A Forward Field Rule Using Only Integer Spatial Counts

A local constitutive rule can use the actual geometric counts directly. There is no need to choose their continuous polynomial interpolation. The following candidate has one common minimum rule, an explicit label selector and exact stationary count-frequency families. All fourteen maintained spatial solutions belong to its smooth local charts.

The minimum rule is a new effective dynamical premise. It is not claimed to follow already from the master interaction. Its forward consequences are derived here, including what it does and does not select.

1. Integer Geometry and One Local Energy Rule

Retain the conditional ordinary spatial sphere and canonical matrix-spinor kinetics from the forward calculation. Define

$$h_l=l+d/2,\qquad S_l=\binom{l+d}{d},\qquad g_l=\binom{l+d-1}{d-1},\qquad l\ge0.$$

The positive spatial shell has eigenvalue h_l/R; its cumulative band has count S_l per spinor seed. Thus S_l-S_(l-1)=g_l for l>=1, and S_(l+1)-S_l=g_(l+1). These counts refer to channels of one classical field rather than occupied mediator particles.

For scalar current s and proper vector norm n, let k=sqrt(n squared-s squared), n>0 and 0<k/n<1. With epsilon=mu_d R, the proposed local potential is

$$\boxed{V(\Phi)=\frac1R\min_{l\ge0} \left[\epsilon nS_l-h_l k\right].} \tag{1}$$

Every term is a Lorentz scalar, homogeneous of degree two in Phi. No new propagating field or physical dimension is introduced. The minimum exists: S_l grows with degree d>=2, while h_l grows linearly. The physical picture encoded by (1) is competition between a count-weighted contact cost and the support contribution. The origin of that competition in one master-field reduction remains to be established.

2. A Geometric Local Label Selector

Put r=k/n. Adjacent costs obey

$$\epsilon S_{l+1}-h_{l+1}r -(\epsilon S_l-h_lr)=\epsilon g_{l+1}-r. \tag{2}$$

The right side strictly increases with l. Consequently the unique global winner is l precisely when

$$\boxed{\epsilon g_l<r<\epsilon g_{l+1}\quad(l\ge1).} \tag{3}$$

The ground winner has 0<r<epsilon g_1=epsilon d. Intersect these intervals with 0<r<1. The open intervals give smooth field laws. At their boundaries, two costs tie and a global evolution prescription would need to specify the nonsmooth dynamics; the local theorem does not silently choose one.

There are only finitely many available winners at fixed positive epsilon, since epsilon g_l eventually exceeds one. This selects a local geometric branch from the current ratio. It does not select the sparse observed list of particle species or infer a particular polarization from mass alone.

For an excited branch the interval width and its fractional width are

$$\epsilon(g_{l+1}-g_l)=\epsilon\binom{l+d-1}{d-2},\qquad \frac{g_{l+1}-g_l}{g_{l+1}}=\frac{d-1}{l+d}. \tag{4}$$

The first relation is Pascal recursion. The electron, muon and tau fractions are respectively 5/18, 1/8 and 9/32. These describe this constitutive polarization interval, not a measured scattering acceptance or an independently derived coupling strength.

3. Forward Spatial Solutions and Their Full Energy

Inside a unique winner chart l, vary (1) with that integer fixed. The complete local field source is

$$G=\left[m\beta+\frac\eta n \left(j^0I-\sum_i j^i\alpha_i\right)\right]\Phi,$$ $$m=\frac{h_l s}{Rk},\qquad \eta=\mu_dS_l-\frac{h_l n}{Rk}. \tag{5}$$

The derivative of the minimum adds no term inside a unique smooth chart. Let E_l be an actual balanced positive Dirac shell and choose any c in its interval (3), with x=sqrt(1-c squared). Then

$$\Phi=\sqrt Q\,(a+b\beta)E_l e^{-i\omega t},\qquad a=\sqrt{(1+c)/2},\quad b=\sqrt{(1-c)/2}$$

has j^i=0, k/n=c and s/n=x. Its spatial pair matrix has scalar source m=h_l x/(R c), vector source eta=mu_d S_l-h_l/(R c), and eigenvalue

$$\boxed{\omega=\mu_dS_l,\qquad E=\omega Q.} \tag{6}$$

The complete pair energy includes h_l c/R from spatial Dirac kinetics and mu_d S_l-h_l c/R from the interaction. Their stationary sum gives (6). There is a continuous polarization family at each fixed clock, without assigning a field amplitude to fit the frequency.

The previous polynomial reconstruction chose c=epsilon P_d'(h_l). Strict convexity puts that derivative between the backward and forward finite count differences, so it lies strictly inside (3). At l=0, P_d'(h_0)=H_d<d for every maintained depth. Thus every previous stationary profile also solves (1), with the same first source and frequency. The old c preparation used the inferred assignment; its recovery is a consistency check, not an independent prediction of that assignment.

4. Energy Controls Distance to the Stationary Family

Retain the complete fixed spatial shell h=h_l, and describe its normalized pair by Bloch coordinates (x,y,c), x squared+y squared+c squared=1. Now r=sqrt(c squared+y squared), which coincides with c only on the positive real stationary family. The complete radius-scaled pair energy is

$$\mathcal E=h_lc+\min_k[\epsilon S_k-h_kr].$$

Since the minimum is no greater than its k=l term and c<=r,

$$\mathcal E\le\epsilon S_l.$$

Equality holds at y=0, c=r in the closure of branch l's winner interval. Write that closed interval as I_l and the energy deficit as Delta=epsilon S_l-mathcal E. Comparing with the adjacent k=l-1 and k=l+1 costs gives the quantitative bound

$$\boxed{\Delta\ge h_l(r-c)+\operatorname{dist}(r,I_l),\qquad y^2\le\frac{2\Delta}{h_l}.} \tag{7}$$

For the second inequality use y squared=(r-c)(r+c) and r+c<=2. These are exact inequalities on the whole Bloch sphere. Energy-preserving pair evolution with small Delta remains close to this maximizing family. This is conditional confinement near a set, rather than persistence near one selected member. At interior stationary points the real-polarization curvature is zero and the relative-phase curvature is -h_l/c. The local linearized pair can therefore drift along the family.

Equation (7) does not supply existence or uniqueness through tied charts. For excited assignments I_l is away from r=0; small Delta stays away from that singular current endpoint. The ground-family closure includes r=0, so its energy bound alone does not supply the same domain protection. No full spatial persistence or observed lifetime follows from this two-mode result. Those are subsequent questions for this specified action.

5. Reproducibility and Physical Scope

An exact ground spatial comparison with its report retains the new scalar Hessian and complete current variation. At the common diagnostic c=3/5 and epsilon_d=c/H_d, the first rotational spatial harmonic has a complex squared-frequency pair at every occupied depth. Exact rational characteristic polynomials and real-root counts verify this growing spatial disturbance. It does not negate the exact stationary-family or pair-energy results, and it is not a test of all assigned excited particles. The first report incorrectly counted the zero endpoint as a negative root; the corrected report removes exact zero factors and separately counts nonreal roots. Both exact sources and the correction record are preserved.

discrete-count-action.py and its report retain exact integer counts and rational winner intervals at all fourteen assignments in depths 2,3,4,5,6 and 10. A direct finite search agrees with the all-label adjacent-difference proof. Three distinct polarizations per assignment satisfy the complete pair equation at 90-digit precision. Arbitrary full Clifford matrix variations independently check the actual minimum potential's source; their maximum discrepancy is below 1.3e-10. Independent full-Bloch checks also verify the energy bounds.

The result gives a forward model using geometric integer data and one explicit local variational premise. It improves the transparency of the count mechanism without deriving the premise's microscopic origin. The ordinary sphere, common radius within a sector and sector frequency unit remain supplied. It does not establish localization, complete boost inertia, observed electric charge or a preference for the inferred depths.