Closed-Circle Count Fields with Spatial Concentration and Persistence

One elementary spatial construction now realizes several of the conditional count model's properties together. It has an exact positive Dirac shell, the prescribed count clock and a calculable concentration of classical energy around a closed circle. Suitable orientations retain the full observer half-spin transformation. Adding a small nodeless mode permits the existing preferred-time nonlinear persistence theorem to apply.

This is a construction on the assumed round spatial sector sphere. It does not establish a mobile localized particle in the containing unbounded space, an observed total spin or a derived physical interaction. The count constitutive action and sector scales remain premises. A literal electron trajectory remains distinct from this internal classical field.

1. A Direct Polynomial Construction

Use the minimal-carrier Clifford convention on an ordinary sphere of physical dimension d and radius R. In a global Killing trivialization its grading and unit-radius Dirac operator are

$$\beta(X)=\Gamma_X=X_A\Gamma_A,\qquad D_0=-\Gamma_X\Gamma\cdot\nabla_{S^d}+\frac d2. \tag{1}$$

The d+1 coordinates X parameterize a unit sphere; its physical tangent dimension is d. Including the grading among the Clifford generators does not add an occupied spatial direction. For a homogeneous polynomial F of degree l, equation (1) gives

$$D_0F=\left(l+\frac d2\right)F-\Gamma_X\Gamma\cdot\partial F. \tag{2}$$

Choose an orthonormal coordinate plane (a,b), set

$$w=X_a+iX_b,\qquad J_{ab}=i\Gamma_a\Gamma_b,\qquad P_\pm=\frac{I\pm J_{ab}}2.$$

For a unit balanced seed C, define

$$\boxed{E_l(X)=\sqrt{N_{l,d}} \left(w^lP_-+\bar w^lP_+\right)C,\qquad N_{l,d}=\frac{((d+1)/2)_l}{l!}.} \tag{3}$$

The two Clifford identities

$$(\Gamma_a+i\Gamma_b)P_-=0,\qquad (\Gamma_a-i\Gamma_b)P_+=0$$

make both summands monogenic: Gamma dot partial E_l=0. Equation (2) therefore proves, without shooting or a spatial simulation,

$$\boxed{D_RE_l=\frac{h_l}{R}E_l,\qquad h_l=l+d/2.} \tag{4}$$

Away from w=0, write w=r exp(i theta). The matrix in (3) is

$$r^l\left[\cos(l\theta)I+\sin(l\theta)\Gamma_a\Gamma_b\right]. \tag{5}$$

It is a real magnitude times a spin rotation. The seed's zero vector and bivector moments are consequently retained at every position. Bare scalar density and tangential current vanish, while

$$\rho_E=N_{l,d}|w|^{2l}. \tag{6}$$

The construction works at all six maintained depths and the previously proved minimum column ranks 2, 2, 4, 1, 1, 1. At l=0 it is the uniform Killing field; concentration requires positive l. Standard sphere Dirac eigenspinors and their degeneracies are reviewed by Camporesi and Higuchi. Equations (2) through (6) are verified directly in the present convention.

2. The Same Complete Count Equation

Use the established count coefficients

$$S_l=\binom{l+d}{d},\quad c_l=\epsilon_dP_d'(h_l),\quad x_l=\sqrt{1-c_l^2},\quad \epsilon_d=\mu_dR,$$ $$m_l=\frac{h_lx_l}{Rc_l},\qquad \lambda_l=\frac{h_l}{Rc_l},\qquad \eta_l=\mu_dS_l-\lambda_l.$$

Here m_l is a source coefficient, rather than the observed mass. The massive map

$$\Phi_l=(a_l+b_l\Gamma_X)E_l e^{-i\omega_lt},\qquad a_l^2=(1+c_l)/2,\quad b_l^2=(1-c_l)/2 \tag{7}$$

has rho_Phi=rho_E, s_Phi=x_l rho_E and zero tangential current. On the intended unique winner chart of the convex count envelope,

$$i\partial_t\Phi_l=(D_R+m_l\beta+\eta_l I)\Phi_l, \qquad \boxed{\omega_l=\mu_dS_l.} \tag{8}$$

The complete energy density, including spatial kinetics and interaction, is omega_l rho_Phi pointwise; its integral is omega_l Q. The amplitude normalization Q remains free. This is a forward solution of the stated effective law. That law already contains the geometric cumulative count; the calculation does not independently derive its master-field origin.

At beam nodes, the preferred-time envelope has V(0)=0 and derivative G(0)=0 because every chart is quadratic. Equation (8) therefore extends there. Its nonzero chart ties remain a separate global-evolution issue. A ratio-defined proper-current action cannot automatically be evaluated at vacuum, and no such extension is silently assumed.

3. Exactly Integrated Spatial Concentration

Normalize the ordinary sphere volume to one. For t=|w| squared, uniform volume has the density proportional to (1-t)^((d-3)/2). Multiplying by (6) gives the classical energy measure

$$\frac{t^l(1-t)^{(d-3)/2}}{B(l+1,(d-1)/2)}\,dt. \tag{9}$$

The beta integral proves N in (3). In particular,

$$\boxed{\frac{\langle r_\perp^2\rangle_E}{R^2} =\frac{d-1}{2l+d+1},} \tag{10}$$

where r_perp squared=R squared times (1-t) is the squared transverse embedding distance to the circle. A geodesic tube of half-width R delta around that circle contains the energy fraction

$$F_l(\delta)=1-I_{\cos^2\delta}\left(l+1,\frac{d-1}2\right). \tag{11}$$

These are normalized classical energy integrals, without a Born rule or an electron position probability. Higher l concentrates energy more strongly about the circle under the same geometry.

The field occupies a tube around an entire great circle, with tails on the finite sphere and a nodal set when l>0. It is not concentrated into one point. Neither a physical radius calibration nor a containing-space translation law follows from (10).

4. Full Observer Rotation Depends on Orientation

Let observer rotations act in coordinates 1,2,3, fixing the sphere point p=e_0 and the remaining coordinate directions. In the global frame,

$$\mathcal J_{ab}=-i(X_a\partial_b-X_b\partial_a) -\frac i2\Gamma_a\Gamma_b. \tag{12}$$

This includes the orbital derivative. The grading Gamma_X commutes with the combined action, so the mass map preserves its transformation.

If the circle plane belongs to the observer-fixed complement, its w and projectors commute with all three observer rotations. The actual field then intertwines with its seed and

$$\boxed{\sum_{j=1}^3\mathcal J_j^2\Phi_l=\frac34\Phi_l.} \tag{13}$$

At depth four, the plane (0,4) already provides such a circle. A circle using two fully normal tangent directions, for example (4,5), requires depth at least five. These two conditions must not be confused.

As an explicit control, the circle in the visible plane (1,2) instead has J_12 weights plus and minus (l+1/2), so its J_12 squared is (l+1/2) squared. Concentration alone does not give observed half-spin. Equation (13) is a transformation identity; the balanced mean classical rotation charge is not thereby identified with an observed spin value.

For the compatible equatorial chain of round sector spheres, a fully normal beam vanishes on the observer S3 at every positive l, to normal order l. This possibility exists at five, six and ten. No equation selects it uniquely for neutrinos, chooses the circle orientation or identifies this vanishing with electric neutrality or weak-interaction rates. The plane (0,4) does not vanish on that equator.

5. Concrete Nodeless Fields and Nonlinear Persistence

Let E_z be the normalized positive zonal field in the same shell. In the global frame its map on a unit seed v is

$$Z_l(X)v=\frac{[(A_l-tB_l)I+B_l\Gamma_0\Gamma_X]v}{\sqrt{g_l}}, \qquad t=X_0,$$ $$Z_l^\dagger Z_l=\frac{p_l(t)}{g_l}I,\qquad p_l=A_l^2+B_l^2(1-t^2)>0. \tag{14}$$

Here A_l and B_l are the Gegenbauer polynomials in the minimal-carrier note. At interior points B_l is proportional to A_l prime; the Gegenbauer differential equation prevents a common root. Endpoint A_l is nonzero, and l=0 has A_0=1. Thus Z_l is invertible everywhere and its density has a positive minimum on the compact sphere.

The direct sum in column space

$$\boxed{\widetilde E_l= [\sqrt{1-\delta^2}E_l,\;\delta E_z],\qquad 0<\delta<1} \tag{15}$$

is an explicit globally nodeless solution of the same shell and balance equations at every occupied depth. Its density is the sum of the two densities, with lower bound delta squared times min(p_l/g_l). It retains the count clock and complete energy and, for the orientations in (13), the same observer Casimir. Its tube energy fraction is at least (1-delta squared) F_l. Choosing delta=1/10 illustrates a one-percent zonal energy addition; it is not a calibrated particle parameter.

The density floor and the unique winner interval permit the existing preferred-time graph-norm proof: a sufficiently small H^s disturbance, s>d/2, stays uniformly inside the same winner chart. Its exact evolution there is generated by the self-adjoint operator D_R+m_l beta-lambda_l I. Consequently (15), after the massive map, has conditional nonlinear Lyapunov persistence on the entire sphere. This theorem is not transferred to the proper-current or responding-clock actions, which have different source derivatives.

6. One-Column Existence at Five and Six

At these two depths the spinor module has an antiunitary involution K=B conjugation, with K squared=I and K Gamma_A=-Gamma_A K. Its fixed space W is real eight-dimensional. Hermitian vectors Gamma_A and Hermitian bivectors i Gamma_A Gamma_B both have odd K parity, so all their expectations vanish on every v in W. The full sphere Dirac operator commutes with K because it contains Clifford products of even degree. Thus any real positive-shell section is pointwise balanced. This closes coherent real combinations, beyond the rotor ansatz (5).

Take a fully normal beam, l>0, and unit real seeds C and v. Then

$$E_{l,\delta,v}=\sqrt{1-\delta^2}E_l(C)+\delta Z_l(X)v \tag{16}$$

is a one-column balanced positive-shell field. The beam vanishes at p, and the reproducing-kernel property makes it orthogonal to every zonal field centered at p. Hence (16) has unit integrated norm for unit v. This orthogonality also follows by integrating its nonzero azimuthal Fourier powers; the l=1 terms cancel by the monogenic projector identity.

Every possible zero of (16) requires

$$v=v_{\rm bad}(X) =-\frac{\sqrt{1-\delta^2}}\delta Z_l(X)^{-1}E_l(C). \tag{17}$$

The map v_bad is smooth on a compact d-sphere and belongs to W. Its image has Hausdorff dimension at most d. For d=5 or 6 it cannot cover the seven-dimensional unit sphere in W; its intersection with that sphere has zero seven-dimensional measure. Almost every unit real v therefore gives a nodeless field. This is an analytic existence proof, rather than an inference from a finite grid of field samples.

For any such v, compactness gives a positive separation from the bad image and the exact density formula

$$\rho_{l,\delta,v}(X)=\delta^2\frac{p_l(X_0)}{g_l} \left|v-v_{\rm bad}(X)\right|^2>0. \tag{18}$$

The same nonlinear persistence theorem applies. Both summands also obey (13), so the coherent field retains the full observer Casimir. Its concentration approaches that of the beam as delta approaches zero. For a tube containing beam fraction F_l, the energy fraction outside that tube is at most

$$\left[\sqrt{1-\delta^2}\sqrt{1-F_l}+\delta\right]^2. \tag{19}$$

No particular random seed is certified nodeless by numerical sampling. Equations (17) and (18) prove generic existence. The eight-dimensional real-space proof does not extend automatically to depth ten; (15) already gives a concrete two-column nodeless construction there.

7. Reproducibility and the Remaining Physical Question

The fully varied positive wave action retains the nodeless fields, their count frequencies and conditional local nonlinear persistence, with nonnegative complete energy. Its phase-action normalization differs from the first-order action, and its interaction must be recalculated.

The same-action contact calculation now derives reciprocal orientation energy and a global alignment theorem for distinct branches in orthogonal columns. Strong aligned electron/muon overlap has negative complete energy in that diagnostic action; this is an encounter constraint, rather than a change to the isolated-state persistence proof.

balanced-count-beams.py and its report pass 2,114 evaluations across all fourteen maintained assignments. Exact Gaussian-integer projector products establish monogenicity. Actual spatial jets verify the full Dirac equation, complete count source, pointwise energy and full observer rotation. The spatial shell residual is below 4.8e-16 relatively; the source check uses its large coefficient as scale. Independent beta integrals check concentration and tube widths. The count clock's independent high-precision derivation remains the one in the forward action note.

balanced-beam-persistence.py and its report pass 2,420 evaluations. Exact real-structure identities accompany the direct-sum and coherent spatial checks. A pilot symbolic comparison used floating entries against integer matrices; conversion of the exactly Gaussian-integer Clifford entries fixes that representation error. The generic nodeless theorem uses the dimension argument above, while the persistence theorem uses the stated analytic domain assumptions.

This is a concrete common realization of count frequency, closed-circle energy concentration, appropriate observer transformation and conditional nonlinear persistence. The same law admits other depths and shapes; it does not independently favor the assigned depths. A physical account still needs the effective count action's origin, motion and stress in the containing manifold, and an actual reciprocal interaction. Those questions remain distinct from the verified existence of these fields.