular, none of these internal field constructions replaces the electron's literal deterministic external trajectory. Finite normal envelopes are supplied in both constructions, rather than derived exact confinement to a lower-dimensional layer.

Count labels and the common spatial carrier

For depth $d\in\{2,3,4,5,6,10\}$, let $W_J$ be the complete real Hermite shell of active degree $J$ in $d$ spatial directions, with Gaussian ground profiles in the remaining $10-d$ directions. Use a physical length $\ell>0$. On these fields the symmetric derivative count operator is

$$M_d=\binom{N_P}{d},\qquad M_d|_{W_J}=c_J I, \qquad c_J=\binom Jd.$$

Here $N_P$ is the sum of the $d$ active oscillator number operators; it labels real spatial functions. The identity follows by summing all order-$d$ annihilator products with multinomial weights. The cumulative prefix count $\dim\{N_P\le n-1\}=\binom{n+d-1}{d}$ equals the eigenvalue on $W_K$, $K=n+d-1$, although these are different spatial subspaces. Neither count identity selects occupation.

Both constructions below use these fourteen supplied positive-count labels and arbitrary supplied positive sector units $\mu_d$:

The target frequency is $\omega=\mu_dc$. No exceptional depth-five coefficient is introduced. Bottom's separate proposed geometric mean and a full zero are not ordinary entries in this table.

The unchanged nonlinear spatial action

Fix a finite band $\mathcal H=\bigoplus_{J=d}^{L}W_J$. Four independent unit profiles $f_a\in\mathcal H$, indexed by $a=(i,j)\in\{0,1\}^2$, occupy orthogonal real field components $e_a$. Their scalar spatial shapes need not be orthogonal. Define

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

All four profiles range over their complete unit spheres. Product component weights and a common phase remain explicit ansatz restrictions. Put

$$m=\sum_a p_a\langle f_a,M_df_a\rangle,\quad z=\sum_a p_a\langle f_a,M_d^2f_a\rangle,\quad \mathcal B=m+\eta(m)\frac{z-m^2}{m},\quad \Lambda=\mu_d\mathcal B(1+gw),\quad g>0. \tag{1}$$

The variance includes differences between component mean counts. Two constitutive branches are retained:

$$\eta=128,\quad L=d+99; \qquad \eta(m)=1+\frac{(d!m)^{1/d}}d,\quad\hbox{any chosen finite }L\ge K. \tag{2}$$

For both current laws set $b=1/2$ and

$$w=\frac{1+k^2}{2}x^2-bx+\frac{b^2}{2},\quad f_{\rm cur}(k)=\frac{k^2}{8(1+k^2)}.$$

The primary reference is $(k_0,x_0)=(0,1/2)$. The alternative law is

$$w_0=w-f_{\rm cur}(k_0)-f_{\rm cur}'(k_0)(k-k_0) +\frac14(k-k_0)^2, \qquad(k_0,x_0)=(3/4,8/25). \tag{3}$$

Below $w$ denotes the selected complete law. The alternative is a specified common constitutive repair, not a different adjustment for each particle.

For dimensional material coordinate $Y$, define

$$q_{AB}=\ell^2\sum_a p_a\int\partial_A f_a\partial_B f_a\,d^{10}Y, \quad S_q=I+\frac\chi{a_0^2}q, \quad T=a_0^2(\det S_q)^{-1/10}S_q,$$ $$F\in GL^+(10),\quad h=F^TF,\qquad V_s=\frac\kappa8\operatorname{tr}[(T^{-1}h-I)^2]. \tag{4}$$

The constants $\chi,a_0,\kappa,I_F,\ell$ are positive. Physical position is $X+FY$, and the unitary affine field is $\Psi(X+FY)=(\det F)^{-1/2}\phi(Y)$. The prescribed material connection $D_\tau\Psi=\partial_\tau\Psi-(\partial_\tau U)U^{-1}\Psi$ transports the internal kinetic norm; it is not the bare Cartesian time derivative. The complete action in proper time is

$$\mathscr L=\frac{r'^2}{2} +\frac{r^2}{2}\left[\theta'^2+G_k k'^2+G_x x'^2 +\sum_a p_a\|f_a'\|^2-\Lambda^2\right] +\frac{I_F}{2}\|F'\|_F^2-V_s. \tag{5}$$

Every force is obtained from (5). In particular profile momentum contains $(r^2p_a)'f_a'$, the current equations contain $-r^2\sum_a p_{a,s}\|f_a'\|^2/2$, and differentiation of (1) retains $\eta'(m)$. No current or support coordinate is statically eliminated. There is one conserved phase charge $Q=r^2\theta'$, not four independently fixed component charges.

Fix $Q\ne0$, write $A=|Q|$, and choose $c=c_K$. The compact reduced stationary family is

$$\mathcal M_K=\left\{f_a\in S(W_K),\ (k,x)=(k_0,x_0),\ r_0^2=\frac A\omega,\ F=R\sqrt{T(q)},\ R\in SO(10),\ \hbox{all reduced velocities zero}\right\}, \quad \omega=\mu_dc. \tag{6}$$

Its phase runs at $\theta'=\operatorname{sign}(Q)\omega$, and its complete energy is $E_0=A\omega$. Its reduced dimension is $45+4[\binom{K+d-1}{d-1}-1]$. It includes independent same-shell shapes, not only physical rotations of one selected shape.

A finite-amplitude count inequality

Let $P_J$ project onto the complete shell and introduce the actual weighted measure

$$t_J=\sum_a p_a\|P_Jf_a\|^2,\quad \sum_Jt_J=1, \quad D_J=\frac{c_J-c}{c},\quad a=\sum_Jt_JD_J,\quad b_2=\sum_Jt_JD_J^2, \quad s=\sum_{J\ne K}t_J.$$

Then $m=c(1+a)$, $z-m^2=c^2(b_2-a^2)$, $a^2\le sb_2$, and $s\le b_2/\delta^2$. Here a valid nonzero relative gap is $\delta=d/K$ for $K>d$, or $\delta=d$ when $K=d<L$. If $L=K=d$, the spectral variables vanish and the spectral barrier is omitted.

Choose the following sufficient constants:

Pascal's identity gives $c/(c-c_{K-1})=K/d$. The binomial product gives $(K+1)/d\le\eta(c)\le1+K/d$ for the variable law. Consequently every other shell satisfies

$$D_J+\eta(c)D_J^2\ge R_gD_J^2.$$

Define

$$\alpha=\min\!\left(\frac12,\frac{R_g}{16\eta_U}\right),\quad \sigma=\frac{R_g}{8\eta_U},\quad \beta=\min(\alpha^2,\delta^2\sigma). \tag{7}$$

For $|a|\le1/2$, the exact function $H(a)=\eta(c(1+a))/(1+a)$ obeys $0<H\le2\eta_U$ and $|H'|\le4\eta_U$. Thus throughout $b_2\le\beta$,

$$\begin{aligned} \frac{\mathcal B}{c}-1 &=a+H(a)(b_2-a^2)\\ &\ge[R_g-4\eta_U|a|-2\eta_Us]b_2 \ge\frac{R_g}{2}b_2. \end{aligned} \tag{8}$$

This is an exact nonlinear remainder estimate. Negative contributions from lower-shell first moments, changes in current weights and between-component variance are included. It is stronger than a reference Hessian sign. An alternative sufficient proof bounds the leakage $s$ using the largest finite count gap; that gives a different, cutoff-dependent neighborhood. A tiny occupation of a remote shell can have small $s$ but large $b_2$, so the two neighborhoods are not universally ordered.

Complete energy barriers and all-time finite persistence

Put $v=x-1/[2(1+k^2)]$. Completing the current square gives

$$w\ge\frac{v^2}{2}+C_k(k-k_0)^2, \qquad C_k=1/16\ \hbox{or}\ 7/32. \tag{9}$$

For the second branch the bound follows from $f_{\rm cur}''\ge-1/16$ and Taylor's integral remainder. Use the same chart consistently:

$$|k-k_0|<\rho=\frac{1-|k_0|}{2},\quad |v|<1/8, \quad 1/8<x<5/8,$$ $$p_a\ge p_{\min}=\frac{1-|k_0|-\rho}{130} =\begin{cases}1/260&\hbox{primary},\\1/1040&\hbox{alternative},\end{cases} \quad \overline W=\min(C_k\rho^2,1/128) =\begin{cases}1/128,\\7/2048.\end{cases} \tag{10}$$

The fixed-charge Routh reduction has kinetic energy

$$T_{\rm red}=\frac{r'^2}{2}+\frac{r^2}{2} [G_k k'^2+G_x x'^2+\sum_a p_a\|f_a'\|^2] +\frac{I_F}{2}\|F'\|_F^2$$

and potential $Q^2/(2r^2)+r^2\Lambda^2/2+V_s$. With $e=E-E_0$, the exact decomposition is

$$e=T_{\rm red}+\frac12(A/r-r\Lambda)^2+A(\Lambda-\omega)+V_s \ge T_{\rm red}+E_0\left[\frac{R_g}{2}b_2+gw\right] +\frac12(A/r-r\Lambda)^2+V_s. \tag{11}$$

All terms are nonnegative in the spectral tube. For noncommuting strain matrices,

$$C=T^{-1/2}F^TFT^{-1/2}-I=C^T, \qquad V_s=\frac\kappa8\|C\|_F^2.$$

Therefore $e<\kappa/32$ ensures $\|C\|_F<1/2$ and $T/2<F^TF<3T/2$. Singular $F$ costs at least $\kappa/8$, a finite barrier; an infinite material wall is not assumed. The same energy gives

$$\frac A{\sqrt{2(E_0+e)}}\le r\le\frac{\sqrt{2(E_0+e)}}\omega, \qquad \left|\frac r{r_0}-\frac{r_0}r\right|\le\sqrt{2e/E_0}. \tag{12}$$

For either current law and either coefficient law in its stated domain, take initial profiles on their full spheres, $r>0$, $F\in GL^+(10)$, finite tangent velocities, $b_2<\beta$, and the current chart (10). If

$$0\le e<e_{\min} =\min\left\{\frac{E_0R_g\beta}{2},\ E_0g\overline W,\ \frac\kappa{32}\right\}, \tag{13}$$

the full finite equations have a unique solution for every real proper time. A first exit through the spectral or current boundary would cost the corresponding term in (13), contradicting conserved energy. The material estimate prevents singular deformation; (12) prevents radial collapse or escape. Positive component weights prevent hidden velocity divergence. At each finite cutoff the profile spheres are compact, the material singular values are bounded above and below, and the complete kinetic metric and its inverse stay regular. Energy bounds all velocities, leaving the smooth finite Euler–Lagrange vector field in a compact regular part of its phase space. The finite ODE continuation criterion applies forward and backward in time.

This basin is nonempty: continuity near the compact family (6) supplies a neighborhood satisfying its strict conditions. Small energy alone, without the initial basin, is insufficient because other lower-shell families have lower energy. The three barriers have different scalings. With common $A,\kappa$, the relative material bound is $\kappa/(32A\omega)$; it shrinks at larger clock frequency. A conservative small bound is neither an optimal basin size nor a measured lifetime.

Writing $\epsilon=e/E_0$, the quantitative consequences are

$$b_2\le\frac{2\epsilon}{R_g},\quad s\le\frac{2\epsilon}{R_g\delta^2},\quad |k-k_0|\le\sqrt{\frac\epsilon{gC_k}},\quad |v|\le\sqrt{\frac{2\epsilon}g},\quad \|C\|_F\le\sqrt{8e/\kappa},\quad T_{\rm red}\le e. \tag{14}$$

If additionally $\epsilon<R_g\delta^2p_{\min}/4$, each component leakage is less than $1/2$. Its normalized projection $f_a^*=P_Kf_a/\|P_Kf_a\|$ then exists and

$$\sum_a\|f_a-f_a^*\|^2\le\frac{4\epsilon}{R_g\delta^2p_{\min}}.$$

Choose the polar rotation $R_F$ of $F$, and form $q^*$ from these projected profiles and reference weights. Then $F^*=R_F\sqrt{T(q^*)}$ belongs to the stationary family. On a fixed finite band the gradient map, determinant derivative and positive matrix square root give $\|F-F^*\|\le C_L\sqrt e$. This constant may depend on the cutoff. For example, with $L_c=2L+10$, the profile tensor satisfies $\|q_a(f)-q_a(g)\|_F\le2L_c\|f-g\|$; the square-root estimate follows by solving $\sqrt A X+X\sqrt B=A-B$.

The actual kinetic weights, rather than a replacement unit metric, imply

$$|r'|\le\sqrt{2e},\quad \|F'\|_F\le\sqrt{2e/I_F},\quad |k'|\le\frac{\sqrt{8e}}{r_{\min}},\quad |x'|\le\frac{\sqrt{2e}(1+x_{\max}^2)}{r_{\min}},\quad \|f_a'\|\le\frac{\sqrt{2e}}{r_{\min}\sqrt{p_{\min}}},$$

where $r_{\min}$ is the lower bound in (12) and $x_{\max}=5/8$. Reconstruction of the phase gives the accepted estimate

$$|\theta'-\operatorname{sign}(Q)\omega| \le2\omega\sqrt{\epsilon(1+\epsilon)}. \tag{15}$$

This rate estimate alone does not give a uniform bound on the accumulated phase difference. Shapes and material rotation may also move far along (6). The theorem controls distance to the entire family, modulo common phase, and its transverse velocities; it does not lock the initial shape, orientation or phase trajectory, nor imply asymptotic convergence.

Count moments bound real spatial localization

A stronger localization estimate removes the upper cutoff from second moments. Every $J\ge d$ satisfies

$$J\le d-1+(d!c_J)^{1/d}.$$

Concavity of $u^{1/(2d)}$, applied to the actual weighted spectral measure, gives

$$\begin{aligned} \overline J&=\sum_Jt_JJ\\ &\le d-1+(d!c)^{1/d}(1+2a+b_2)^{1/(2d)}\\ &\le d-1+(d!c)^{1/d}(1+\sqrt\beta)^{1/d} \le d-1+\frac32K. \end{aligned} \tag{16}$$

The final bound uses $\beta\le1/4$ and $(d!c)^{1/d}\le K$. Set $y=Y/\ell$ and use dimensionless normalized profiles in the next integrals. Since $-\Delta_y+|y|^2=2N_{\rm total}+10$,

$$\sum_a p_a\int\left(|y|^2|f_a|^2+|\nabla_y f_a|^2\right)d^{10}y =2\overline J+10\le L_*=2d+3K+8. \tag{17}$$

Position and gradient pieces separately need not be diagonal between shells; their sum is, and both are nonnegative. Thus $\operatorname{tr}q\le L_*$. With $S_*=1+\chi L_*/a_0^2$,

$$\frac{a_0^2}{S_*}\le\lambda_{\min}(T),\quad \lambda_{\max}(T)\le a_0^2S_*,\quad \sigma_{\min}(F)^2\ge\frac{a_0^2}{2S_*},\quad \|F\|_{\rm op}^2\le\frac32a_0^2S_*.$$

Changing variables to actual position $Z=X+\ell Fy$, including the affine Jacobian, now gives

$$\frac{\int|Z-X|^2|\Psi|^2d^{10}Z}{\|\Psi\|^2} \le\frac32\ell^2a_0^2S_*L_*,\qquad \frac{\int|\nabla_Z\Psi|^2d^{10}Z}{\|\Psi\|^2} \le\frac{2S_*L_*}{a_0^2\ell^2}. \tag{18}$$

The first moment uses $F$; the gradient uses $F^{-T}$. The common amplitude cancels in these ratios. The fraction of field norm outside radius $R_{\rm phys}$ about $X$ is at most the first bound divided by $R_{\rm phys}^2$. These are real spatial field estimates, not probabilities for the electron's position.

At fixed target shell and physical parameters, (7)–(8) and (16)–(18) contain no upper cutoff for the variable-coefficient branch. The fixed coefficient retains its hundred-shell domain. Finite Hermite fields also have Gaussian-polynomial tails, with bounds at each fixed cutoff. No uniform higher-derivative Gaussian envelope, infinite-dimensional force regularity or infinite-band evolution theorem follows from (18). The all-time continuation theorem remains finite-dimensional. Its nonlocal global potential has not acquired a local positive energy density merely from a field-norm estimate. Under the separately supplied collective proper-time completion, the isolated center moves at constant velocity with the complete rest energy contributing to its inertia; this is not a derivation of a local relativistic field equation.

Several frequencies with actual overlapping sources

The second action class uses different spatial templates. Define unit one-dimensional functions

$$f(s)=\frac{\sqrt{15}}4(1-s^2)\quad (|s|\le1),\qquad h(s)=\begin{cases}\sqrt{3/8},&|s|\le1,\\ \sqrt{3/8}(2-|s|),&1<|s|<2,\end{cases}$$

both zero outside their stated support, and

$$u_d(X)=\ell^{-5}\prod_{a\le d}f(X_a/\ell)\prod_{a>d}h(X_a/\ell), \quad p_d=u_d^2,\quad s_d=p_d/\sqrt{C_{dd}}.$$

These are actual $H^1(\mathbb R^{10})$ fields. Their spatial integrals are

$$C_{de}=\langle p_d,p_e\rangle =\ell^{-10}(5/7)^{\min(d,e)}(3/8)^{|d-e|}(27/80)^{10-\max(d,e)},$$ $$\langle s_d,s_e\rangle=(\sqrt{7/12})^{|d-e|},\quad \langle u_d,u_e\rangle=(\sqrt{5/8})^{|d-e|},\quad \|\nabla u_d\|^2=\frac{7d+30}{4\ell^2}. \tag{19}$$

Retained templates $u_{d_i}e_i$ have orthogonal real field components, explaining an initial identity retained kinetic Gram despite scalar spatial overlap. A finite-rank source $Bq=\sum_i b_iq_i$ acts into another component of the same field, with $b_i=gp_{d_i}$ or $gs_{d_i}$ as explicitly chosen. Its adjoint is the actual spatial integral $(B^*y)_i=\langle b_i,y\rangle$. The source kernel is a supplied nonlocal finite-rank law, not a derived local density interaction. Fourteen repeated-depth columns have rank six, not fourteen. The compact templates differ from the first class's Gaussian profiles and do not inherit its nonlinear persistence theorem.

Initially consider

$$\mathcal A=A_0\int dt\left[\frac12\dot q^TM\dot q+\frac12\|\dot y\|^2 -\frac12q^TK_0q-\frac12\|L^{1/2}y\|^2+\langle Bq,y\rangle\right], \tag{20}$$

with $M>0$, positive self-adjoint response $L$, and fixed physical action unit $A_0>0$. Reversing the source sign reverses the responding component and leaves the frequency and positivity statements unchanged. The equations are $M\ddot q+K_0q-B^*y=0$ and $\ddot y+Ly-Bq=0$. The complete energy is

$$E/A_0=\frac12(\dot q^TM\dot q+\|\dot y\|^2+q^TSq) +\frac12\|L^{1/2}(y-L^{-1}Bq)\|^2, \quad S=K_0-B^*L^{-1}B. \tag{21}$$

Prescribe $\lambda_i=h_i^2>0$, retained vectors $e_i$, and $\lambda_i\in\rho(L)$. Let $Y_i=(L-\lambda_i)^{-1}b_i$, $W_i=(e_i,Y_i)$. The retained row requires $K_0e_i=\lambda_iMe_i+B^*Y_i$. Subtracting its transposed pair and using the resolvent identity yields

$$(\lambda_i-\lambda_j)\big(M_{ij}+\langle Y_i,Y_j\rangle\big)=0. \tag{22}$$

This is full-mode orthogonality, not optional normalization. With diagonal $M$, flat $L=lI$, distinct targets and overlapping off-pole sources, $\langle Y_i,Y_j\rangle=\langle b_i,b_j\rangle/[(l-\lambda_i)(l-\lambda_j)]\ne0$. No symmetric direct counterterm alone fixes all rows. Small nonzero coupling still fails an exact equality. For repeated identical sources below any common response gap, the spectral integral for this inner product is strictly positive as well. This restriction holds only with those prescribed retained vectors, source and response law; a nonflat response can have orthogonal responses to overlapping sources.

For example, let $\lambda=(1,4)$, $M=I$, and $c_e=(48-4\sqrt{95})/7$. In an orthonormal physical response basis set

$$B=\begin{pmatrix}4&c_e\\c_e&4\end{pmatrix},\quad L=K_0=\begin{pmatrix}5&c_e\\c_e&8\end{pmatrix}.$$

The source columns have normalized overlap $7/12$, realizable by the actual $s_2,s_4$ with a common strength. Since $L-B=\operatorname{diag}(1,4)$ and $L+B>0$, the complete modes $(e_i,e_i)$ have the desired frequencies, full norm two and positive energy. This is a witness against a universal overlap prohibition, not the repeated-source fourteen-label construction.

A common changed-metric construction and its velocity alternative

Let $\boldsymbol\Lambda=\operatorname{diag}(\lambda_i)$, and let $D>0$ commute with it. Repeated eigenvalues permit positive blocks in $D$. Use a real parameter $\vartheta$, with the required resolvents defined, and set

$$Y_i=(L-\vartheta\lambda_i)^{-1}b_i,\quad G=Y^*Y,\quad B=LY-\vartheta Y\boldsymbol\Lambda,$$ $$M=D+(1-2\vartheta)G,\quad C=-(1-\vartheta)Y,\quad K_0=D\boldsymbol\Lambda+Y^*LY -\vartheta(G\boldsymbol\Lambda+\boldsymbol\Lambda G),$$ $$M_f=\begin{pmatrix}M&C^*\\C&I\end{pmatrix},\quad K_f=\begin{pmatrix}K_0&-B^*\\-B&L\end{pmatrix},\quad \mathcal A=\frac{A_0}{2}\int(\dot z^TM_f\dot z-z^TK_fz)dt, \quad z=(q,y). \tag{23}$$

These are physical constitutive terms before variation. In particular $C$ gives reciprocal acceleration terms in both equations. Direct substitution gives

$$K_fW=M_fW\boldsymbol\Lambda,\qquad W=\binom IY, \qquad W^*M_fW=D. \tag{24}$$

The full norm includes support and mixed kinetic energy. Positivity is controlled by

$$D-\vartheta^2G>0,\quad T_c=I-\vartheta^2YD^{-1}Y^*>0,\quad V_c=L-\vartheta^2Y\boldsymbol\Lambda D^{-1}Y^*>0. \tag{25}$$

The first two conditions are equivalent. The constant shears $r_c=y-Yq$ and $\zeta=q+\vartheta D^{-1}Y^*r_c$ turn the complete kinetic and potential forms into

$$\dot\zeta^*D\dot\zeta+\langle\dot r_c,T_c\dot r_c\rangle, \qquad \zeta^*D\boldsymbol\Lambda\zeta+\langle r_c,V_cr_c\rangle. \tag{26}$$

Thus all complementary modes remain. In finite dimension the characteristic determinant is $\det D\prod_i(\lambda_i-z)\det(V_c-zT_c)$. If $L\ge l_0I$, $l_0>\lambda_{\max}$, $0\le\vartheta\le1$, then $V_c\ge l_0T_c$; the complementary squared frequencies lie above that common gap. This is sufficient, not necessary. Healthy off-pole examples with $L=I$, targets $(4,9)$ and weak nonzero overlapping sources satisfy (25). At a pole use the full equations; an inverse cannot be continued through an unsatisfied response row.

For $\vartheta=1$, $C=0$ and $M=D-G$. This is the common changed-retained-metric construction: it holds $B,L$ fixed and changes the kinetic and direct matrices. Its static Schur form is

$$S=D\boldsymbol\Lambda-\boldsymbol\Lambda Y^*L^{-1}Y\boldsymbol\Lambda. \tag{27}$$

For positive targets, $S>0$ and $V_c>0$ are equivalent Schur tests. Writing $B=gB_0$, $G=g^2G_0$, $Y^*L^{-1}Y=g^2V_0$, the exact strict conditions are

$$g^2\|D^{-1/2}G_0D^{-1/2}\|<1,\qquad g^2\|D^{-1/2}\boldsymbol\Lambda^{1/2}V_0 \boldsymbol\Lambda^{1/2}D^{-1/2}\|<1. \tag{28}$$

Below the response gap the kinetic condition suffices for potential positivity. At a kinetic equality the metric is singular; a potential equality creates an additional zero. These are limits of this realization, not universal exclusions.

For $\vartheta=1/2$, the retained block stays $M=D$ but $C=-Y/2$. The explicit new term is $-\langle Y\dot q,\dot y\rangle/2$, and $Y_i=(L-\lambda_i/2)^{-1}b_i$, not the old denominator. The metric is positive precisely when $D-G/4>0$, with potential complement $L-Y\boldsymbol\Lambda D^{-1}Y^*/4>0$. Keeping the retained diagonal fixed therefore does not keep the full kinetic form fixed. At $\vartheta=0$, the construction is a pullback of decoupled oscillators; that endpoint is not a new interaction mechanism. Constant diagonalizing coordinates prove consistency but do not establish a physical origin. Moving coordinates require their transport derivatives.

A distinct fixed-kinetic polar-response construction

One can instead hold the full block-diagonal kinetic form $\operatorname{diag}(M_0,I)$ fixed, with positive diagonal $M_0$, and redesign the response operator. For arbitrary actual $B$, let $H_B=(B^*B)^{1/2}$, possibly singular. Extend its polar partial isometry to $O^*O=I$, $OH_B=B$. Explicitly,

$$O=BH_B^+ +ZP_{\ker H_B},$$

where $Z$ maps the kernel isometrically into response functions orthogonal to $\operatorname{ran}B$. Fourteen rank-six source columns require eight additional response directions. Smooth functions supported outside the compact source support provide them in the same ordinary $\mathbb R^{10}$; their selection is additional physical embedding data.

For $c_p>0$, prescribe on $\operatorname{ran}O$

$$L=O(\boldsymbol\Lambda+H_B/c_p)O^*,\qquad K_0=M_0\boldsymbol\Lambda+c_pH_B. \tag{29}$$

Add a specified positive $L_\perp$ on its orthogonal complement. In coordinates $r_p=O^*y$, the entire active potential is

$$2V=q^*M_0\boldsymbol\Lambda q+r_p^*\boldsymbol\Lambda r_p +c_p(q-r_p/c_p)^*H_B(q-r_p/c_p). \tag{30}$$

For positive targets it is strictly positive at every source strength, including singular $H_B$. The selected full modes $(e_i,c_pOe_i)$ have squared frequencies $\lambda_i$ and norms $(M_0)_{ii}+c_p^2$. They solve every row directly, even at an accidental response pole; the displayed solution supplies the required source solvability. The source itself remains $B$, although its responses become orthogonal.

The complementary columns $(-c_pM_0^{-1},O)^T$ have norm $N=I+c_p^2M_0^{-1}$ and stiffness $N\boldsymbol\Lambda+NH_BN/c_p$. Their whitened stiffness is

$$\boldsymbol\Lambda+N^{1/2}H_BN^{1/2}/c_p\ge\lambda_{\min}I. \tag{31}$$

There are fourteen additional active modes in the fourteen-target construction, plus the chosen $L_\perp$, whose own lower bound is needed for a uniform full gap. No relative ordering above all targets is asserted. The price of this alternative is a changed response span and a copy of the supplied target spectrum in $L$. It is neither the mixed-velocity law nor a coordinate rescaling of a fixed flat response.

Simultaneous controls, zero modes and response norms

The following common choices are separate examples, not interchangeable numerical margins:

• With raw densities $b_i=gp_{d_i}$, $\ell=1$, fourteen positive targets, $D=I$, $L=lI$, $l=4\lambda_{\max}$, and $g=l/(8\cdot14)$, the changed-metric construction gives $\operatorname{tr}G<1/504$, $M>(503/504)I$, and $\boldsymbol\Lambda^{-1/2}S\boldsymbol\Lambda^{-1/2}>(2015/2016)I$. Each protected full mode has norm one. The polar alternative using this same $B$, $M_0=I,c_p=1$, gives norm two and needs eight extra response directions.
• With normalized sources, measure frequencies in units $H_* =\max_i h_i$, so $\widehat\lambda_i=(h_i/H_*)^2\le1$. Add one dressed zero with source $s_2$, use $D=I_{15}$, $L=4I$ on the actual six-source space, and common $g=1/8$. For $\vartheta=1$, $\operatorname{tr}G\le5/192$ and the kinetic margin is $187/192$. For $\vartheta=1/2$, $\operatorname{tr}G\le15/784$ and $D-G/4\ge(3121/3136)I$. Both have fifteen full target norms equal to one and six complementary modes at squared frequencies at least four.

The proofs allow arbitrary supplied positive $\mu_d$. Executed count controls use equal sector units solely for exact arithmetic, not empirical calibration. Degeneracies from other supplied units are handled by commuting norm blocks. Repeated-depth sources remain identical, and all depth-five labels obey the same law.

A pure zero is an adjoined source-free shift with positive kinetic coefficient. In (23) a sourced zero instead has $Y_0=L^{-1}b_0$, and shifts both retained and response coordinates. Strict complementary positivity makes it the only zero when exactly one target is zero. Its free canonical momentum permits drift; it is not a bounded zero oscillator or an identified photon. For the polar construction, adjoin a pure zero separately: placing both a zero target and zero source column inside (29) would add an unwanted response zero. The separate conditional target $h_b=\mu_3\sqrt{816\cdot969}=\mu_3\sqrt{790704}$ can be appended using the same finite-set rule and a recomputed common range; this does not identify bottom's mechanism or occupation.

For (23), the correct off-pole forced-response pencil is

$$\mathcal F(z)=K_0-zM-(B^*+zC^*)(L-z)^{-1}(B+zC). \tag{32}$$

If $y=(L-z)^{-1}(B+zC)q$, then

$$-q^*\mathcal F'(z)q=q^*Mq+2\langle Cq,y\rangle+\|y\|^2. \tag{33}$$

This is the full lifted kinetic norm; its positivity follows from $M_f>0$, not the sign of a self-energy. For flat response, $\vartheta=1$ and $D=I$, an off-diagonal entry in fixed physical retained coordinates is

$$\mathcal F_{ij}(z)=-G_{ij}\frac{(z-\lambda_i)(z-\lambda_j)}{l-z},\quad i\ne j. \tag{34}$$

It is generically nonzero, including at zero probe frequency for two positive targets. This forced response is compatible with protected normal frequencies; it is not by itself a force between moving particles.

A spatial-gradient response is also possible: $L=\Omega_0^2-c_s^2\Delta$ on $L^2(\mathbb R^{10})$, operator domain $H^2$, form domain $H^1$, with $c_s>0$, $\Omega_0^2>\lambda_{\max}$, and the resolvent and positivity conditions in (23)–(25). Then

$$\widehat Y_i(p)=\frac{\widehat b_i(p)}{\Omega_0^2+c_s^2|p|^2-\vartheta\lambda_i}.$$

The Gram uses the product of the two distinct denominators; the raw overlap divided by one gap is not a substitute. The full response is not restricted to the six-source span. The positive finite-rank corrections define linear energy evolution on these domains, but their nonlocal retained source and inverse terms do not establish whole-system finite propagation. The polar construction instead supplies a finite reducing response space and $L_\perp$; it is not automatically a local gradient PDE.

Physical canonical momentum, phase source and force limits

For fixed $A_0$, the physical canonical momentum and Hamiltonian of (23) are

$$P=A_0M_f\dot z,\qquad H_{\rm phys}=\frac{P^*M_f^{-1}P}{2A_0}+\frac{A_0}{2}z^*K_fz. \tag{35}$$

The vector $M_f\dot z$ is normalized momentum $p=P/A_0$; using it in an unscaled physical canonical one-form loses the action factor. On a real mode $z=aW_i\cos(h_it)$, full norm $Z_i=W_i^*M_fW_i$ gives $I_i=A_0a^2h_iZ_i/2$ and $E_i=h_iI_i$. Doubling the entire action into two quadratures gives

$$Q=A_0(z_1^*M_f\dot z_2-z_2^*M_f\dot z_1),\qquad Q=\pm A_0a^2h_iZ_i,\quad E=h_i|Q|$$

on a circular mode. The response and mixed kinetic energy remain included. Several different normal-mode energies add, together with complementary and zero-mode energy; there is no single universal $hQ$ for a mixed-clock state. Source sign, temporal sign, circular charge sign and geometric deformation sign are distinct. This phase charge is not a derived electric charge, and a common occupied action magnitude remains an input.

Let a physical parameter $\xi$ vary the actual source, operator, metric or spatial inclusion smoothly. Differentiate (35) at fixed physical $(z,P)$ first. For constant $A_0$,

$$H_\xi=\frac{A_0}{2}[-\dot z^*M_f'\dot z+z^*K_f'z].$$

For a smoothly continued reference mode $W$, write $\lambda=h^2$, $Z=W^*M_fW$, $m'=W^*M_f'W$. Differentiating its full eigen-equation yields $W^*K_f'W=\lambda'Z+\lambda m'$; the terms containing $W'$ cancel using the reference equation. On $z=aW\cos\varphi$,

$$H_\xi=\frac{A_0a^2}{2} [-\lambda m'\sin^2\varphi+(\lambda'Z+\lambda m')\cos^2\varphi], \quad \langle H_\xi\rangle=Ih',\quad I=\frac{A_0a^2hZ}{2}, \tag{36}$$ $$\frac{\hbox{coefficient of }\cos2\varphi}{I} =\frac{\lambda'Z+2\lambda m'}{2hZ}.$$

The force conjugate to $\xi$ is $-H_\xi$. This is a reference-mode probe, not the solution for arbitrary time-dependent geometry. All physical derivatives of $B,L,Y,D$, bases, measures and inclusion must be included. Smooth polar differentiation requires controlled rank and a smooth chosen isometry. Varying $A_0$ would introduce extra terms not assumed here.

If the repaired law follows the geometry while all protected $h_i$ remain constant, the averaged pure-mode geometric source is zero at fixed modal action. A changed full kinetic metric can leave a twice-frequency term proportional to $m'$. With a genuinely fixed full kinetic form, as in the polar alternative in fixed physical coordinates, $m'=0$ and the pure-mode geometric source vanishes instantaneously. Fixing only the retained diagonal in the velocity alternative does not imply that result. Two circular quadratures cancel the oscillatory source for a constant target.

Multimode cross terms can beat at sums and differences of frequencies. Their full-torus or infinite-time average vanishes for distinct frequencies; finite-time beats need not vanish. In constant-frequency degenerate blocks the projected $K_f'-\lambda M_f'$ vanishes but oscillatory kinetic terms can remain. Internal restoring forces, reciprocal support acceleration and the nonzero forced-response pencil (34) are distinct from a nonzero geometric pair force. A dynamical separation/orientation sector, complementary excitation or relaxed protection requires its own full action calculation. No completed particle-pair interaction or rate is asserted here.

For a separate conditional adjacent-pair target $h=\sqrt{ab}$, deforming $a'=F_0$, $b'=\pm F_0$ gives mean factors $R_+=(a+b)/(2\sqrt{ab})$ and $R_-=(b-a)/(2\sqrt{ab})$, with $R_+^2-R_-^2=1$. For adjacent counts they are $(2n+d)/(2\sqrt{n(n+d)})$ and $d/(2\sqrt{n(n+d)})$. These retain both deformation signs as comparisons; they do not reassign ordinary particles to geometric means or establish an odd-depth force rule.

Evidence and physical scope

The nonlinear result has 6,708 original-construction checks and 6,336 independent-construction checks, each replayed unchanged. A separate 1,883-check governing comparison verifies rational radical enclosures, off-reference four-profile measures and sparse occupations at distant shells, including $J=1000,10000$, as well as the localization bound. A further 38-check bounded verification supports the spatial and phase estimates. These are finite mathematical diagnostics supporting the analytic proof; they are not time integration, observed lifetimes or an infinite-band simulation.

The response constructions have 509 original and 343 independent checks, replayed unchanged, plus a separate replayed 122-check governing comparison. The latter includes a singular-source polar witness, complementary-mode identities and physical $A_0$ source derivatives. Raw-density and normalized-source examples remain separate. The rank-six fourteen-column polar theorem uses its analytic square-root construction; no numerical fourteen-dimensional radical diagonalization is claimed.

The standalone spatial checker and its dated results contain the independent 6,336-check certificate. Its sufficient leakage basin differs from the governing moment basin (7)–(13); it must not be advertised as executing the governing 1,883 checks or the original 6,708. The response checker and results contain the independent 343-check construction. The provenance record distinguishes those copies from the original and governing evidence and pins the exact source versions.

Together these are stronger compatibility results at all maintained depths, including six and ten. They provide a finite nonlinear spatial persistence theorem and positive simultaneous linear response actions under explicit supplied data. They do not experimentally favor those depths, select observed occupations or sector units, infer lifetimes, derive electric charge or spin, establish photon or bottom mechanisms, or complete the physical mass operator. In particular, the response construction's prescribed templates have not been made autonomous localized solutions by the separate persistence theorem. The origin of the material connection, kinetic repairs and inverse response law remains a physical derivation problem.