Finite-Amplitude Localized Coefficients and Their Clock Window

This is a candidate continuation of the phase-complete positive wave field. Let U=Psi/R0 now range over the full complex chiral coefficient space, with R0 an unknown physical vacuum norm. Keep the flat positive coefficient metric, the common combined generator G=(N1+N2)/2, beta=16 lambda R0 squared, kappa=T beta and the full determinant kinetic term. The norm is dynamical rather than constrained. The model still has second-order positive wave kinetics and does not claim Lorentz Dirac dynamics, observed spin/statistics or particle-frequency selection.

Globally Controlled Polynomial Potential

Put s=U dagger U, z=U dagger G squared U and

$$V=\kappa\left[\frac{M^2}{4}(s-1)^2+\frac12(m^2z-bz^2)\right], \qquad 0\le z\le s,\quad 0<b<m^2.$$

The constant M squared is a common norm stiffness, not a fitted particle amplitude. For fixed z, minimizing over s>=z gives s=1 for z<=1 and s=z for z>=1. The first range is nonnegative. The second range is the quadratic

$$\frac{V_{\min}}\kappa=\frac{M^2-2b}{4}z^2+\frac{m^2-M^2}{2}z+\frac{M^2}{4}.$$

It is nonnegative for all z>=1 when M squared>=2b if m squared>=2b. When m squared<2b, the sharp nonnegativity threshold is M squared>=m to the fourth/[2(m squared−b)]. A strict inequality in the latter case avoids an additional zero-potential vacuum at larger norm. Exact differentiation and the quadratic minimum verify these statements. A simple sufficient bound for either range is M squared>=m to the fourth/[2(m squared−b)], which also implies M squared>=2b.

For the numerical comparison m squared=1 and b=1/4, M squared>=1/2 already suffices for global nonnegativity. Localizing solutions may demand much greater stiffness; positivity is not existence or stability.

Two Independent Radial Equations

Use U=(a(r),c(r) exp(i omega t),0,...), with a tending to 1 and c tending to 0. The symbol c here is a radial coefficient, not the speed of light or the quartic coefficient b. Let

$$A=1+\beta(a'^2+c'^2),\quad B=1-\beta\omega^2c^2,\quad C_r=\sqrt{B/A},\quad C_0=\sqrt{A/B}.$$

The complete radial equations are

$$\operatorname{div}_r(C_r a')=M^2(a^2+c^2-1)a,$$

$$\operatorname{div}_r(C_r c')= \left[M^2(a^2+c^2-1)+m^2-2bc^2-C_0\omega^2\right]c, \qquad \operatorname{div}_r X=X'+(d-1)X/r.$$

They must be solved together. Inserting a=cos F, c=sin F from the constrained model leaves a nonzero radial norm force at finite M. The constrained profile is an initial numerical preparation, not an exact solution of this action.

Set P_a=Cr a', P_c=Cr c' and S=B−beta(P_a squared+P_c squared). Then a'=P_a/sqrt(S), c'=P_c/sqrt(S), and the right-hand sides above give P derivatives with C0=1/sqrt(S). Both B>0 and S>0 are monitored. Complete energy, rotation charge and pressures are

$$e=T(\sqrt{A/B}-1)+V,\quad q=\kappa\omega c^2\sqrt{A/B},$$

$$p_r=T(1-\sqrt{B/A})-V,\qquad p_t=T(1-\sqrt{AB})-V.$$

The total stress identity and ordinary-sector boost response must be recomputed from these complete fields. The exterior contains a radial norm mass sqrt(2)M and the charged coefficient exponent sqrt(m squared−omega squared). The norm tail is also forced by c squared, so an independent homogeneous massive-tail boundary condition on a−1 would generally be inappropriate.

One Positivity Identity Still Applies to Every Unused Coefficient

An unused complex coefficient of G weight w has rotating-frame spatial potential

$$W_w=M^2(a^2+c^2-1)+w^2(m^2-2bc^2-C_0\omega^2).$$

The complete flat coefficient metric adds no projective subtraction. The active radial equations give L0 a=0 and L1 c=0. If the obtained a and c are strictly positive at finite radius, their ground-state transformations imply nonnegative endpoint forms. Therefore

$$\mathfrak q_w=(1-w^2)\mathfrak q_0+w^2\mathfrak q_1\ge0 \quad\hbox{for every }0\le w\le1.$$

The same argument that controlled the constrained sphere thus extends to finite amplitude under its explicit positivity and boundary hypotheses. This includes every available nonpure q4 direction. It does not settle the coupled two-amplitude radial Hessian, its fixed-charge constraint or nonlinear global stability. The added stiffness and selected generator remain constitutive inputs, and no dimensional preference or physical charge identification is claimed.

Timelike Homogeneous Core Diagnostic

A thick saturated rotating core with a=0, c squared=x approximately obeys

$$g(x)=(M^2-2b)x+m^2-M^2-\frac{\omega^2}{\sqrt{1-\beta\omega^2x}}=0.$$

For M squared>2b its maximum in the timelike range occurs at

$$B_*^{3/2}=\frac{\beta\omega^4}{2(M^2-2b)},\qquad g_{\max}=\frac{M^2-2b}{\beta\omega^2}(1-3B_*)+m^2-M^2.$$

If this maximum is negative, no homogeneous saturated core of that kind exists. This does not exclude every spatially varying localized branch. It prevents blindly treating an arbitrarily weak norm potential as equivalent to the constrained sphere. At beta=1, m squared=1, b=1/4, omega=0.97 and M squared=100, this maximum is approximately -1.83156, while M squared=400 gives 11.95430. Norm-potential nonnegativity alone would not reveal this distinction.

Initial Finite-Amplitude Execution

The first execution passes 12 exact checks and solves twenty localized boundary problems: common M squared 400 and 1600, each at omega 0.95/0.97 and ordinary depths 2,3,4,5,6. Depth ten is explicitly empty under the retained normal-generator prescription. Every case is recomputed at radius 80/tolerance 1e-7 and radius 110/tolerance 1e-9. Maximum reported energy/charge/center change is below 6e-9 relative, normalized virial below 3.7e-11 and lapse above 0.0504. Both active radial coefficients stay positive at finite radius. The maximum norm-squared change is approximately 0.009239 at M squared 400 and 0.002150 at 1600. These are dynamically solved amplitudes under one common stiffness, not fitted observed particle masses.

The twenty positive solutions satisfy the hypotheses of the endpoint ground-state transformations for unused polarizations. Their complete coupled two-amplitude radial Hessian and fixed-charge response have now been recomputed independently of the constrained sphere's radial index.

Complete Coupled Radial Stability Test

Let u=(a,c), p=u-prime and R_omega=T(sqrt(A B)−1)+V. The second variation contains the three complete matrices R_uu, R_up and R_pp. Dividing by kappa, write H=I−beta p p-transpose/A. The radial gradient coefficient is Cr H and the angular coefficient is Ct H, with Ct=sqrt(A B). Both are positive. For beta=1, the mixed matrix has only its second row nonzero:

$$\frac{R_{up}}\kappa=\begin{pmatrix}0&0\\-\omega^2cp_1/\sqrt{AB}&-\omega^2cp_2/\sqrt{AB}\end{pmatrix}.$$

The potential Hessian is

$$\frac{R_{uu}}\kappa= \begin{pmatrix} M^2(s-1+2a^2)&2M^2ac\\ 2M^2ac&M^2(s-1+2c^2)+m^2-6bc^2-\omega^2\sqrt A/B^{3/2} \end{pmatrix}.$$

The finite-element calculation retains every mixed derivative term rather than substituting a scalar radial potential. It uses ordinary positive lumped radial mass to count the energy-form index. These eigenvalues are not physical oscillation or growth frequencies; the actual positive kinetic matrix is different. The index is invariant under replacing the positive mass used for that comparison.

Independent symbolic differentiation verifies all entries of R_uu, R_up and R_pp. The full radial calculation has 17 exact action/Hessian conditions, and the independent potential/mixed-Hessian checker supplies eight further conditions. Numerical charge sensitivity solves all four differentiated radial equations, including the tail-boundary derivative, and satisfies E'=omega Q' to better than approximately 3.1e-12 relative in these cases.

Every sampled coupled radial Hessian has one negative computed eigenvalue and a positive second eigenvalue. Its spatial angular-momentum-one translation mode converges to zero. Four selected cases, including the stiffest depth-six profile, are additionally refined from 2200 to 4400 and 8800 intervals. For depth six, omega 0.97 and M squared 1600, the lowest radial eigenvalue changes from -0.1098108 to -0.1102502 and -0.1103518; the translation eigenvalue changes from 0.0005191 to 0.0001209 and 0.00002969. No extra negative direction appears in these controls. The higher angular sectors add a positive matrix centrifugal term. These observations support the tested finite-domain index; they do not certify the continuum operator.

The fixed-charge rank-one update acts on both amplitude functions. Its fixed-shape susceptibility is integral[kappa c squared sqrt(A)/B^(3/2)]>0. Under the computed one-negative-direction hypothesis, a negative complete Q' removes that negative direction. Eighteen of the twenty cases pass. Both depth-three omega 0.97 cases retain a negative constrained-energy direction: Q'=818.63868 at M squared 1600 and 936.29513 at 400. All other cases have Q'<0. This establishes neither a lifetime for those two cases nor global nonlinear stability for the other eighteen.

Representative M squared 400 values are:

Nine of twenty profiles remain above the E/Q=1 free charged-wave threshold. Crossing that threshold differs from the local fixed-charge Hessian test. The finite-amplitude depth-six omega 0.97, M squared 400 sample is just below the threshold, whereas its M squared 1600 and constrained counterparts are just above it. No sector identification or physical binding claim follows from this small model-dependent difference.

A Necessary Clock Bound and the Mass-Frequency Limit

The simple radial family has only one rotating temporal frequency, omega, and a static supporting coefficient. Its root frequency window can be bounded without fitting amplitude to a measured mass. At zero spatial slope let v=V/kappa and z=c squared. Its rotating potential is U_omega=V+T(sqrt(1−beta omega squared z)−1). A nonconstant localized radial critical point at d>=2 requires U_omega negative somewhere: if it were nonnegative everywhere, its dilation derivative would be strictly positive. The gradient contribution to that derivative is T sqrt(B)[d(sqrt(A)−1)−(A−1)/sqrt(A)]>=0, strictly positive at a nonzero radial slope.

Negativity requires beta v<1 and

$$\omega^2>\frac{2v-\beta v^2}{z},\qquad \omega^2<m^2$$

where the upper inequality comes from the charged exterior. For m squared=1, b=1/4 and beta=1 there is a useful exact lower bound. If z<=1, minimizing over s gives v−z/3=z(4−3z)/24>=0. If z>=1,

$$v-\frac z3=\frac{M^2-2/3}{4}(z-1)^2+\frac{(z-2)^2}{24}\ge0 \qquad(M^2\ge2/3).$$

Thus throughout either tested finite-stiffness action, U_omega<0 implies omega squared>1/3. Any two such rotating clocks under one common m=1 obey omega_high/omega_low<sqrt(3). Numerically minimizing the exact stationary quartic sharpens this upper ratio to approximately 1.28121076 at M squared 400 and 1.28107157 at 1600. The fixed-norm phase-complete action has the exact upper ratio 8/sqrt(39); the orientation-only action has 16/sqrt(183).

The maintained electron/muon target frequency ratio is3838380/18564=3515/17, approximately 206.7647. Therefore identifying this simple family's rotation directly with their mass-frequencies fails under one common action and common coefficients. Complete positive inertia and field localization do not supply the simplex spectrum. This is a scoped failure of that identification, not a rejection of geometric resonance, other field actions, additional spatial-mode mechanisms or finite-lived particles. The construction supplies a complete positive classical wave object that can serve as a testbed; a different frequency-selection mechanism is required for IDWT's observed family.

Reproduction and Remaining Physical Premises

The finite-amplitude driver, coupled stability calculation, independent Hessian check, selected spectral refinement and clock-window calculation retain their executed source hashes. Their reports are pilot, stability, Hessian, refinement and clock window. The common audited radial solver is a declared source dependency. Each execution writes a fresh dated report.

The constitutive action still assumes its polynomial potential, common stiffness, selected normal/phase generator and declared ten-parent normal carriers. It solves the finite-amplitude positive wave comparison without an imposed coefficient norm. It does not derive Lorentz Dirac behavior, observed spin/statistics, electric charge, a strong-force law, motion through unused containing directions or the depth-ten charged lepton. Complete stress gives positive ordinary-sector boost inertia; no measured particle amplitude or mass is fitted. The exact cross-polarization tool and local spatial construction are reusable, while their frequency limitation points the research back toward the missing same-field spectral operator.