A Direct Variational Derivation of the Conditional Count Clock

The reconstructed current interaction admits a short forward derivation. Its algebraic feedback label is varied independently; the spatial field equation then identifies that label with a Dirac eigenvalue. The resulting clock is the geometric cumulative count multiplied by the sector unit. No separate coefficient is adjusted for each particle.

This is a conditional theorem for a specified effective action. Its counting constitutive form was found by inverse research. Writing that form explicitly makes its consequences easy to derive and verify; it does not independently derive the interaction from the master action. The auxiliary representation removes an implicit inverse evaluation, without reducing the model's physical premises.

1. Geometry Supplies the Counting Function

On the conditional ordinary round spatial sphere of radius R, positive Dirac shells have dimensionless eigenvalue h_l=l+d/2. Their multiplicity per spinor seed is g_d(l)=binomial(l+d-1,d-1). Therefore the cumulative spatial band through l has

$$S_l=\sum_{k=0}^l\binom{k+d-1}{d-1}=\binom{l+d}{d}.$$

Define the minimal-degree polynomial agreeing with these values,

$$P_d(\chi)=\frac1{d!}\prod_{j=1}^d(\chi+j-d/2),\qquad P_d(h_l)=S_l. \tag{1}$$

The count is derived from the specified spatial bands. Its use as a local constitutive energy is the additional dynamical premise below. Available channels are being counted; no assumption that every channel contains a separate particle or an equal amount of energy is required.

2. An Explicit Local Action

Use the same commuting matrix-spinor carrier and canonical Dirac kinetic action as in the inverse construction. With signature (+,-,...,-), define

$$s=\operatorname{tr}\bar\Phi\Phi,\quad j^\mu=\operatorname{tr}\bar\Phi\gamma^\mu\Phi,\quad n=\sqrt{j^\mu j_\mu},\quad k=\sqrt{n^2-s^2}.$$

Work on n>0, k>0 and a convex chart of (1). Set epsilon=mu_d R and introduce a real algebraic variable chi, with no derivative term. The action is

$$\boxed{\mathcal L=\mathcal L_{\rm Dirac}-V_d,\qquad V_d=\mu_d nP_d(\chi)-\frac{\chi}{R}k.} \tag{2}$$

Chi is an auxiliary constitutive coordinate, not a propagating field or mediator. Both terms are Lorentz scalars and homogeneous of degree two in Phi. The field amplitude remains free. Spatial dimensions are ordinary physical dimensions; this internal classical field does not replace an electron's deterministic six-dimensional trajectory.

Variation of chi gives

$$\boxed{\epsilon P_d'(\chi)=k/n.} \tag{3}$$

The full field source, including the proper-current derivative, is

$$G_\Phi=\left[m_\chi\beta+ \frac{\eta_\chi}{n} \left(j^0I-\sum_i j^i\alpha_i\right)\right]\Phi,$$ $$m_\chi=\frac{\chi s}{Rk},\qquad \eta_\chi=\mu_dP_d(\chi)-\frac{\chi n}{Rk}. \tag{4}$$

Here m_chi is a scalar source coefficient, not the observed particle mass. The complete equation is i partial_t Phi=D_R Phi+G_Phi. Holding n fixed when varying the field would omit part of (4).

3. Two Stationary Equations Give the Clock

Let E_l be an actual balanced positive spatial shell field, normalized by integral tr(E_l dagger E_l)=1, with

$$D_RE_l=(h_l/R)E_l,\qquad E_lE_l^\dagger=\rho_l I/r.$$

The opposite Dirac sign is beta E_l. Write

$$\Phi=\sqrt Q\,(a+b\beta)E_l e^{-i\omega t},\qquad |a|^2+|b|^2=1,$$ $$c=|a|^2-|b|^2>0,\quad x=2\operatorname{Re}(a^*b),\quad y=2\operatorname{Im}(a^*b),\quad c^2+x^2+y^2=1.$$

Balanced matrix density makes j^i=0 pointwise, n=Q rho_l and s=x n. The complete energy per canonical action, in radius-scaled units, is

$$e(c,\chi,y)=R E/Q =h_lc+\epsilon P_d(\chi)-\chi\sqrt{c^2+y^2}. \tag{5}$$

The term h_l c is the spatial kinetic energy. It is retained throughout. On the real pair y=0, the independent stationary equations are simply

$$\boxed{\partial_c e=h_l-\chi=0,\qquad \partial_\chi e=\epsilon P_d'(\chi)-c=0.} \tag{6}$$

The relative-phase derivative also vanishes at y=0. Thus

$$\chi=h_l,\qquad c=\epsilon P_d'(h_l),\qquad a=\sqrt{(1+c)/2},\quad b=\sqrt{(1-c)/2}.$$

Substituting into the complete energy gives

$$\boxed{E/Q=\omega=\mu_dP_d(h_l) =\mu_d\binom{l+d}{d} =\mu_d\binom{n_{\rm index}+d-1}{d}.} \tag{7}$$

Here n_index=l+1 is the particle mode index, distinct from the proper current n. The temporal field equation follows independently from (4): on the pair its matrix is

$$\frac{h_l}{R}\sigma_3+m_\chi\sigma_1+\eta_\chi I.$$

Its positive mixed eigenvalue is h_l/(Rc)+eta_chi=mu_d P_d(h_l). Because the interaction is degree two in Phi, its Euler contraction equals V_d and the integrated complete energy is omega Q. There is no omitted energy of chi because it has no independent kinetic degree of freedom.

The cancellation of the two h_l c terms in (5) follows after variation. It is not an instruction to discard the spatial kinetic operator.

4. Exact Relation to the Reconstructed Law

On a chart where P_d''>0, equation (3) determines chi uniquely. Eliminating it from (2), with x=s/n and c=sqrt(1-x squared), gives

$$V_d=\frac nR f_d(x),\qquad f_d(x)=\epsilon P_d(\chi)-\chi c,\qquad c=\epsilon P_d'(\chi).$$

This is exactly the original Legendre reconstruction. In particular, the reduced pair curvatures are

$$e_{cc}=-\frac1{\epsilon P_d''(h_l)},\qquad e_{yy}=-\frac{h_l}{c}.$$

The auxiliary Hessian before elimination has entries e_cc=0, e_cchi=-1 and e_chichi=epsilon P_d''. Eliminating the algebraic variable gives the displayed negative polarization curvature. The auxiliary coordinate is a constraint, not an additional dynamical unstable mode. The original restricted pair persistence result therefore applies.

This is the raw proper-current action. Its known full ground-shell instabilities remain. The flattened interaction and responding-clock continuation have changed Hessians; their stronger persistence conclusions are not transferred to (2).

5. Verification and Scope

A convex dispersion continuation now replaces the continuous count energy by a smooth maximum of count-tangent massive dispersion curves. Unique auxiliary elimination preserves every eligible single-shell count clock and gives open affine scalar-source charts. This preferred-time variant has an explicit finite validity window; its positive-wave persistence result does not transfer to the proper-current action above.

A positive-wave continuation of the continuous preferred-time source now retains the count clocks without envelope smoothing. Its complete ground spatial linearization excludes exponential growth under an exact source norm and spectral-gap bound. The preferred-time density, local constitutive domain and ground-versus-excited distinction remain explicit; the proper-current instability findings are not overwritten.

The minimal-carrier theorem weakens square-matrix balance to vanishing Clifford vector and bivector seed moments. It preserves these full equations and every maintained count clock, while permitting one spinor column at five, six and ten. The same actual fields have exact dimension-dependent local rotation forms.

The geometric convex envelope now replaces arbitrary affine patch shapes and widths by the maximum of the count tangents. All maintained assignments have unique open quadratic charts. The preferred-time realization retains the earlier nonlinear spatial persistence proof, with the current dynamics of covariant density choices treated separately.

Two forward continuations clarify this conditional model. An explicit count-energy curvature correction preserves every stationary clock and source while setting the scalar response Hessian. A minimum over actual integer counts avoids choosing a continuous polynomial interpolation and gives a local label selector with exact fixed-clock polarization families. Their constitutive premises and persistence conclusions remain distinct.

forward-count-action.py and its report pass 258 evaluations over all fourteen maintained assignments at depths 2,3,4,5,6 and 10. The calculation independently solves the three stationary equations from displaced polarization, auxiliary label and relative phase. It enumerates the spatial shells, verifies complete pair frequencies at 85-digit precision, varies arbitrary actual full Clifford matrix fields, checks Lorentz boosts and evaluates actual covariant spatial jets.

One diagnostic epsilon_d=3/[10 max P_d'(h_l)] per occupied sector keeps every maintained shell inside 0<c<1. It is a comparison choice, not a mass-fitted amplitude or a derived physical radius. Arbitrary joint variation errors are below 1.3e-10; actual spatial Dirac residuals are below 2.3e-15 relatively. Complete floating field residuals are below 2.1e-12 after scaling by the large source coefficient. The separate high-precision pair test verifies the selected clock relatively to 1e-65, so source-scale conditioning is not misreported as mass-frequency accuracy.

The result supplies a compact conventional variational proof for the effective model. Geometry fixes the available channel count and spatial label; the stated constitutive interaction turns that count into a clock. Deriving this interaction from one master-field reduction, the sector units, localized support and sparse occupation remains open. This all-depth construction does not independently favor the assigned sector depths and supplies no new observed particle property by itself.