Spatial and dynamical stability

A spatial pattern can have positive high-frequency inertia and causal characteristics while its lower-frequency shape still grows. A purely imaginary linear spectrum can coexist with secular drift unless neutral modes and the evolution norm are controlled. A stable pattern on a prescribed closed ball need not remain localized when its support and exterior are varied. Stability results must therefore identify the action, background, varied fields, domain, norm and duration they actually cover.

IDWT does not require every particle candidate to be infinitely stable. Electron-like candidates need substantial persistence; short-lived branches may be localized resonances. A negative energy direction or a growing linear disturbance is not yet a decay process: its timescale, outgoing channels and complete energy, momentum and charges remain to be calculated. Conversely, an energy-minimum theorem is a sufficient stability condition, not a universal definition of a particle.

Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Field components, harmonic coefficients and perturbation spaces do not add physical dimensions or a second physical manifold. Internal pattern stability does not replace the electron's deterministic four-direction external trajectory around a nucleus. No probabilistic position law or mediator particles are introduced.

Reviewed Connections as of 5 November 2026

Recent results occupy different levels of this stability taxonomy. The completed-support selector has finite-chart internal-pattern stability under its strict gap hypotheses; it is not an attractor or physical formation theorem. The centered-window minimum is an existence and constrained-energy result, without general-data uniqueness or relaxation. Positive initial phase-kick response does not prove positive energy curvature after all shapes relax at fixed momentum.

The six-depth density profiles have finite-domain numerical corroboration rather than a proved lifetime. A separate rank-twelve centered-only family is exactly stationary above its minimum; that ordering supplies no decay rate. At positive gradient strength, small exterior window weights are not an exact absence or a proof of physical count stability.

Levels of a stability claim

For a stationary background at particle frequency $\omega$, a common linearized structure is

$$M\ddot q+G\dot q+Kq=0,\qquad M=M^*>0,\quad G^*=-G,\quad K=K^*.$$

The background particle frequency is distinct from the disturbance frequencies of this equation. Retaining the complete gyroscopic term $G$ is essential. Positive kinetic energy alone does not make the spectrum stable, and an indefinite $K$ need not force growth when gyroscopic coupling is present.

Several different conclusions are useful:

• Component stability: perturbations in a retained finite component space do not grow exponentially. Spatially varying disturbances may still couple outside that space.
• Full spatial spectral stability: the complete specified generator has no off-axis spectrum. Neutral Jordan chains and norm growth still need attention.
• Bounded linear evolution: the initial-value problem has a group bounded in a stated complete norm, including neutral modes and high frequencies.
• Nonlinear persistence: the exact evolution stays near a family in a norm that also preserves the action's domain and supporting geometry.
• Physical persistence or decay: a localized complete object retains or loses its structure through the actual exterior with accounted energy–momentum and charges.

The constructions below reach different entries in this list. Their conclusions are not transferred between actions.

A finite nonlinear bound for compact shapes

The compact law in localization and support now has a finite support-energy estimate with all angular directions retained. On its fixed ball put $y=(x-X)/R$, $s=1-|y|^2$, $f=F_R(1+h)$, $v=sh$, and normalize the measure $d\mu=s^5d^{10}y/Z_5$. The domain requires exact $\int f^2=1$, $|h|\le\epsilon\le1/100$, the ten conditions $\langle v,y_a\rangle_\mu=0$, and the stated weighted form and boundary limits. These conditions fix a mathematical translation slice; it is neither a density-centroid condition nor a claim that any deformed boundary can be put in this form.

The natural Hessian operator and its normalized constraint are

$$\mathcal A_0=-s\Delta_y+12y\cdot\nabla_y-12,\qquad Q[v]=\langle v,\mathcal A_0v\rangle_\mu, \qquad 2\langle s^2,v\rangle_\mu+\langle sv^2\rangle_\mu=0.$$

The Hessian has one negative direction, ten translation zero modes and a positive constrained complement. The exact nonlinear constraint retains the fourth-order negative contribution; replacing it with its tangent condition would lose that term. With

$$b=\mathcal A_0^{-1}s^2=\frac{s^2+7s/4-7/2}{84},\qquad D=\langle s^2,b\rangle_\mu=-\frac{123}{16016},$$

the decomposition $v=z+tb$, $t=\langle s^2,v\rangle_\mu/D$, gives $Q[v]=Q[z]+t^2D$. The full tangent gap is $123/14$. Exact normalization and $\|b\|\le1/24$ therefore yield

$$Q[v]\ge\left[\frac{123}{14}\left(1-\frac{\epsilon}{48|D|}\right)^2 -\frac{\epsilon^2}{4|D|}\right]\|v\|_\mu^2 \ge8\|v\|_\mu^2.$$

The singular boundary remainder is controlled by the weighted divergence identity $\operatorname{div}(ys^5)=20s^5-10s^4$. If $G=\int s|\nabla v|^2d\mu$, this gives $\int v^2/s\,d\mu\le4\|v\|^2+G/25$. The fractional-power Taylor remainder has absolute value at most $(34/5)\epsilon Q[v]$. Together these establish the finite, independently reviewed bound

$$E_{\rm sh}[f^2]-\frac{104\gamma}{7R^2} \ge\frac{7\gamma}{R^2}\int\frac{(f-F_R)^2}{s}\,d^{10}x.$$

This is stronger than a tangent Hessian test, but it is not a nonlinear evolution theorem. The pointwise relative bound, support ball, inverse-coordinate domain and slice have not been proved invariant. New lobes, support bridges and arbitrary energy-small perturbations are outside the theorem. In the new cluster Hamiltonian the bound supplies a restricted rest-mass lower bound; equality requires compatibility with every conserved charge. No selected count, lower-depth confinement or physical decay time follows from it.

A complete count-clock stability theorem on a prescribed ball

The material pullback action varies both the wave and its clamped material carrier. Its weighted ball operator has degree eigenvalues $N(N+d+1)$, and the supplied positive count polynomial gives

$$\omega_N=\mu_dD_N,\qquad D_N=\binom{N+d}{d}, \quad d\in\{2,3,4,5,6,10\}.$$

Balanced real quadrature pairs solve the complete stationary carrier equation. For a general real field $\Psi=A(x)\cos\omega t+B(x)\sin\omega t$, that cancellation requires

$$\nabla(|A|^2-|B|^2)=0,\qquad \nabla(A\cdot B)=0.$$

A complex spatial profile packages two real quadratures and can carry a stationary current. A single rotating scalar dipole is a different preparation and generally drives the carrier.

For each fixed nonzero profile in each of the fourteen maintained degree spaces, the complete pullback theorem gives a common sufficiently small-amplitude regime with imaginary spectrum and uniformly bounded linear evolution. A sufficient material inequality is

$$\boxed{\kappa[\mu_d(D_N-1)]^2<\Lambda_1(L_{\rm el}),} \qquad L_{\rm el}=-\Delta-2\nabla\operatorname{div}.$$

With the prescribed elastic coefficients, clamping gives $\Lambda_1\ge d\pi^2/4$. One sufficiently small positive carrier inertia $\kappa$ satisfies the strict condition for the finite collection of all fourteen fixed profiles. This is a common constitutive regime. The theorem proves existence of an amplitude interval, not its physical value or a certified numerical width.

The actual domain and negative-energy branches

Let $w=\sqrt{1-r^2}$, $P=P_d(L)$, and let $q=(a,h,v)$ contain both wave quadratures and every clamped carrier displacement. The form spaces are

$$X=L^2_w\oplus L^2_w\oplus L^2(B;\mathbb R^d), \qquad V=D(P^{1/2})\oplus D(P^{1/2})\oplus H^1_0(B;\mathbb R^d).$$

For background $\alpha u$, $u=u_R+iu_I$, the complete positive mass form is

$$m(q,q)=\|a+\alpha v\cdot\nabla u_R\|_w^2 +\|h+\alpha v\cdot\nabla u_I\|_w^2 +\kappa\|v\|_2^2.$$

The stiffness is $K=\operatorname{diag}(\mu_d^2P-\omega_N^2,\mu_d^2P-\omega_N^2,L_{\rm el})$. It has

$$n_-(K)=2\binom{N+d-1}{d}$$

negative directions, interpreted as zero at $N=0$. These are retained. The weighted gyro forms are bounded on these actual spaces; the proof handles the singular derivative of $w$ using the clamped Hardy estimate.

A complex profile adds a carrier gyroscopic term

$$G_{cc}(v,z)=-\omega_N\alpha^2\int w\,j\cdot [(v\cdot\nabla)z-(z\cdot\nabla)v], \quad j=\operatorname{Im}(u^*\nabla u).$$

Dropping this term would discard part of the physical two-quadrature perturbation. It does not change the conserved energy or negative index, but belongs to the full collision argument.

The exact count search has one opposite-sign wave collision among the fourteen labels:

$$(d,N,m,l)=(2,75,30,102),\qquad (D_m,D_N,D_l)=(496,2926,5356).$$

The proof treats the complete $31+103=134$ dimensional collision space. Its carrier-source map is injective on both entire polynomial shells, and the leading positive source-Gram form separates the opposite signatures at order $\alpha^2$. A single zonal sample would not establish that conclusion. The exceptional conditioning affects the unknown width of the small-amplitude interval.

Why the linear evolution is actually bounded

On $Z=V\oplus X$, the precise generator is

$$\mathcal A(q,p)=(p,-M^{-1}(Kq+Gp)), \quad D(\mathcal A)=\{(q,p)\in V\times V:Kq+Gp\in X\}.$$

The domain retains possible cancellation between $Kq$ and $Gp$; it does not demand that each separately lie in $X$. A positive stiffness shift proves generation, while compact form embeddings supply the spectral realization. The stationary kernel is the entire two-quadrature occupied shell, not only the phase tangent. Its restricted gyro is invertible,

$$G_E=\begin{pmatrix}0&2\omega_N\\-2\omega_N&0\end{pmatrix} \otimes I_{V_N},$$

which excludes a zero Jordan chain and gives a bounded stationary projection $P_0$.

The finite invariant negative space $W$ exhausts the negative index. If $P_W$ is its energy-orthogonal projection and $\mathscr H$ the conserved indefinite energy form, the theorem constructs the positive norm

$$\|z\|_\sharp^2=\mathscr H(z,z) -2\mathscr H(P_Wz,P_Wz)+\|P_0z\|_*^2.$$

It is equivalent to the complete form norm and conserved. Hence

$$\|T(t)z\|_*\le C\|z\|_*,\qquad t\in\mathbb R.$$

This includes high frequencies and neutral modes. It is stronger than observing imaginary eigenvalues in a truncated matrix. For a finite collection of fourteen fixed profiles, one may take a common amplitude interval and the largest of their finite bounds. Uniformity over arbitrary profiles in every shell is not proved.

The carrier is controlled in $H^1_0$, not in $\|Dv\|_\infty$. At these physical depths that norm does not preserve invertibility of $\mathrm{id}+v$. Therefore the theorem does not establish global nonlinear material evolution, a free boundary, observed particle lifetimes or complete localization. The high-order count dispersion also retains unbounded high-frequency group speed; this stability theorem is not a causal completion.

Forced response and finite persistence

The same complete linear norm gives, for a load $F(t)\in Z$,

$$\|z(t)\|_\sharp\le\|z(0)\|_\sharp +\int_0^t\|F(s)\|_\sharp ds.$$

A mechanical load belongs directly to this statement when its force is in $X$; a force known only in $V^*$ or applied at the boundary needs an additional admissibility argument.

Constant forcing can generate $tP_0F$ drift along the stationary family even though the unforced zero mode is semisimple. A harmonic load grows linearly if it has a nonzero resonant projection. At a fixed nonresonant frequency the response is bounded by its inverse detuning. Near the exceptional collision, detuning can scale as $\alpha^2$, so forced response bounds may grow as $\alpha^{-2}$ while the unforced evolution remains uniformly bounded.

If initial size is $\epsilon_0<\rho$ and $\|F\|_\sharp\le f_*$, a sufficient condition to stay inside a specified linear departure radius is

$$\epsilon_0+tf_*\le\rho.$$

In $n$ complete particle-frequency cycles this becomes $\epsilon_0+2\pi nf_*/\omega_N\le\rho$. These are conditional lower bounds on linear persistence. The actual same-field surroundings must determine the load, phase, projection and duration. No noise floor, physical amplitude, survival distribution or decay width is inferred.

Full variation in the homogeneous-current comparison

The spatial current analysis concerns the separate positive action $L_+=\|\dot\Psi\|^2-\|\mathcal F[\Psi]\|^2$. At a stationary $\mathcal F[\phi]=\lambda\phi$, the full real derivative $L=D\mathcal F$ gives the directional Hessian

$$\frac12\delta^2\|\mathcal F\|^2[h,h]=\|Lh\|^2.$$

For the stated nodal fields this is a directional quadratic-form result, justified using the source's Lipschitz bound and its almost-everywhere nonzero background. It does not establish a uniform Frechet Hessian at the field origin or nonlinear well-posedness.

Angular projection must follow the complete square. If $P$ is the retained angular projection, $Q=I-P$, and $N$ the local real current derivative, then

$$PL^2P=(A_P+N_P)^2+(QNP)^*(QNP).$$

The positive omitted-square term is retained. The rotating perturbation equation is

$$\ddot\eta-2i\omega\Sigma\dot\eta+ (PL^2P-\omega^2I)\eta=0.$$

The actual spectral pencil includes the gyro term. Positivity of $PL^2P$ alone does not prove rotating stability.

A useful complete-angular bound for the underlying linear component chain is

$$\operatorname{gap}_{\rm fast}\ge G\sin\frac\pi{d+1}-\frac{c(d-1)}{2R}-|C|.$$

It controls all but the lowest signed pair when positive, before including the real current derivative. In the finite component fixed-point problem, the secular criterion $1-f''\epsilon_{GG}>0$, together with the stated bare gap assumptions, prevents fast eigenvalues entering the particle gap. This component result does not replace the full spatial test.

For the depth-two form carrier, all 36 tested combinations of maintained indices, two polarizations, azimuth shifts and cutoffs had no resolved growth at the illustrative common strength $gR/c=1000$. Spinor channels at all six maintained depths likewise had no resolved growth in their tested axial truncations. The depth-ten squared-matrix resolution was only about one percent of the particle frequency in the stated control. These are finite channel tests, not full nonlinear stability or a proof for every spatial azimuth.

Conditional geometric support response

The conformal Hessian comparison separates fixed-profile geometric stiffness from a complete supporting state. In its specified current action, a patterned same-field inverse response cancels the reference metric force and mixed field–metric derivative while preserving the reference frequency. The retained geometry/background block at unit phase action is

$$\begin{pmatrix}G&-F/v\\-F^T/v&a\end{pmatrix}.$$

When $G>0$, its exact Schur condition is $av^2>F^TG^{-1}F$. At general phase action $Q$, the right side is multiplied by $Q$; at unit wave norm it is instead multiplied by $2\omega$. Those normalization choices are not interchangeable.

A pointwise argument makes the bare fixed-profile conformal spinor block positive for all smooth conformal directions under its stated coefficient inequalities. The depth-two form result has only the finite zonal scope. Neither result covers every traceless metric change, the complete field block, a causal supporting law or compatible freely localized physical geometry. Its supplied curved comparison geometry is not imposed as IDWT's physical containing space. The full energy/inertia volume explains why such a partial Hessian cannot identify complete particle inertia.

Actual local principal health of the common material exit

The common material/shape/phase action described in same-field actions has a different, accepted local result. On an actual smooth stationary origin exit at the physical down clocks, its complete constrained frozen principal symbol has positive time inertia, positive spatial stiffness, real complete characteristics and both signed physical cone inequalities on a nonzero inner interval. It includes all ten labels, current, both phases after the actual rho constraint, every empty real carrier quadrature, amplitude and all readout quadratures.

The true rho constraint is eliminated before forming the symbol. Along the stationary uniform-clock origin subclass it reduces to $\rho=(\omega-F)/B_e$; off that subclass the horizontal phase contributions remain. The actual determinant is above $1/2$ on the short solved interval, so the determinant gate is identically one with all derivative-gate partials zero there. The amplitude gate still varies and remains in the full lower-order equations. This is a property of an actual solved branch.

For any unit physical spatial covector $k$, write the reduced characteristic polynomial as

$$\mathcal P(\sigma,k)=K\sigma^2+2B(k)\sigma-S(k), \qquad K>0,\quad S(k)>0.$$

The explicit companion matrix is symmetrized by $\operatorname{diag}(S,K)$, giving real complete characteristics. Checking both $\mathcal P(+1,k)\ge0$ and $\mathcal P(-1,k)\ge0$ bounds all characteristic speeds by one. A radial/transverse check alone would not cover every physical angle or polarization.

The material/current/phase and empty-carrier blocks have strict subluminal margins on a sufficiently short interval. Five exactly decoupled real Cartesian directions are luminal:

$$\mathcal P_{\rm light}=\epsilon k_v(\sigma^2-1) (|\delta a|^2+|\delta Z|^2+|\delta Y|^2), \qquad \epsilon=1-\chi>0.$$

Thus the complete cone margin is nonnegative, not strictly positive. The full real readouts have four quadratures, and amplitude supplies the fifth direction. They cannot be deleted as algebraic locks inside the open exit.

The later common positive mixed vacuum square changes both label/amplitude forces and derivative-gate loads. The accepted local construction rechecks the short determinant-plateau branch; its nonnegative complete cone scope survives there. No global determinant-transition health is inferred from the earlier action or from a positive vacuum potential. Both physical current signs, both polynomial branches and both common gate-pressure branches remain in this local result. It does not certify the other thirteen anisotropic interfaces or physical depth-three confinement of the ten-dimensional origin support.

The rank-changing endpoint remains a separate problem

The five Cartesian kinetic weights $\epsilon k_v$ vanish flatly at the material edge. Dividing each frozen symbol by that positive coefficient gives a standard light cone at every interior point, but the transformation by $\sqrt{\epsilon k_v}$ is not bounded through the endpoint. In the variable-coefficient PDE it introduces derivatives of $\log\epsilon$ that can diverge.

An endpoint evolution theorem therefore still needs its physical weighted norm, invariant data class, algebraic lock compatibility, complete time-dependent transmission conditions and control of singular lower-order terms. Stationary source-free conormal matching does not supply that PDE theorem. A singular transformed potential likewise does not by itself establish physical confinement or a unique endpoint domain.

The frozen principal Hamiltonian is distinct from raw energy $E$, both charge Hessians and the rotating generator $E-\omega Q_\theta-\Omega Q_\psi$. Imposing first-order charge constraints alone does not remove second-order charge variations or the curvature of a path on the nonlinear rho shell. No complete physical perturbation norm, lower-order persistence theorem or globally normalized particle follows from the local principal pass.

What the present results support

The positive count-wave and convex dispersion constructions also provide conditional nonlinear persistence within their open affine source charts on an assumed ordinary sphere. Their conserved graph norms and density floor retain the chart under sufficiently small regular disturbances. They do not supply localization in the unbounded containing space. Other source laws have their own full spatial growth results; those are not transferred to these stable candidates or treated as measured decay times.

The useful current conclusion is a collection of scoped positive results: complete bounded linear evolution for a clamped count-clock action; component and finite spatial-channel checks for a homogeneous-current action; conditional geometric stiffness; and full all-angle frozen principal health on an actual local material exit. They test the same maintained depths without a special depth-five stability coefficient. None establishes that the complete folded depth-five hypothesis behaves identically to a restricted one-plane control.

A physical continuation still needs an actual localized spatial solution, a complete energy–momentum and charge balance, the appropriate supporting-state perturbations and the intended persistence interval. The localization volume and pair-encounter volume explain how those requirements enter free support and reciprocal unlike-clock interactions. Finite tau-like persistence remains viable; an off-reference unstable mathematical branch is not a rejection of an actual maintained branch without a dynamical accessibility argument.

Patterns uniform along a free direction

These calculations use the density law $i\Psi_t=-i\sigma_v\partial_v\Psi+m_0\sigma_z\Psi-\delta\rho^{1/9}\Psi$ with untwisted preparations, in which the vibrating profile is copied along each free direction. They establish stability and growth rates, not lifetimes. They say nothing about a free extension that ends at contact or does not carry the vibration, whose stability is open (volume 2).

Necking. Every pattern of depth below ten that is uniform along a free direction is linearly unstable to necking, at depths 2 through 6, for both seeds and at $\omega/m_0=0.99,\,0.9,\,0.7$. The largest growth rates are 0.12 to 0.17 in units of $\delta$. The long-wave law is exact:

$$\Gamma^2=-\frac{k^2 S}{m_0\,dN/d\omega},$$

with $S$ the stress along the free direction and $N(\omega)$ the norm along the family. Necking needs room: there is a shortest unstable wavelength, about 23 to 35 in units of $1/m_0$ at depth 6 and $\omega/m_0=0.9$, so a span shorter than that has no necking mode. Changing the exponent to $p=0.4$, above the critical $2/D$, removes necking but produces fast growth at $k=0$, with rate 4.04 $\delta$.

Past the linear stage the pattern pinches repeatedly and does not settle. At depth 6 the peak density relative to its start reaches 12.27, 14.80, 9.08 and 12.39 at $t=523,\,762,\,1034,\,1325$. Wall and resolution controls leave this unchanged.

The bound directions. The seed decides what happens within the bound directions. The negative seed ($\sigma=-1$) has real instabilities at depths 2, 4, 6 and 10. The positive seed has none found near the gap. In the plane-rotation sector at $\omega=0.7$, the negative seed grows at:

The positive seed does not grow. Nonlinearly, the depth-4 shares swing repeatedly between 0.50/0.50 and 0.67/0.33. At depth 6 the concentration moves from plane to plane, with spacings 405 and 499, and no breakup.

Splitting. In the full channel, a depth-2 negative-seed pattern splits into two moving lumps, 99.7% in the positive-$\beta$ component. At depth 4 the $(0,2)$ rate of 0.0912–0.0928 agrees with the eigensolver's 0.0890. At depth 3 with nothing reduced, the $m=2$ mode grows at 0.1096 $\delta$, about five times the plane-rotation rate, and two lumps leave tangentially; the positive seed shows no growth through $t=380$.

The negative seed's large components are vortices, one per bound plane: one at depth 2, three at depth 6, five at depth 10. Its splitting fits the known breakup of ring vortices in nonlinear media (Desyatnikov, Torner and Kivshar; Cuevas-Maraver et al., Phys. Rev. Lett. 116, 214101). At equal norm the negative-seed pattern lies above one and two positive-seed patterns in energy.

The fragments are not plain patterns of the other seed. Their coherence with one is 0.002 at $t=200$ and 0.42 at $t=300$, and each keeps part of its parent's texture. They carry two large components on two clocks, 0.67 and 0.72. Internal angular momentum pays for the orbital motion: total $J/N$ holds at $+0.5000$, and in the larger box the orbital part rises from 0.482 to 0.563 while the spin part falls from $+0.018$ to $-0.063$ by $t=399$.

Depth 10's positive seed is the only tested case with no growth found. It has no free direction within the matter domain, so it has no necking channel there.

A Charge-Preserving Finite-Width Trial Minimum

The finite-source construction separates a boundary charge label from a metric-dependent energy. Its smooth radial orientation can retain the inherited charge while the source moments change. A separate positive-gradient, cubic/quintic comparison energy on a two-width Gaussian trial has an independently assessed aspect minimum and the reduced form

$$E(a)=\frac{K}{a^2}-\frac{G}{a^{10}}+\frac{H}{a^{20}},\qquad K,G,H>0.$$

Under its stated two-root inequality, the smaller positive stationary width is a radial minimum and the aspect direction is minimized. This is a concrete restricted variational result, not a full field stability theorem. The Gaussian is a trial core rather than the exact stationary far tail; a core-to-tail solution still has to be found. This comparison changes the energy law from the critical first-order Dirac branch, so their successful properties cannot be combined as if one solved particle already possessed both.

The real-spatial span calculation is another distinct trial. Its positive arclength coefficient in a chosen co-rotated preparation does not prove relaxed stable elasticity under arbitrary orientation changes. The contact and path account gives the exact trial functional, reduced path equation and remaining endpoint and outgoing-evolution requirements.

A Sampled Trial Maximum and an Exact Saddle Are Different Claims

The bent-profile control checks an exact trial-energy identity on a prescribed family. Its sampled quadratic bounce has the largest of seven amplitudes, but the exact arc functional has a nonzero first derivative there. This has an independent analytic proof. The measured trial agreement is retained without promoting it to a solved graph saddle or a relaxed field solution.

The later finite tanh envelopes are likewise specific preparations inside periodic boxes, not actual contact-ended boundaries. Their fixed initial-density threshold lengths, cell-density rms widths and hump counts must retain their definitions. Both stated grand-potential drift bounds fail, the short norm bound fails, and no refinement study is supplied. The frozen observations are useful controls; they do not establish a universal finite-length instability rule or a physical lifetime.