Localized Rotation and Constrained Stability in the Orientation Law

Projective rays here describe field coefficients. Particle positions remain ordinary spatial coordinates in the infinite-dimensional containing space.

Result and Added Structure

One explicit extension of the determinant orientation action supports nonconstant stationary rotating radial profiles at every occupied nonempty normal carrier. The common finite calculation covers ordinary spatial depths two, three, four, five and six, with depth ten recorded as an empty carrier. It tests full spatial localization instead of extending a two-dimensional defect uniformly along additional directions.

The extension adds a smooth bounded potential and a selected ordinary normal rotation generator H. They are constitutive premises, whose same-field origin remains to be derived. No coefficient is retuned between depths. The profiles are solutions of an orientation-field candidate; they are not identified observed particles, complete Dirac spinors, or a derived simplex frequency family.

One Global Target Potential

In a real four-direction normal block choose quaternionic complex structures I_a with Frobenius norm squared four. Let H=I_3/2 on this block and zero on the remaining normal directions. Its Frobenius norm is one at every nonempty carrier. On SO(2q)/U(q), define its Killing field K(J)=[H,J] and

$$y(J)=\frac14\|K(J)\|_F^2,\quad 0\le y\le1, \qquad V(J)=\frac\kappa2(m^2y-b y^2),\quad 0<b<m^2.$$

The bound follows from norm([H,J])_F at most 2 norm(H)_F norm(J)_op = 2. The potential is globally smooth and nonnegative. Its fixed H transforms with the ordinary normal frame; it is genuine additional geometric data, not a coordinate trick. Selecting its physical origin is an open problem.

The target root sphere J=n_a I_a plus constant remainder is totally geodesic. Both the metric and the first potential variation preserve this block. Thus solutions restricted to it solve the full target equation, although their transverse stability still needs checking.

Write n=(sin F cos alpha,sin F sin alpha,cos F), whose metric is G=4(dF squared+sin squared F d alpha squared). Let alpha=omega t and F=F(r), with r the radius in ordinary R^d. Put

$$\beta=4\lambda,\quad \kappa=T\beta,\quad A=1+\beta F'^2,\quad B=1-\beta\omega^2\sin^2F.$$

The complete stationary determinant is Delta=A B. Its time and radial target derivatives are orthogonal, so the timelike lapse is exactly B. The stationary Lagrangian is

$$\mathcal L=T(1-\sqrt{AB})-\frac\kappa2(m^2\sin^2F-b\sin^4F).$$

Equation, Exterior and Continuous Rotation Window

The radial Euler equation, including every determinant variation, is

$$F''=A\left\{ \sin F\cos F\left[(m^2-2b\sin^2F)\frac{\sqrt A}{\sqrt B}-\frac{\omega^2}{B}\right] -\frac{d-1}{r}F'\right\}.$$

At a regular center, F'(0)=0 and F''(0) is the first brace's nondamping term at F(0), divided by d. The exterior has exponent sqrt(m squared−omega squared), so subthreshold omega is used. This rotation clock is supplied by the candidate action and its branch; it is not the known particle mass-frequency or the sector fundamental.

The stationary rotating potential at zero spatial slope is

$$U_\omega(F)=V(F)+T\left(\sqrt{1-\beta\omega^2\sin^2F}-1\right).$$

A negative region permits a finite charged bubble to balance its gradients. In the dimensionless common comparison beta=1/4, T=4, m squared=1, b=1/4 and kappa=1. The equator has U_omega negative when omega squared is greater than 183/256, while the exterior decays for omega squared less than one. The tested omega=0.95 and 0.97 satisfy both conditions and keep B at least 1−beta omega squared>0 everywhere.

For numerical continuation use the bounded radial momentum P=F' sqrt(B)/sqrt(A). Then

$$F'=\frac P{\sqrt{B-\beta P^2}},\qquad P'=\sin F\cos F(m^2-2b\sin^2F) -\frac{\omega^2\sin F\cos F}{\sqrt{B-\beta P^2}} -\frac{d-1}{r}P.$$

The calculation monitors B−beta P squared separately from B, so a momentum-chart breakdown would not be mistaken for a missing physical branch.

Complete Energy, Charge, Stress and Inertia

The Noether charge Q belongs to this target rotation, not a derived electric charge. Its density and the complete rest energy density are

$$q=\kappa\omega\sin^2F\frac{\sqrt A}{\sqrt B},\qquad e=T\left(\frac{\sqrt A}{\sqrt B}-1\right)+V.$$

Both include the full supporting field. Their Routhian is e−omega q=T(sqrt(A B)−1)+V. The physical radial and tangential pressures are

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

The radial equation gives p_r'=−(d−1)(p_r−p_t)/r. Consequently a localized stationary solution satisfies integral[p_r+(d−1)p_t]=0. Its momentum density vanishes because the time and radial target derivatives are orthogonal. Spherical symmetry and this integral give the complete von Laue condition. A boost of the whole field within its ordinary d-sector therefore has E(v)=gamma E(0) and P(v)=gamma v E(0), with positive complete inertia E(0) in c=1 units. This is not a derivation of translation through the unused containing directions or of the observed particle mass law.

The normal curvature is genuinely electric for this ansatz:

$$\widehat F_{0r}=\frac\omega2\sin F F' J_{\rm block},\qquad \widehat F_{ij}=0.$$

It is generated by J, not prescribed independently. Its traceless fraction is again 1−2/q: absent at q2 and nonzero at q3/q4. No independent Gauss constraint, electric-charge identification or confinement follows.

Boundary and Residual Verification

The script first brackets regular shooting data, then solves the complete boundary problem by collocation. At the outer radius it uses the exact linear decaying radial-tail Robin coefficient sqrt(m squared−omega squared) times K_(d/2) divided by K_(d/2−1). It compares radius 80 at collocation tolerance 1e−8 with radius 110 at tolerance 2e−10, recomputing energy, charge, stresses and the center. This does not replace a continuum existence proof; it provides a bounded, residual-controlled numerical construction with analytic exterior behavior.

The refined calculation additionally solves the linear branch-sensitivity equation, including the frequency derivative of the tail boundary condition. Independent derivatives of the complete energy and charge obey dE/domega=omega dQ/domega. This tests the normalization and continuation without fitting energy to a desired frequency.

The Complete Target Directions

At the north vacuum, H acts on a complex normal vector with diagonal weights 1/2,1/2,0,…,0. A target root with indices a,b has rotation weight h_a+h_b. Its potential Hessian has mass squared m squared times that weight squared. The exact target comparison is:

Thus one minimal root rotation gaps every orientation direction at q3 while q4 retains a commuting two-dimensional target component. Rank two is also gapped but has only central canonical curvature. This is a conditional consequence of the selected generator and potential, not a unique geometric selection law. Other generators can give different gaps, and two gapless scalar orientation coordinates do not derive photon polarization.

Stability Test Rather Than Principal Causality Alone

The derivative-free potential leaves the determinant principal cone unchanged. For this radial ansatz its radial speed squared is B/A and every tangential speed squared is B. All target polarizations share that cone and stay causal throughout the tested profiles.

For the root perturbations, let R_omega(F,F')=T(sqrt(A B)−1)+V. Its amplitude Hessian has gradient coefficient R_pp=kappa sqrt(B)/A^(3/2) and potential R_FF−div(R_Fp). Its phase spatial form is nonnegative, with radial coefficient kappa sin squared F/sqrt(A B). A translation mode F' belongs to the spatial angular-momentum-one amplitude operator and has eigenvalue zero. Since the observed radial profiles are monotone, this mode has no interior node. Higher spatial angular sectors add a nonnegative centrifugal term.

After eliminating the uniform rotation velocity at fixed Q, the amplitude Hessian is the rank-one update

$$L_{+,Q}=L_++\frac{q_F\otimes q_F}{Q_{\omega}|_{\rm fixed\ shape}}, \qquad Q_{\omega}|_{\rm fixed\ shape}>0.$$

If L_plus has exactly one negative direction, the update removes it precisely when the complete branch derivative dQ/domega is negative. Here q_F denotes the functional derivative of the integrated charge, including its spatial-gradient dependence. This is a constrained-energy criterion, not a rule for declaring every negative unconstrained direction a decay channel.

To see the sign condition directly, differentiate the stationary equation delta R_omega=0. It gives L_plus F_omega=q_F and hence Q'=Q_omega|fixed shape+inner(q_F,L_plus inverse q_F). The rank-one update changes its determinant sign by Q'/Q_omega|fixed shape. Starting with exactly one negative direction and adding a positive rank-one form removes that direction when this ratio is negative. This argument is restricted to the radial subspace with invertible L_plus; translations and the common phase are handled as separate symmetry zero modes.

The extra q3 directions can be checked explicitly because its full ray target is CP3 with metric sixteen times Fubini–Study. Write the rotating root ray as s=(cos(F/2) exp(−i omega t/2),sin(F/2) exp(i omega t/2),0,0) and a transverse complex coordinate zeta orthogonal to its first two components. To quadratic order,

$$ds_{\rm FS}^2=g_0+|D\zeta|^2-g_0|\zeta|^2,\quad D_t\zeta=\partial_t\zeta+i\frac\omega2\cos F\,\zeta,\quad D_r\zeta=\partial_r\zeta.$$

The same equation holds for each overlap root at q4. Its fixed-rotation quadratic spatial form, after removing the common positive factor, has radial coefficient C_r=sqrt(B/A), angular coefficient C_t=sqrt(A B), time coefficient C_0=sqrt(A/B), and potential

$$W_\perp=\frac14\left[(1-2\sin^2F)(m^2-2b\sin^2F-C_0\omega^2)-C_rF'^2\right].$$

A positive lowest eigenvalue of this spatial form controls its gyroscopic time equation energetically. At q4 the remaining disjoint-root component is a commuting product sphere with no potential and a positive massless wave form on the effective metric. It has no infinite-volume spectral gap. At q2 there are no target directions outside the root sphere.

The checker derives this transverse form directly from the normalized ray jets, including the Berry connection and the Killing variance, rather than checking only a proposed final potential. Each q4 overlap lies in a rank-three normal subspace and therefore has the same CP3 calculation; the stabilizer symmetry gives four identical complex overlap directions. The disjoint root uses the independent real four-direction block. These account for all twelve real q4 target directions, all six q3 directions and both q2 directions.

The numerical checker assembles the root and overlap quadratic forms using radial finite elements with 1,100 and 2,200 intervals. It calculates the four lowest eigenvalues, the translation mode, the branch derivative and the complete stress identity. The reported finite-domain indices and their mesh convergence are numerical stability evidence under this action. They are not a nonlinear global-existence theorem, a certified infinite-domain spectrum, a proof against finite-amplitude fission or an observed particle lifetime.

Representative Calculated Values

All values use the same dimensionless action parameters above. E and Q include the ordinary sphere-area factor and are not fitted particle masses or electric charges. Lambda_0 is the lowest unconstrained radial shape eigenvalue; Lambda_perp is the lowest overlap eigenvalue on the refined mesh. A dash means that the overlap direction is absent, not a zero eigenvalue.

Every sampled radial Hessian has one negative computed eigenvalue and a positive second eigenvalue. Every computed complete Q' is negative. Translation eigenvalues approach zero with the expected second-order mesh error. Every tested overlap form has a positive lowest eigenvalue. The largest change in E or Q upon enlarging the radius and tightening collocation is approximately 6.1e-10 relative; the normalized virial residual is below 5.5e-13. These figures describe this bounded numerical execution rather than certified error bounds.

The root vacuum wave has mass per rotation charge m=1. In particular the depth-three omega=0.97 profile has E/Q greater than one. Its favorable local constrained test therefore does not imply global energetic binding against a distant collection of free charged waves. The other nine samples pass this particular free-wave threshold, which is still insufficient to rule out all competing finite-amplitude states.

Single-Field and Frequency Limits

These radial states have no spatial orientation winding. A local normalized spinor lift (cos(F/2),sin(F/2) exp(i omega t)) tends to a constant at spatial infinity and has finite spatial and temporal gradient integrals. It avoids the phase-energy obstruction of the earlier degree-one defect. Nevertheless adding the lift's phase and amplitude kinetics changes the J equation and its supporting stress. The J-only action is not thereby a complete spinor action.

The additional fixed normal axis, restoring potential, continuous conserved rotation charge and continuous frequency window remain explicit. The calculation does not give the occupied simplex harmonics, their shared sector units, spin statistics, physical source forces, or the depth-ten charged-lepton branch. It supplies a geometrically consistent localized orientation construction and tests its complete inertia and constrained stability, without treating those unresolved mechanisms as already derived.

The fixed H also reduces the target's global rotation symmetry. Its polarization gaps cannot simultaneously be presented as unbroken SU3 color dynamics. Their value is a conditional cross-sector comparison, not evidence that the chosen axis or restoring law is physically selected.

Context and Reproduction

Rotation-supported localization is an established Q-ball mechanism. Kuniyasu, Sakai and Shiraishi, Q-balls in DBI type k field theory (2016), study a square root of a kinetic trace with polynomial amplitude potentials. That kinetic law differs from the full multifield determinant here. Its existence and stability results are not imported. The contribution of this calculation is the explicit determinant construction on the retained normal-orientation geometry, with its complete transverse forms and common occupied-sector comparison.

The single reproducible script and recorded report contain 47 exact checks and ten localized numerical cases, plus the empty depth-ten control. The exact checks cover the radial equation, complete energy and stress, every vacuum target weight, normalized-ray transverse jets and actual quaternionic curvature. Run with system Python, SymPy, NumPy and SciPy available. A new timestamped report is written beside the executed source, retaining its SHA-256. Boundary refinement, branch sensitivity and both spectral meshes are run in one execution under common inputs.