A Convex Dispersion Rule Gives Exact Count States and Persistence

The count-state construction has a direct convex forward form. Take the maximum of massive-Dirac dispersion curves fixed by the geometric count data, smooth that maximum in its energy arguments, and eliminate one strictly convex algebraic variable. This gives one compatible constitutive law across every eligible shell. Each count node retains its exact value, slope and an open massive-dispersion neighborhood.

In the preferred-time positive-wave realization, these neighborhoods become exactly affine scalar-source charts. The complete energy is nonnegative, and the existing full-spectrum argument gives conditional nonlinear persistence for nodeless fields at all fourteen maintained assignments. The finite feedback-label window is explicit: omitting it would admit extra single-shell frequencies beyond the count range.

This is a simpler effective realization, not a derivation of its constitutive premise from the master field. The ordinary sphere, sector units and count tangency data remain supplied. No new preference for the assigned depths follows from the fact that the construction works there.

1. The Geometric Data

Use radius units R=1 on the conditional ordinary round spatial sphere. Its positive Dirac shells have

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

Set epsilon_d=mu_d R. On the existing convex count domain let

$$c_l=\epsilon_dP'_d(h_l),\quad x_l=\sqrt{1-c_l^2}, \quad m_l=\frac{h_lx_l}{c_l},\quad \eta_l=\epsilon_dS_l-\frac{h_l}{c_l}. \tag{2}$$

The eligible set contains every nonnegative integer l satisfying 0<c_l<1 and

$$h_l\epsilon_dP_d''(h_l)>c_l(1-c_l^2). \tag{3}$$

At fixed epsilon this set is finite and contiguous in the maintained comparison. It includes unassigned shells, not merely the observed particle list. The coefficient m_l is a source coefficient, distinct from the target mass-frequency epsilon_d S_l.

Equation (3) is the convexity condition of the count-tangent construction. At d>=3 it holds throughout h>=d/2 with 0<epsilon P'(h)<1. At d=2 it imposes the genuine lower convexity boundary. These are mathematical domain conditions, not an empirical occupation law.

2. Massive Dispersion Curves Touch the Count Curve

For every eligible l define

$$F_l(\chi)=\sqrt{\chi^2+m_l^2}+\eta_l,\qquad \chi>0. \tag{4}$$

At its own shell,

$$F_l(h_l)=\epsilon_dS_l,\quad F_l'(h_l)=c_l,\quad F_l''(h_l)=\frac{c_l(1-c_l^2)}{h_l}>0. \tag{5}$$

These are the usual massive-Dirac dispersion functions with coefficients specified by the count tangency data. Their mathematical use does not identify m_l with the observed mass.

There is a short proof that the intended branch wins. Write F_count(h)=epsilon_d P_d(h), c(h)=F_count prime and m(h)=h sqrt(1-c(h) squared)/c(h). Differentiation gives

$$m'(h)=\frac{c(1-c^2)-hc'}{c^2\sqrt{1-c^2}}<0 \tag{6}$$

by (3). Since c(h)=h/sqrt(h squared+m(h) squared), for h>h_l the slope of F_count exceeds that of F_l, and for h<h_l it is smaller. Their difference has its unique minimum zero at h_l. Thus

$$F_l(h)<F_{\rm count}(h)\quad(h\ne h_l) \tag{7}$$

on the continuous convex count interval. In particular, among all the curves (4), l is the unique winner at h_l.

Let

$$F_{\max}(\chi)=\max_{l\in L_d}F_l(\chi). \tag{8}$$

This is one strictly convex energy function with the desired count values and slopes. It is generally not differentiable at crossings; section 4 supplies an explicit compatible smoothing.

3. Crossings and Winner Gaps Have Exact Rational Checks

All h_l, c_l, eta_l and m_l squared are rational under the diagnostic scale prescription. For adjacent l and l+1 put

$$\delta_l=\eta_{l+1}-\eta_l>0,\qquad \Delta_l=m_l^2-m_{l+1}^2>\delta_l^2.$$

Their unique positive crossing has

$$\boxed{\chi_l^2= \left(\frac{\Delta_l-\delta_l^2}{2\delta_l}\right)^2 -m_{l+1}^2,\qquad h_l^2<\chi_l^2<h_{l+1}^2.} \tag{9}$$

Its energy is rational as well: F_cross=(Delta_l-delta_l squared)/(2delta_l)+eta_(l+1). The ordered adjacent crossings establish the full winner ordering; no unseen nonadjacent curve can displace the winner between them.

A convenient rigorous lower bound for any node gap uses

$$A_{lk}=\epsilon_dS_l-\eta_k,\qquad D_{lk}=A_{lk}^2-h_l^2-m_k^2.$$

For k!=l, A_lk and D_lk are strictly positive. Rationalization gives

$$F_l(h_l)-F_k(h_l) =\frac{D_{lk}}{A_{lk}+\sqrt{h_l^2+m_k^2}} >\frac{D_{lk}}{2A_{lk}}>0. \tag{10}$$

Thus every node and every competitor can be certified by rational arithmetic, without numerical subtraction of nearly equal roots.

4. Smooth the Maximum While Preserving Open Winning Arcs

Let N be the number of eligible curves, and let Delta_min be the minimum of the rational lower bounds in (10). Use the common width

$$\zeta_d=\frac{\Delta_{\min}}{8N}. \tag{11}$$

This width uses every eligible node. It is a specified regularization rule, not a fitted physical scale or a uniquely derived choice.

Convolving absolute value with the normalized compact kernel 15(1-t squared) squared/16 on [-1,1] gives the explicit smooth absolute

$$H_\zeta(z)= \begin{cases} |z|,&|z|\ge\zeta,\\ \displaystyle\frac{\zeta}{16}(5+15u^2-5u^4+u^6), &|z|<\zeta,\quad u=z/\zeta. \end{cases} \tag{12}$$

It is convex and C^3, with |H prime|<=1 and H double prime nonnegative. Define

$$M_\zeta(a,b)=\frac{a+b+H_\zeta(a-b)}2. \tag{13}$$

This is convex and nondecreasing in each argument, and equals max(a,b) exactly whenever |a-b|>=zeta. Its excess over max is at most 5 zeta/32. Fold (13) over (4) in increasing shell order to define F_s.

The accumulated excess before any winning node's own curve is inserted is less than N zeta/2. Equations (10) and (11) leave a gap much larger than zeta, so insertion selects the correct curve exactly. Every later competitor remains below it by more than zeta. Continuity supplies an open neighborhood of h_l on which

$$\boxed{F_s(\chi)=F_l(\chi).} \tag{14}$$

Strict convexity is also retained. At every two-curve smoothing step,

$$(M_\zeta(a,b))''= \frac{1+H_\zeta'}2a''+ \frac{1-H_\zeta'}2b''+ \frac{H_\zeta''}2(a'-b')^2>0. \tag{15}$$

Each branch has positive curvature; the weights are nonnegative and sum to one. Its slope lies strictly between zero and one for chi>0. The same argument applies recursively.

5. A Unique Algebraic Feedback Law

A finite set of curves has a lowest and a highest member. Without a constitutive label restriction, its outer arcs extend to additional integer spatial shells and assign them non-count frequencies. This is a real limitation of an unrestricted finite envelope.

Use the explicit common validity interval

$$I_d=(h_{\min}-1/2,\ h_{\max}+1/2). \tag{16}$$

The half-spacing is a common domain convention, not a fitted particle coefficient. It contains every eligible integer shell and no others. Restrict c to the image F_s prime(I_d), and x=sqrt(1-c squared)>0. For each such x there is exactly one chi in I_d satisfying

$$F_s'(\chi)=c. \tag{17}$$

This follows from (15). Define the preferred-time potential by

$$\rho=\operatorname{tr}\Phi^\dagger\Phi,\quad s=\operatorname{tr}\Phi^\dagger\beta\Phi,\quad x=s/\rho,$$ $$V(\Phi)=\rho f(x),\qquad f(x)=F_s(\chi)-\chi\sqrt{1-x^2}. \tag{18}$$

Equivalently, eliminate chi from rho F_s(chi)-chi sqrt(rho squared-s squared). The domain requires rho>0 and the stated open ratio interval. The constitutive coordinate is algebraic, not an extra propagating field, mediator or physical dimension.

The complete source is

$$G=[f'(x)\beta+\eta(x)I]\Phi,\qquad f'(x)=\frac{\chi x}{c},\quad \eta(x)=F_s(\chi)-\frac{\chi}{c}. \tag{19}$$

It retains numerator and denominator variations. This uses the chosen time density rho. Replacing rho with a proper current norm or a responding clock gives different source derivatives and does not inherit this result.

By (14), local elimination around x_l yields exactly

$$\boxed{f(x)=m_lx+\eta_l} \tag{20}$$

on an open interval. Thus f double prime vanishes there, not merely at its center. The count and response conditions coexist in one globally compatible convex feedback construction.

The function is locally smooth on the open domain in (18). Pure-grading endpoints, vacuum crossings and a global physical continuation outside the finite label interval remain separate questions.

6. Two Forward Equations Give Every Count Clock

Let E_k be an actual balanced positive Dirac shell, D E_k=h_k E_k. Use the real normalized pair (a+b beta)E_k, with c=a squared-b squared and x=2ab>0. Its complete first-order generating energy per norm, before elimination, is

$$e(c,\chi)=h_kc+F_s(\chi)-\chi c.$$

The two stationary equations are simply

$$\boxed{h_k-\chi=0,\qquad F_s'(\chi)-c=0.} \tag{21}$$

Therefore chi=h_k. Because h_k must lie in (16), k is eligible; (14) then gives c=c_k and

$$\boxed{\omega_k=F_s(h_k)=\epsilon_d\binom{k+d}{d}.} \tag{22}$$

Independent substitution into D+G yields the same eigenvalue: sqrt(h_k squared+m_k squared)+eta_k=epsilon_d S_k. This classifies the balanced positive single-shell eigenstates of D+G within the stated constitutive domain. It does not classify arbitrary mixed-shell nonlinear fields or every standing wave of the second-order completion. Sparse observed occupation remains open.

7. Positive Complete Energy and Nonlinear Spatial Persistence

Use the fully varied positive-wave action

$$\mathcal F(\Phi)=D\Phi+G(\Phi),\qquad L_+=\|\dot\Phi\|^2-\|\mathcal F(\Phi)\|^2,\qquad \ddot\Phi+(D\mathcal F[\Phi])^*\mathcal F(\Phi)=0. \tag{23}$$

Its real adjoint and all field dependence are retained. Homogeneity and phase covariance imply that each solution of mathcal F(phi_k)=omega_k phi_k gives the exact standing wave Phi_k=exp(-i omega_k t)phi_k. Complete energy and canonical phase action are

$$E_+=\|\dot\Phi\|^2+\|\mathcal F(\Phi)\|^2\ge0,\qquad \mathcal Q_+=-2\operatorname{Im}\langle\Phi,\dot\Phi\rangle.$$

For a count state of norm N_k,

$$E_+=2\omega_k^2N_k,\quad \mathcal Q_+=2\omega_kN_k,\quad E_+=\omega_k\mathcal Q_+. \tag{24}$$

Canonical phase action is not electric charge. Its common physical normalization is not fixed by these identities, and amplitudes remain free.

On the open chart (20), the actual nonlinear equation reduces exactly to

$$\ddot\Phi+H_k^2\Phi=0,\qquad H_k=D+m_k\beta+\eta_k. \tag{25}$$

Its complete spatial spectrum is eta_k plus or minus sqrt((j+d/2) squared+m_k squared), j>=0. A zero exists precisely when

$$(j+d/2)^2=h_k^2-2(h_k/c_k)\omega_k+\omega_k^2. \tag{26}$$

The right side is rational, and comparison with the adjacent integer shells decides the entire infinite spectrum. All fourteen assignments have a nonzero gap. No harmonic cutoff is used in this decision.

For any nodeless balanced background, the conserved graph energies of (25), ellipticity and Sobolev embedding control sufficiently regular small perturbations uniformly in position for all time. Choose the initial position and velocity disturbance small enough to preserve the positive density floor and the open chart (20). The exact wave evolution then never leaves that chart and solves (23) for all time.

This is the positive count-wave persistence argument with the newly constructed common source. It includes every spatial variation and the actual minimum carriers when their stated balance conditions hold. It is conditional nonlinear persistence on the assumed sphere, not containing-space localization or a complete particle.

The principal wave operator and the local positive energy-flux identity also remain those of (23), with speed one in radius units. Preferred-time constitutive dependence still prevents identifying this alone with a complete Lorentz-covariant particle or proving its translational inertia.

8. All-Sector Comparison

Use the same diagnostic prescription as the preceding count-state work, epsilon_d=(3/10)/max_assigned P_d prime. The scalar energy smoothing width zeta differs from a scalar-ratio convolution width.

Every maintained assignment retains its exact target frequency and its same balanced field. The first omitted higher shell at each depth supplies an explicit unrestricted-envelope control: its last-arc clock is not the intended count clock, and (16) excludes that shell.

These are shared mathematical properties, not an additional depth-five selection or an independent empirical test of the inferred assignments. The minimal carriers' previously derived rotation forms remain available because the fields themselves are unchanged.

Reproduction and Remaining Physical Work

dispersion-count-envelope.py and its report pass 96,422 conditions. Exact rational inequalities test every eligible node against every competing curve, and locate every adjacent crossing. Independent 100-digit elimination and scalar differentiation verify the source value, slope and vanishing curvature at all fourteen assigned fields. The script checks every joining band, the compact smoothing's C^3 matching, and the exact all-spectrum zero condition. Count-energy errors are below 4e-97 in the displayed dimensionless comparison.

The global winner, smoothing and nonlinear persistence conclusions also use the proofs above. Check counts are coverage, not empirical evidential weight. No particle trajectory or time-domain simulation enters these calculations.

This construction replaces a chosen analytic shell-correction shape or a nonsmooth maximum of field potentials with a convex dispersion feedback rule. It preserves the mass targets, exact source, positive complete energy and chart-local persistence together. Its count tangency data, finite validity domain, ordinary spatial carrier and preferred-time density remain effective premises. A master-field derivation, physical support and motion, normalization, reciprocal encounters and sparse occupied states remain to be established.