Radial freedom in the gapped simplex carrier

The equally spaced polarization values of the normal Dirac carrier do not require the unbounded coefficient W=z^n. They follow from a radial boundary identity that remains valid for a much larger profile family. This permits a bounded spatial coefficient while preserving the simplex numerator. The profile is a coefficient of the inverse spatial operator; its physical source and total supporting inertia remain open.

Use the four-component ordinary planar Dirac operator defined in the holomorphic carrier, now with W=w(r) exp(i n theta), where w is real, positive away from zero, regular as r^n at zero, and supports decaying zero modes. Write q=n-1. The negative-chiral zero equations are

$$U_l'=lU_l/r-wV_l,\qquad V_l'=(q-l)V_l/r-wU_l. \tag{1}$$

Normalize each four-component mode by 2 pi integral r(U_l^2+V_l^2)dr=1. Direct differentiation gives

$$\frac{d}{dr}[r^2(U_l^2-V_l^2)] =2r[(l+1)U_l^2-(n-l)V_l^2]. \tag{2}$$

When the boundary term vanishes, this proves

$$\|U_l\|_2^2=\frac{n-l}{n+1},\qquad \|V_l\|_2^2=\frac{l+1}{n+1},\qquad P(\sigma_3\otimes I)P=-\frac{2J_z}{n+1}. \tag{3}$$

Here (J_z)_ll=l-q/2. Equation (2) also excludes every integer label outside l=0,...,q for a nonzero square-integrable mode. The n regular labels each have one decaying radial solution under the stated profile conditions. Their count refers to available modes, not their occupation.

Two other exact identities survive the radial change. Ordinary momentum has zero within-kernel projection, P p_x P=P p_y P=0, because its adjacent-label radial integral is the boundary derivative

$$rw(U_{l+1}V_l+V_{l+1}U_l)-2(q-l)V_{l+1}V_l =-\frac{d}{dr}(rV_{l+1}V_l). \tag{4}$$

The complementary inverse of the polarization source is still the positive-chiral vector r(V_l,U_l)/(n+1), with the angular phases of the original carrier. These identities concern this unconnected radial operator. The connected extension has different weighted polarization and momentum formulas.

A smooth bounded profile and a full gap

One explicit choice is

$$w(r)=\frac{r^n}{\sqrt{1+(r/R)^{2n}}},\qquad R=2. \tag{5}$$

It approaches the finite positive value R^n at infinity. The positive-chiral square has diagonal -Delta+w^2 and off-diagonal magnitude |w'-nw/r|. Its full quadratic form therefore obeys

$$Q^\dagger Q\ge-\Delta+(1-\epsilon_n)w^2, \qquad \epsilon_n=\frac{\sqrt n\,(n-1)^{(n-1)/(2n)}}{R^{n+1}}. \tag{6}$$

To obtain (6), maximize |w'-nw/r|/w^2. Its maximum occurs at (r/R)^(2n)=n-1. Put a=(1-epsilon_n)/2. Then the scalar potential in (6) is at least a r^(2n) inside r=R and a R^(2n) outside.

Use a positive test function equal to exp(-eta r^2/2) inside R and exp(-eta R r+eta R^2/2) outside. Its value and first derivative agree at R. The ground-state transform gives the full-space lower bound

$$g_n^2\ge\min\left\{ 2\eta-(n-1)n^{-n/(n-1)}a^{-1/(n-1)}\eta^{2n/(n-1)},\; aR^{2n}-\eta^2R^2\right\},\qquad n\ge2. \tag{7}$$

For eta=1 and R=2, the lower bounds on the nonzero Dirac frequencies are approximately 1.16350 at n=5, 1.15149 at n=6 and 1.11670 at n=10. Thus the entire complementary spectrum has a positive gap. This is an analytic operator inequality, not a finite-box eigenvalue estimate.

The bounded multiplier h=z/sqrt(1+r^2) still has fixed positive adjacent integrals h_l. Their relative changes from the monomial profile are about 0.00059 at n=5, 0.00018 at n=6 and 0.0000013 at n=10. They change the shared frequency unit, while the diagonal product is unchanged:

$$\omega_n(\beta)= \frac{\prod_{l=0}^{n-1}(\beta+l-(n-1)/2)} {\gamma^{n-1}\prod_l h_l}[1+O(\gamma^{-2})]. \tag{8}$$

The existing assignments n=d and beta=L+(d-3)/2 give the numerator d! binomial(L+d-2,d). No particle indices or field magnitudes were adjusted. The full normal correction remains bounded by the earlier chirality-based resolvent argument with the new gap (7).

The complementary response must also change

For (5), the positive-chiral Green inverse is a coupled radial two-component inverse. The monomial carrier's separate scalar Green functions are no longer valid. The complete coupled calculation retains every forced angular sector and both spinor components.

At n=6 and gamma=10000, the refined partial wave momentum response divided by frequency is approximately (265079.42,-265077.76) for the electron branch and (44061.20,-44059.54) for the muon branch. These signed responses belong to a wave in prescribed coefficients. They do not describe negative physical particle inertia. The phase of a partial response and the momentum of the complete sustaining field are separate calculations.

The report bounded-normal-20260914T130117_970869Z.json passes 37 checks of the kernel identities, analytic bounds, coupled field residuals and second-order radial refinement. The finest-grid complete first-equation residual is about 1.26e-6 relative to its source. Source hashes and raw profiles accompany the report.

Bounded coefficients improve the inverse operator, but their boundedness alone does not establish finite physical supporting energy. The next linked construction supplies a finite geometric functional as well; its reciprocal wave source must still be included in a complete model.