A Self-Supported Real Harmonic Core

An exactly harmonic spatial operator can arise from the field's own density, without an imposed external well or a quadratic approximation near a center. One positive-filter specialization of the entropy-difference action has an exact Gaussian stationary profile, a global fixed-norm energy minimum, and a complete formal linear spectrum without growing modes. Its real excitation spaces have the simplex dimensions used in the mass law. It supplies no finite cutoff and does not turn those dimensions into particle frequencies.

All positions are ordinary coordinates in a common $\mathbb R^{10}$. The two real canonical field components describe internal field motion; they do not replace deterministic electron center trajectories.

Action and Exact Spatial Solution

Let $\rho=q^2+p^2$, $J=\int\rho$, and $T=\int(|\nabla q|^2+|\nabla p|^2)$. For Gaussian convolution $K_s$ of covariance $s^2I$, define

$$\mathcal D_s(\rho)=\operatorname{Ent}(\rho)-\operatorname{Ent}(K_s\rho), \quad \operatorname{Ent}(f)=\int f\log f, \qquad H=\frac12[\mu J+\alpha T-\beta\mathcal D_s(\rho)], \quad \alpha,\beta,s>0.$$

Use the real canonical action $\int p q_t-H$. Its equations are $q_t=A_\rho p$, $p_t=-A_\rho q$, where

$$A_\rho=\mu-\alpha\Delta-\beta[\log\rho-K_s\log(K_s\rho)].$$

The outer convolution is part of the full density variation. The action is homogeneous of degree two in field amplitude; $J$ is uncalibrated. This normalization halves both the energy and symplectic term of the linked entropy note, preserving its equation and time frequencies.

If $2\beta s^2>\alpha$, put

$$b=\frac\beta\alpha-\frac1{2s^2}>0,\qquad r=\frac1{1+2s^2b},\qquad \alpha b=\beta(1-r).$$

For $u^2=J(b/\pi)^5e^{-b|x-X|^2}$, direct Gaussian integration gives

$$\log u^2-K_s\log(K_su^2) =5\log(1+2s^2b)+10s^2br-b(1-r)|x-X|^2.$$

The potential is exactly quadratic throughout space, with coefficient $\beta b(1-r)=\alpha b^2$. Consequently

$$q=u\cos\omega t,\quad p=-u\sin\omega t,\qquad \omega=\mu+5[\alpha b-\beta\log(1+2s^2b)],\quad H=J\omega/2$$

is an exact localized solution. The common offset $\mu$ can make its complete energy positive without changing its shape. It is a constitutive choice, not a particle-specific mass fit. This isotropic solution has one support clock and no preferred active plane.

Global Energy Minimum and Evolution

The sharp heat-flow bound in ambient dimension $D$ is

$$\frac{\mathcal D_s(\rho)}J\le \frac D2\log\left(1+\frac{4s^2T}{DJ}\right).$$

With $y=T/J$, it follows that

$$\frac{2H}J\ge f(y)=\mu+\alpha y -\frac{\beta D}{2}\log(1+4s^2y/D).$$

This strictly convex function has its unique positive minimum at $y=Db/2$. The isotropic Gaussian saturates the inequality there, proving a global fixed-$J$ minimum of the full field energy. Equality in the Fisher heat inequality forces an isotropic Gaussian density; equality in the initial Fisher bound forces constant quadrature direction. Thus the minimizer is unique modulo translation and constant phase, at fixed norm. If $2\beta s^2\le\alpha$, spreading approaches the infimum and no nonzero finite-width minimizer of this type exists.

The existing global evolution theorem applies to this one positive filter: data in $H^1\cap L^2(|x|^2dx)$ have unique global finite-energy evolution, with conserved norm and energy, bounded gradient norm, and continuity in that weighted energy space. General second moments can grow. This theorem does not say that every initial pattern stays localized.

Complete Linear Spectrum

The rotating-frame phase and amplitude operators are

$$L_-=\alpha[-\Delta+b^2|x-X|^2-Db],\qquad L_+=L_--2\beta(I-\mathcal R),\qquad \mathcal R a=uK_s\left[\frac{K_s(ua)}{K_su^2}\right].$$

On a real Hermite mode of total degree $k$, $\mathcal R$ has multiplier $r^k$, so

$$\lambda_-(k)=2\alpha bk,\qquad \lambda_+(k)=2\beta(1-r)\sum_{j=0}^{k-1}(1-r^j).$$

Both are positive for every $k\ge2$. Degree zero supplies amplitude and phase; degree one supplies translations and their velocity partners. The Hermite basis is complete, and the bounded operator $\mathcal R$ leaves no additional continuous spectral sector. The formal internal rates are

$$\nu_k=2\alpha b\sqrt{k\left[k-\frac{1-r^k}{1-r}\right]}.$$

This proves the sign of the complete formal linear spectrum. It does not establish a uniform Fréchet linearization of the nonlinear evolution near a decaying profile or nonlinear orbital stability.

The distinction is substantive. Two far-separated scaled Gaussian copies can approach the same minimum energy at fixed total norm. Put $q_R=\sqrt a\,u(x)$, $p_R=\sqrt{1-a}\,u(x-Re_1)$. Their norm is exactly $J$, and the entropy overlap bound gives

$$0\le H(q_R,p_R)-H(u,0) \le2\beta J\sqrt{a(1-a)} e^{-bR^2/[4(1+2s^2b)]}.$$

At fixed $0<a<1$, these near-minimizers stay away from every single Gaussian as $R\to\infty$. Taking one small packet also defeats a uniform quadratic energy-gap bound in unweighted norms. This prevents a routine energy-minimum proof of nonlinear orbital stability; it is not a proof of instability.

Simplex Counts in the Same Ordinary Space

Choose a $d$-plane in the common ten-dimensional space and restrict the Hermite excitations to it, retaining the transverse Gaussian ground factor. Then

$$g_d(k)=\binom{k+d-1}{d-1},\qquad \sum_{k=0}^{N}g_d(k)=\binom{N+d}{d},\qquad N=n-1.$$

These are actual real spatial functions of the self-generated harmonic operator. All fourteen maintained assignments give their exact target counts. The same construction works at depths two, three, four, five, six and ten; no depth-five coefficient or suppression is introduced.

The subspaces are invariant under the linearized operator, with no proved nonlinear invariance or selected orientation, depth or cutoff. The complete harmonic tower is infinite: there is no finite rim. For $N\ge1$, only $\binom{N+d}{d}-1-d$ modes have positive internal frequencies; the remaining degrees include the neutral profile and translations. The down assignment $N=0$ contains no positive-frequency internal oscillator. The full simplex dimension and the number of oscillating disturbances must therefore be kept distinct.

Nor are $\nu_N$ the required count frequencies. For the one common control $\alpha=\beta=s^2=1$, $\nu_k^2=k[k-2+2^{1-k}]$. Its upper-mode muon/electron ratio is about $3.0110$, not $3515/17$; tau/electron is about $1.9148$, not $3475.1261$. The carrier clock remains the common $\omega$.

The Actual Carrier Phase Requires Its Own Equation

For a degree-two real Hamiltonian, $\int(pq_t-qp_t)=2H$. In the exact local phase chart

$$q+ip=e^{-i\theta}[\sqrt{J-\|w\|^2}\,u_0+w],\quad w=a+iz, \quad a,z\perp u_0,$$

where the paired notation only abbreviates two real fields,

$$\dot\theta=\frac{2H}J- \frac1J\int(z\dot a-a\dot z).$$

For small internal oscillations with quadratic energy $Q=(\langle a,L_+a\rangle+\langle z,L_-z\rangle)/2$, the shape term is $2Q$. Its contribution cancels the $2Q/J$ added to $2H/J$. Thus equal small internal-mode energies do not automatically shift the carrier rate in proportion to their count. Higher-order dynamics, resonances and the actual measured signal require further calculation. This does not alter mass-frequency equivalence; it identifies the missing dynamical step.

Motion and Evidential Scope

The original action has exact Galilean translation with $P_v=P_0+Jv/(4\alpha)$ and $H_v=H_0+v\cdot P_0+J|v|^2/(8\alpha)$. Its positive inertia is $J/(4\alpha)$. Frequency-following motion is a separate changed action; neither relativistic inertia nor finite propagation follows from this constant-gradient law.

The useful result is a self-generated harmonic spatial structure, exact localization and complete formal persistence compatible with real simplex mode counting. Occupied-band selection, count-proportional particle clocks, spin, charge and physical identification remain unresolved. The accompanying checker contains 1,421 exact rational and integer checks; the all-degree positivity result follows from the displayed positive sum, not a finite scan.

Reproduce the finite identities with the exact single-filter checker. Its frozen report records the original inputs and source hashes.

Exact Motion for the Self-Supported Core

An explicit changed action combines the exact Gaussian with frequency-following motion. Let $E[\rho]$ be the preceding energy restricted to constant phase, $w=\delta E/\delta\rho$, and write the two real fields as $(q,p)=\sqrt\rho(\cos\theta,\sin\theta)$. Replace the original phase-gradient energy by

$$H_{\rm mov}=E[\rho]+\int\rho\left[ \sqrt{w^2+c^2|\nabla\theta|^2/4}-|w|\right].$$

Use the same canonical symplectic form. At the exact Gaussian, $w=\omega_0/2>0$. For every constant spatial vector $k$,

$$\Omega=\sqrt{\omega_0^2+c^2|k|^2},\qquad v=c^2k/\Omega, \qquad(q,p)=u(x-X-vt)(\cos(k\cdot x-\Omega t),\sin(k\cdot x-\Omega t))$$

solves the full variational equations against every compact smooth real-field variation. This is a whole-space solution, not merely a collective-coordinate ansatz. To justify the density-dependent source variation, put $g[u]=\delta E[u^2]/\delta u$. For compact $\eta$, the Gaussian convolution ratio $K_s(2u\eta)/(K_su^2)$ is bounded by a constant times $\exp[-a|x|^2+R|x|/s^2]$, where $a=[2s^2(1+2bs^2)]^{-1}>0$. Thus the logarithmic source has a strong $L^2$ directional derivative $B\eta$. Self-adjointness and homogeneity give $Bu=\omega_0u$, so $\int u(B\eta-\omega_0\eta)=0$. This cancels the full source-response term in the motion variation; freezing the source is unnecessary.

The complete conserved quantities on this history satisfy

$$E_{\rm mov}=J\Omega/2,\quad P=Jk/2,\quad E_{\rm mov}^2-c^2|P|^2=(J\omega_0/2)^2,\quad \left.\frac{\partial P}{\partial v}\right|_0 =\frac{J\omega_0}{2c^2}I=\frac{E_{\rm rest}}{c^2}I.$$

The added energy is nonnegative. The constant-phase Gaussian therefore retains its global fixed-$J$ minimum on the admitted energy domain. At rest, $L_+$ is unchanged and $L_-$ is multiplied by $\kappa=c^2/(2\alpha\omega_0)>0$. The full formal generator is similar to the original generator multiplied by $\sqrt\kappa$; every Hermite spectral sign and neutral multiplicity is retained. The common example $\alpha=\beta=s^2=1,\mu=2$ has positive $\omega_0$ in ten dimensions.

The changed action's general Cauchy theory, nonlinear orbital stability, finite-speed propagation and Lorentz covariance remain open. The exact traveling history exists for all times, which does not settle those broader questions. This result supplies no distinct particle clocks or selected cutoff.

Sector Scale as Energy per Counted State

The maintained IDOS law has the direct interpretation

$$E_{d,N}=\varepsilon_d D_{d,N},\qquad D_{d,N}=\operatorname{IDOS}_d(N)=\binom{N+d}{d},\qquad \varepsilon_d=\mu_dc^2.$$

The sector scale is the energy per counted state in this interpretation, rather than an additive energy beneath the oscillator. The cumulative count includes every level through the cutoff, not just its highest shell. This restates the maintained mass law; equal contributions are a dynamical hypothesis, not a consequence of counting alone.

For a concrete classical comparison, independent canonical oscillations have $H=\sum_j\nu_j I_j$. Equal energies require $I_j=\varepsilon_d/\nu_j$, whereas equal actions give energy weighted by $\nu_j$. Neither condition follows from the number of states. In the self-supported core, neutral states also prevent interpreting the entire IDOS as a collection of positive-frequency disturbances with equal energies. The useful next target is a common same-field occupation or interaction rule that yields uniform energy contributions and the complete particle clock, without fitting a separate amplitude for each particle. The carrier-phase identity above remains the explicit check on that proposed mechanism.

The IDOS energy and coherence continuation derives the exact filled-ball trace, distinguishes it from coherent local energy, and tests conservative equal-energy occupations with their supporting dynamics. Its additional modal and scale actions retain their own stated assumptions.

Exact Breathing and Rotating Spatial Patterns

The same original entropy-filter equation also supports exact time-dependent Gaussian shapes. Let $A=J/2$, let $G$ be a positive symmetric matrix of squared widths, and let $C$ be a symmetric quadratic phase matrix. In the ordinary ten-dimensional coordinates, write

$$q+ip=\sqrt J\,\pi^{-10/4}(\det G)^{-1/4} \exp[-x^TG^{-1}x/2+ix^TCx/2-i\theta].$$

The notation pairs two real fields. The density covariance is $G/2$. Gaussian convolution gives the exact source

$$\log\rho-K_s\log(K_s\rho) =\tfrac12\log\det(I+2s^2G^{-1}) +s^2\operatorname{tr}(G+2s^2I)^{-1} -x^T[G^{-1}-(G+2s^2I)^{-1}]x.$$

Matching the full field equation, including its constant terms, yields

$$\begin{aligned} \dot G&=2\alpha(CG+GC),\\ \dot C&=2\alpha(G^{-2}-C^2) -2\beta[G^{-1}-(G+2s^2I)^{-1}],\\ \dot\theta&=\mu+\alpha\operatorname{tr}G^{-1} -\tfrac\beta2\log\det(I+2s^2G^{-1}) -\beta s^2\operatorname{tr}(G+2s^2I)^{-1}. \end{aligned}$$

These are exact solutions of the original PDE, including noncommuting $G,C$, rather than trial-profile equations. Their complete energy and spatial angular momentum are

$$H=A\mu+\frac A2[\alpha\operatorname{tr}(G^{-1}+C^2G) -\beta\log\det(I+2s^2G^{-1})],\qquad L_{ij}=\frac A2[G,C]_{ij}.$$

The commutator is conserved directly by the two matrix equations. A block-diagonal preparation preserves its active $d$-plane and leaves the other $10-d$ directions at $G=g_0I,C=0$, where $g_0=2\alpha s^2/(2\beta s^2-\alpha)$. This holds for every maintained depth. Cross-plane perturbations require their own dynamics; block invariance is not universal protection against encounters.

Put $W(g)=\alpha/g-\beta\log(1+2s^2/g)$, $e_0=W(g_0)<0$, $\omega_c=\mu+5e_0$, and $K=H-A\omega_c$. Since

$$\frac KA=\frac12\left[\alpha\operatorname{tr}(C^2G) +\sum_a(W(g_a)-e_0)\right],$$

the sufficient condition $0\le K/A<-e_0/2$ bounds every eigenvalue $g_a$ away from zero and infinity and bounds $C$. Thus the exact Gaussian matrix evolution exists for all time and stays uniformly localized in this energy window. This is a theorem on the invariant Gaussian family, not unrestricted nonlinear orbital stability.

For commuting matrices, write $G_{aa}=L_a^2$. Each width obeys

$$\ddot L_a=\frac{4\alpha^2}{L_a^3} -\frac{8\alpha\beta s^2}{L_a(L_a^2+2s^2)},\qquad C_{aa}=\frac{\dot L_a}{2\alpha L_a}.$$

Synchronized motion in $d$ directions has field-derived radial inertia $A d/(4\alpha)$ and potential $A d[\alpha/L^2-\beta\log(1+2s^2/L^2)]/2$. Every radial energy strictly between its minimum and zero gives bounded periodic breathing. Its small-amplitude rate equals the previously derived degree-two disturbance rate $\nu_2$.

A closed-form rotating example uses the one common control $\alpha=\beta=s^2=1$. Let

$$h=\frac{13+\sqrt{6001}}{54},\quad G_b=\operatorname{diag}(2h,h),\quad \Omega^2=\frac{9(h-1)}{5h^2(h+1)},\quad C_b=\begin{pmatrix}0&\Omega/6\\\Omega/6&0\end{pmatrix}.$$

Conjugate both matrices by the ordinary planar rotation through angle $\Omega t$, and retain stationary transverse widths $g_0=2$. The matrix equations reduce to $27h^2-13h-54=0$, so this is an exact anisotropic rotating spatial pattern. Its illustrative values are $\Omega\simeq0.40235168$ and $K/A\simeq0.02527266$. The latter lies below the sufficient bound $-e_0/2\simeq0.09657359$. No measured particle mass or amplitude was fitted. Independent rotating blocks and scalar breathing directions can coexist under the same equation.

The complete phase reconstruction is

$$\dot\theta=H/A+\tfrac14\operatorname{tr}(G\dot C),\qquad \Delta\theta=(H/A)T-\tfrac14\oint\operatorname{tr}(C\,dG).$$

Projection onto the stationary isotropic real Gaussian never vanishes: its Gaussian integral has a matrix with positive definite real part. Its additional phase returns on a closed shape cycle. Consequently the displayed cycle phase is also that of an actual linear field projection. For the rotating ellipse it is constant, $\dot\theta=H/A-\Omega L_{12}/A$.

For a synchronized breathing cycle with shape action $B=(2\pi)^{-1}\oint P_L\,dL$ and rate $\nu_b$, the mean shift from the stationary carrier is

$$\delta\Omega=\frac{K-\nu_b B}{A},\qquad 0<\delta\Omega<K/A$$

at every nonconstant bounded amplitude. To establish its strict sign, set $x=L/\sqrt{2s^2}$, $a=\alpha/(2\beta s^2)\in(0,1)$, and $w(x)=a/x^2-\log(1+x^{-2})$. For $q=x^2$, put $a_{\min}=q(3q+1)/[3(q+1)^2]$. At $a=a_{\min}+t$, $t>0$,

$$5w'''^2-3w''w''''=\frac{8}{x^{10}(q+1)^6} \left[90(q+1)^6t^2+3q(q+1)^2(35q^2+126q+19)t +10q^2(5q+1)^2\right]>0.$$

At $w''=0$, $w'w'''=-16(5x^2+1)/[3x^4(x^2+1)^5]<0$. The Schaaf criterion stated in Rojas, Theorem 2.1 therefore makes the period strictly increasing throughout the positive-width bounded well. Since $dB/dK=1/\nu_b$, differentiation gives $d\delta\Omega/dK=-(B/A)d\nu_b/dK>0$, proving the bound. The first shift is fourth order in a small width displacement.

These histories give nonlinear continuations of the real quadratic spatial disturbances, with shell dimension $d(d+1)/2$, across all six depths. Their finite deformations generally involve infinitely many fixed-width Hermite coefficients. They do not fill a finite IDOS band or turn its count into a carrier frequency. The phase subtraction quantifies the remaining clock mechanism. These breathing results belong to the original entropy action; they are not automatically solutions of its changed motion action.

Reproduce the identities with the Gaussian matrix checker, dilation and breathing checker, and breathing phase checker.

Polarization and the Complete Pair Kinetic Structure

The entropy action extends to an even-dimensional real component field $\psi$ with a fixed orthogonal skew matrix $J$: $J\psi_t=A_\rho\psi$, $\rho=\psi^T\psi$. Every constant unit polarization $e$ gives the exact core $\psi=e^{-\omega Jt}u e$. In the rotating frame its Hessian is

$$L=L_- I-2\beta e(I-\mathcal R)e^T.$$

The plane spanned by $e,Je$ retains the scalar amplitude/phase calculation. The orthogonal component complement is $J$ invariant and evolves by $\eta_t=-JL_-\eta$. Each extra Hermite pair therefore has rate $2\alpha b k$. Ground polarization changes are semisimple zero modes; positive degrees oscillate. This is the complete formal component linearization, not nonlinear orbital stability. Fixed invariant component subspaces remain empty if initially empty, but the action does not select one minimal spin module.

Replacing the kinetic term by $m\int|\psi_t|^2/2$, $m>0$, is a different action. For $\omega>0$ it retains the Gaussian circular core with $\Omega^2=\omega/m$ and complete energy $N\omega$, where $N=\int\rho$. Its transverse rotating-frame frequencies are $\pm\sqrt{\Omega^2+2\alpha b k/m}\pm\Omega$. The longitudinal polynomial is $m^2\lambda^4+[m(\lambda_++\lambda_-)+4m^2\Omega^2]\lambda^2+ \lambda_+\lambda_-=0$. The displayed positive branch has no exponentially growing formal mode, while changing both clock and phase-space dimension.

For two Gaussian amplitudes $f,g$ with real polarizations $a,b_0$, the actual shared density is $\rho=f^2+g^2+2(a\cdot b_0)fg$. Orthogonal polarizations remove interference but not the entropy interaction. The excess over isolated cores is nonnegative in that orthogonal control, vanishes at coincidence and at infinite separation, and forms a barrier. For equal norms $N_0$, with $v=1/(2b)$, its small-separation value is

$$V_{\perp}(R)=\frac{\beta(2N_0)s^2}{16v(v+s^2)}R^2+O(R^4).$$

This is an energy calculation. Converting its derivative into motion requires the full field kinetic term. With $z=q+ip=z_1+z_2$ and $z_i=\sqrt{2A_i}\phi_{X_i}e^{ik_i(x-X_i)-i\vartheta_i}$, the exact collective one-form is $-\operatorname{Im}\int z^*dz/2$. Its eight-coordinate matrix is $F_{ab}=-\operatorname{Im}\int(\partial_a z)^*\partial_bz$ and the variational equation is $F\dot\lambda=\nabla H$. Both the energy and this matrix retain interference.

For $R=X_2-X_1$, let $S=e^{-bR^2/4-(k_2-k_1)^2/(4b)} e^{i[\vartheta_1-\vartheta_2-(k_1+k_2)R/2]}$. The conserved phase action and translation momentum are

$$A_{\rm total}=A_1+A_2+2\sqrt{A_1A_2}\operatorname{Re}S,$$ $$P=A_1k_1+A_2k_2+\sqrt{A_1A_2} [(k_1+k_2)\operatorname{Re}S+bR\operatorname{Im}S].$$

The matrix is nondegenerate for positive actions and distinct complex Gaussian labels $bX_i+ik_i$. Coincident labels make the coordinates redundant, not the physical field force infinite. Individual label momenta omit an interference contribution and need not follow raw separation-energy slopes. Nine initial controls using the actual entropy Hamiltonian conserve total momentum to a maximum derivative residual of $1.3\times10^{-10}$; some label responses have the opposite sign to that raw slope. These are variational initial velocities on a fixed-width family, not exact PDE encounters. All isolated scalar fundamentals have the same clock; these are explicitly equal-frequency controls.

The polarized-core report contains 820 checks and independent momentum formulas. The kinetic and initial-velocity scripts add 31 and 45 checks, including independent quadrature. Active spatial depth, unlike selected clocks, full encounter deformation and IDOS occupation remain open.

A Regular Local Carrier Penalty

For a fixed active spatial decomposition, write the real field in its Schur matrix components, with $g=X^\dagger X/2$, $n=\operatorname{Re}\operatorname{tr}g=|X|^2/2$, and a fixed skew clock matrix $j^2=-I$. The following added energy is local:

$$W(g)=\frac B4[n^2-\operatorname{Re}\operatorname{tr}g^2] +\frac C4\|[g,j]\|^2,\qquad V(X)=\begin{cases}W(g)/n&n>0,\\0&n=0,\end{cases} \qquad H_{\rm sel}=H_0+\int V(X),\quad B,C>0.$$

Here $H_0$ is the real half-energy entropy action and $N=\int n$. The matrix notation describes internal classical field components, not extra physical dimensions. All active-depth and representation premises remain explicit inputs. With $M=B(nI-g)/2+C[j,[g,j]]/2$, complete variation gives

$$\nabla_XV=X\left(\frac M n-\frac{W}{n^2}I\right).$$

The quotient term cannot be omitted. The function is nonnegative, homogeneous of degree two, and smooth away from zero. Its off-zero Hessian is homogeneous of degree zero and bounded on the component unit sphere. Thus $V$ is $C^{1,1}$, with a globally Lipschitz gradient extending by zero at the origin. It generally is not $C^2$ there. These statements concern the added force; they do not alone establish evolution for the complete entropy equation.

On a constant compatible rank-one polarization, both $V$ and its gradient vanish. The exact Gaussian therefore retains its clock $\omega_s$ and energy $N\omega_s$ at every amplitude. In a pure real-Schur block, $W\ge c n^2$, $c=\min(B/8,C/2)$, so $H_{\rm sel}/N\ge\omega_s+c$; a minimizing constant polarization times the Gaussian attains the bound. For the common coefficients $B=24,C=6$, the separation is exactly $3$. Real-type clocks survive at the higher rate. A small real admixture into a compatible block has a small cost, not the full pure-type gap.

At a compatible Gaussian, the complete added component Hessian commutes with the clock structure and has possible stiffnesses $c_a=0,B/2,2C$. Multiplicity depends on the representation. The longitudinal amplitude/phase block is unchanged; transverse Hermite rates are $2\alpha b k+c_a$ for first-order kinetics. For the separate positive second-order action they are $\pm\sqrt{\Omega^2+(2\alpha b k+c_a)/m}\pm\Omega$, with $\Omega^2=\omega_s/m$. The latter has circular energy $2N\omega_s$. These are formal spectral statements, not nonlinear orbital stability.

The force bound also gives $|V(X+Y)-V(X)-V(Y)|\le L|X||Y|$. Hence the integrated added energy becomes additive for widely translated $L^2$ fields. The integrated Gram alternative lacks this clustering property, while an unnormalized local quartic penalty loses its finite gap under fragmentation. The entropy part still has its own nonlocal convolution and does not acquire finite propagation.

Local compatibility is weaker than a single fixed carrier for the whole pattern. In a compatible multiplicity-two complex block, let $X_\epsilon=u(x)a(\cos\epsilon x_1,\sin\epsilon x_1)$, where $a$ is a constant unit left factor and $u$ is the core Gaussian. Pointwise $V=0$, but

$$H_{\rm sel}[X_\epsilon]/N=\omega_s+\alpha\epsilon^2,\qquad \operatorname{spec}\int g= \left\{\frac N2(1+e^{-\epsilon^2/b}), \frac N2(1-e^{-\epsilon^2/b})\right\}.$$

Every nonzero $\epsilon$ occupies two fixed copies at arbitrarily small excess energy. Exact minimizers remain constant compatible polarizations; there is no positive threshold excluding every textured pattern. Fixed compatible subspaces are invariant, with velocity in the same subspace for second-order kinetics. A freely varying rank-one factorization is generally not invariant because its Laplacian creates normal components. No generic purification or depth-five interaction suppression follows.

Independent exact checks verify the quotient force, noncentral quaternionic Hessian, texture example and both kinetic branches.

Direct Field Encounters and Shape Accuracy

The original entropy equation admits an exact reduction to one varying spatial coordinate times nine stationary transverse Gaussian factors. This retains ordinary ten-dimensional support. Direct integration of that reduced PDE provides an independent check of the pair approximation, including interference and shape deformation. All following preparations have equal isolated scalar clocks; they are not unlike-particle or selected-IDOS encounters.

With $\alpha=\beta=s^2=1$, $\mu=3$, transverse $b=1/2$, and initial packet actions one, the isolated clock is $2.0342640972$. Four histories were integrated through $t=2$: initial separation four with relative phase zero, $\pi/2$ or $\pi$, and an incoming control at separation five with momenta $\pm0.3$ and phase $0.7$. The fixed-width variational fields have final relative $L^2$ errors $0.10837$, $0.13579$, $0.15117$ and $0.07249$ against the direct PDE. Excellent conservation within a reduced family does not make that family an accurate field evolution.

Adding independent widths and quadratic phase curvatures gives six real coordinates per Gaussian. The same original energy and full symplectic pullback then reduce the in-phase error to $0.017326$ and the opposite-phase error to $0.071543$. A global phase adjustment changes these only slightly. For one isolated packet the enlarged family reproduces exactly

$$\ddot L=\frac{4\alpha^2}{L^3} -\frac{8\alpha\beta s^2}{L(L^2+2s^2)},\qquad c=\frac{\dot L}{2\alpha L},\quad b=L^{-2}.$$

Its independent breathing integration agrees to $2.26\times10^{-13}$ over the tested interval. At $t=2$ the in-phase center labels approach slightly while their widths broaden. These labels describe the sum ansatz, not a unique decomposition of the full field into objects.

More coordinates also aggravate near-coincident redundancy. At identical center, momentum, width and chirp, the twelve-coordinate symplectic matrix has rank six. An independent 70-digit calculation gives condition numbers $2.34\times10^7$ at separation one and $2.66\times10^{16}$ at separation $1/8$ in the stated coordinates. This is a coordinate limitation, not an infinite physical force. Neither tested history reaches the conditioning gate.

The direct PDE uses Fourier kinetic propagation, positive Gaussian convolution and Strang splitting. Refining space and time changes the final field by $4.78\times10^{-7}$ in phase and $2.32\times10^{-5}$ in opposite phase, well below the approximation errors. Separate time, space, box and density-floor checks corroborate the comparison; these are numerical convergence tests, not rigorous continuum error bounds. Exact zeros use the limiting products $z\log|z|^2=0$ and $\rho\log\rho=0$.

The shape calculation passes 44 independent algebraic and numerical checks. Longer encounters, richer shape channels and different selected particle clocks require further work.

Nonlinear Evolution with the Local Selector

For a fixed finite real component space and a constant orthogonal skew clock $J$, the first-order entropy equation with the local selector has a unique global solution for every datum in $\Sigma=H^1\cap L^2(|x|^2dx)$:

$$X\in C(\mathbb R;\Sigma)\cap C^1(\mathbb R;H^{-1}).$$

More generally this holds for any nonnegative $C^{1,1}$ added density $V$ with $V(0)=\nabla V(0)=0$, globally Lipschitz gradient of constant $L_V$, and pointwise clock-phase invariance. A finite list of entropy filters with real coefficients $\beta_j$ is allowed. Norm and energy are conserved, the gradient stays bounded, and

$$\|X(t)-Y(t)\|_2^2\le e^{(4\sum_j|\beta_j|+2L_V)|t|}\|X(0)-Y(0)\|_2^2, \qquad \|xX(t)\|_2\le\|xX(0)\|_2+2\alpha|t|\sup_\tau\|\nabla X(\tau)\|_2.$$

The solution map is continuous in $\Sigma$ on every bounded time interval. This permits growing second moments; it is not uniform localization or nonlinear orbital stability. Promoting the fixed decomposition or clock to a field-dependent quantity changes the equation and is outside this theorem.

A useful domain improvement is that the entire nonlinear entropy force is $L^2$-valued and continuous under strong $L^2$ convergence on bounded $\Sigma$ sets. For $f=K_s|X|^2$, Jensen and self-adjointness give

$$\int |X|^2|K_s\log f|^2\le\int f(\log f)^2.$$

For $f\le1$ and any $a>0$, $f(\log f)^2\le a|x|^2f+(16/e^2)e^{-\sqrt a|x|/2}$. Choosing $a$ small and then a large spatial cutoff controls tails using bounded second moments alone. A spare Sobolev exponent controls the positive local logarithm. This also handles the zero-field limit. The Laplacian, not the nonlinear force, prevents a general $L^2$ time-derivative claim.

The construction uses Hamiltonian finite-dimensional approximations with a small positive Gaussian density background, norm conservation, Fisher coercivity, strong local compactness and moment-controlled tails. The vector logarithmic inequality gives uniqueness. Energy lower semicontinuity first gives an inequality; restarting the unique flow in the opposite time direction gives equality. Strong gradient and weighted continuity then follow from the energy and moment identities. The full proof and independent audit are preserved in selector-evolution/report.md and evolution-audit/report.md under the current run directory in MariaDB; the audit sharpens the original report's nonlinear-force domain statement.

For the separate positive second-order kinetic law, positive core energy gives a conditional bound on the $H^1$ field and $L^2$ velocity of an existing energy-conserving solution. A complete wave existence and uniqueness theorem is still open. The nonlinear logarithmic force is not locally Lipschitz at fields with empty regions, although that fact alone does not rule out other uniqueness methods. Nor does $\Sigma\times L^2$ preserve a finite displacement moment even for the linear wave comparison: disjoint velocity packets at rapidly diverging centers can have finite total $L^2$ norm and infinite displacement moment at every sufficiently small positive time. Stronger weighted data or a different phase space are needed for such a wave theorem.

Actual Field Variance During an Encounter

For the original scalar entropy action, put $M=\int\rho$, $T=\int|\nabla z|^2$, $P_{\rm raw}=\operatorname{Im}\int z^*\nabla z$, and $\mathcal V=M^{-1}\int|x-X|^2\rho$. The canonical momentum used above is $P=P_{\rm raw}/2$. Under the stated decay and regularity conditions,

$$\ddot{\mathcal V}=8\alpha^2 \left[\frac TM-\left|\frac{P_{\rm raw}}M\right|^2\right] -\frac{4\alpha\beta s^2}{M}\int\frac{|\nabla K_s\rho|^2}{K_s\rho}.$$

This follows from $K_s(x\rho)=xK_s\rho+s^2\nabla K_s\rho$ and the full nonlinear force moment. Independently, norm-preserving spatial compression has entropy derivative $s^2 I(K_s\rho)$ and gives the same identity. The tensor identity replaces kinetic and Fisher scalars by their directional matrices. Stationary transverse Gaussian contributions cancel exactly.

Both curvature signs occur: in the common unit control an unchirped Gaussian with inverse squared width $1/4$, $1/2$ or $1$ has curvature per direction $-1/3$, $0$ or $4/3$. The four direct encounter preparations above have positive initial curvatures $0.0750803$, $0.777548$, $1.928106$ and $1.159200$. The incoming preparation nevertheless initially contracts, with variance derivative $-3.001587$. The in-phase field variance grows from $4.523188$ to $4.560741$ through $t=2$, despite decreasing variational center separation. Shape broadening and coherent interference must be included before interpreting center labels as motion of the complete field.

The 63-check field-virial script verifies the force moment, independent dilation, Gaussian matrix closure, both signs and direct-PDE short-time refinement. The identity does not establish long-time capture, scattering or selected particle clocks.

Carrier Selection Changes Actual Pair Dynamics

In a compatible complex two-by-two component block with scalar clock $j=i$, the invariant diagonal preparation $X=\operatorname{diag}(z_1,z_2)$ gives

$$p=|z_1|^2,\quad q=|z_2|^2,\quad \rho=p+q,\qquad V=\frac{Bpq}{4\rho},\qquad (\nabla V)_1=\frac B2\left(\frac q\rho\right)^2z_1, \quad(\nabla V)_2=\frac B2\left(\frac p\rho\right)^2z_2.$$

The force extends by zero at vacuum. Each component norm is conserved in this preparation, so its center is an unambiguous field moment rather than a Gaussian label. The exact axial reduction retains nine stationary transverse Gaussian factors in the common ordinary ten-dimensional space.

For two equal-action Gaussian components at separation $R$, the added energy is

$$U_B(R)=\frac{BA}{4}e^{-bR^2/4} \mathbb E_{h_b}[\operatorname{sech}(bRx)],\qquad h_b=\sqrt{b/\pi}e^{-bx^2}.$$

The expectation denotes a normalized classical integral, not a position probability law. Together with the original entropy interaction its curvature at coincidence is $Ab(B_{\rm crit}-B)/4$, where $B_{\rm crit}=4\beta(1-r)=4\alpha b$, $r=(1+2bs^2)^{-1}$. For $\alpha=\beta=s^2=1$, $b=1/2$, this is $B_{\rm crit}=2$. The complete initial field force agrees with the restricted energy derivative, $\ddot R=-4\alpha U'(R)/A$, to below $9\times10^{-13}$ in the numerical checks.

Direct PDE evolution releases the shape restriction. From $R=0.2$, $A=1$, and zero initial phase slopes, the separation at $t=1$ is $0.109402$ for $B=0$, $0.199576$ for $B=2$, and $2.215573$ for $B=24$. In the last case each component variance contracts from $1$ to $0.534967$, while total variance grows from $1.01$ to $1.762158$. Attraction, component contraction and total spreading are distinct diagnostics. At separation four every tested $B=0,1,2,4,24$ initially repels. These are equal-clock module controls, not different observed particle branches or a depth-five coupling prediction.

The coincident matrix Gaussian $X_0=uI/\sqrt2$ is itself an exact nonminimal core with clock $w=\omega_s+B/8$ and energy $Nw$. Its complete eight-real-component formal spectrum explains the threshold. In the orthonormal Hermitian Pauli basis, the trace amplitude/phase pair retains $L_+,L_-$. Each of three traceless polarization pairs has $L_--B/2,L_-$. Thus at Hermite degree $k$, with $q_k=2\alpha b k$,

$$\lambda^2=q_k(B/2-q_k).$$

There is exponential growth precisely when $B>4\alpha b k$, neutrality at equality and oscillation below. At $B=24$ the reference first-order system has growing degrees one through eleven, with maximum rate six at degree six. The constant traceless modes have phase shear rather than exponential growth. Full left and right matrix charges $Q_L=\int XX^\dagger/2$ and $Q_R=\int X^\dagger X/2$ are conserved. Fixing them removes the uniform charge-changing amplitudes, but exact norm-preserving translated paths show that the spatial growing modes remain accessible.

For the separate second-order kinetic action, $m\Omega^2=w>0$ and the traceless exponent polynomial is instead

$$m^2\lambda^4+m(2q_k+4\omega_s)\lambda^2+q_k(q_k-B/2)=0.$$

The positive-$\omega_s$ branch has the same onset and different rates. If $\omega_s<0$ but $w>0$, a uniform traceless instability also exists; at $\omega_s=0$ the zero-mode Jordan structure changes. Its conserved matrices are angular momenta, not the first-order Gram charges. The two kinetic laws must not be identified by transferring their stability formulas.

Shape Relaxation Removes the Apparent Nearby Minimum

The fixed-width energy has a positive quartic coefficient at $B=2$ and shallow displaced minima for approximately $2<B<2.0835063$. The full stationary calculation changes that conclusion. In a real reflection-symmetric diagonal preparation, put $z_1=u\cos(\pi/4+v)$, $z_2=u\sin(\pi/4+v)$, with $u$ even, $v$ odd and $J_\rho=\int u^2$. The exact reduced energy is

$$H=E[u]+\tfrac12\int u^2(v')^2+\tfrac B{16}\int u^2\cos^2(2v).$$

At the reference threshold write $v=ax+a^3w+\cdots$ and $u=u_0[1+2a^2(x^2-1)-4a^4+\cdots]$. Solving the full forced density equation cancels the quartic coefficient exactly. Eliminating the next odd shape gives $w=-4(x^3-3x)/3$ and

$$H-H_{\rm coinc}=J_\rho\left[\frac{2-B}{4}a^2-\frac{32}{3}a^6 +O(a^8+|2-B|a^4)\right].$$

The nearby nonzero stationary branch therefore lies below threshold: $2-B=128a^4+O(a^6)$, with center separation $R=4a+O(a^3)$. It has a negative constrained energy direction. Forty independently refined stationary solves corroborate the expansion; at a degree-one coefficient $0.005$, they give $B=1.99999992004136$ and $R=0.02000125$. This is formal local analysis and converged stationary numerical evidence, not a function-space bifurcation theorem or a classification of remote branches. It excludes the apparent fixed-width minimum near threshold as a full energy minimum. The exact initial-force calculation remains valid.

The direct flow has 44 histories and 50 assertions; the complete spectrum has 23 symbolic checks and sixteen independent rational certificates. No observed lifetime, stable binary or mass-frequency selection is inferred.

Both Sides of the Common Smoothing Parameter

The preceding sixth-order result is specific to $r=1/2$. Keeping the same action and allowing its common smoothing ratio to vary gives a different outcome. In the unit-Gaussian coordinate $\xi=\sqrt{2b}x$, write $v=a\xi+O(a^3)$ and $u/u_0=1+a^2f+O(a^4)$. The complete forced density correction is $f=(\xi^2-1)/(1-r)$. Eliminating it gives

$$\frac{H-H_{\rm coinc}}{J_\rho} =\frac{B_{\rm crit}-B}{4}a^2+q a^4+ O(a^6+|B-B_{\rm crit}|a^4),\qquad q=\frac{2\alpha b(1-2r)}{1-r}.$$

Thus $B-B_{\rm crit}=8q a^2+O(a^4)$ and $R=4a/\sqrt{2b}+O(a^3)$. For $r<1/2$ the branch is supercritical with positive reduced separation curvature; for $r>1/2$ it is subcritical with a negative direction. The degenerate boundary retains the negative sixth-order term derived above. This varies common constitutive parameters, not independently fitted particle parameters.

At $\alpha=1,b=1/2$, the controls $r=1/3$ and $2/3$ have respectively $q=1/2$ and $-1$. Thirty-six stationary solves at three Hermite cutoffs confirm both branches. All eighteen tested $r=1/3$ Hessians are positive on the real reflection-paired norm tangent; all eighteen $r=2/3$ cases have a negative direction. The amplitude constraint used to locate each stationary solution is released when forming this Hessian. At coefficient $0.12$, the favorable control gives $B=2.04026868598704$, $R=0.490895270685$ and minimum constrained stiffness $0.0210463921$. The complete component and off-axis spectrum requires a separate check. The exact expansion, original entropy-integral verification and stationary coefficients are preserved in general-pair-bifurcation/ under this run.

Matrix Charges and the Global Energy Infimum

For a central-clock complex matrix block and $B>0$, fix any two positive semidefinite matrices $Q_L=L,Q_R=R$ with common trace $N>0$. Then

$$\inf_{Q_L=L,\,Q_R=R}H=N\omega_s.$$

The bound follows from the sharp entropy minimum and nonnegative selector. To approach it with exact charges, decompose both matrices into equal-weight rank-one factors using a finite Fourier frame. If their ranks are $r_L,r_R$, $K=\max(r_L,r_R)$ factors suffice, and the rank bound proves this is the minimum number of separated constant rank-one pieces. Place those factors on disjoint smooth normalized truncations of the Gaussian. Both charges are then exact at finite separation; the selector vanishes pointwise. Increasing the truncation radius recovers each isolated minimum, and increasing their separation removes the smoothed entropy cross term exponentially.

Equality is attained only when both charge matrices have rank one, because equality in the support bound requires a constant-polarization Gaussian and zero selector requires rank one. Even that charge level has noncompact near-minimizing sequences formed from separated pieces of the same carrier. Thus a positive local Hessian and the global fragmentation infimum are compatible. This proves neither dynamical fission nor nonlinear instability. For $B=0$, attainment instead permits matching nonzero charge spectra. Noncentral clocks and second-order angular-momentum charges require their own analysis. The complete proof and 248 exact checks are preserved in matrix-charge-fragmentation/ under this run.