Higher angular response of the localized current field

Summary: Homogeneous closed and coclosed polynomial forms give complete angular blocks for the even-depth matrix field; independent Cartesian checks and expanded electron and muon scans retain the distinction between angular classification and full spatial stability.

Three different indices

The earlier localized response calculation used constant Clifford forms. Those angular families include translations but only a bounded range of ordinary spherical harmonics. Here $d$ is an even ordinary spatial depth, $k$ is a Clifford form degree, and $\ell$ is a polynomial angular degree. Neither $k$ nor $\ell$ is the occupied mass grade $N$. The electron and muon retain $N=12$ and $N=34$ at $d=6$.

At even depth, the full complex $r\times r$ matrix space, $r=2^{d/2}$, is represented by all Clifford forms. No spinor or partner directions are discarded by using that matrix basis.

Let $P_{\ell,k}(x)$ be a Hermitian Clifford $k$-form whose coefficients are homogeneous polynomials of degree $\ell$, satisfying

$$\sum_a\alpha_a\partial_aP_{\ell,k}=0,\qquad \sum_a(\partial_aP_{\ell,k})\alpha_a=0. \tag{1}$$

These are simultaneously closed and coclosed polynomial differential forms. A concrete nonzero example for $1\leq k\leq d-1$ uses $z=x_1+ix_2$:

$$P_{\ell,k}(x)=i^{k(k-1)/2} \left[\operatorname{Re}(z^\ell)\alpha_1 -\operatorname{Im}(z^\ell)\alpha_2\right] \alpha_3\cdots\alpha_{k+1}. \tag{2}$$

The final product is empty for $k=1$. Direct differentiation proves both equations (1). Real linear combinations and rotations provide the other polarization patterns in the same polynomial space.

A closed radial block at every angular degree

Write $x=r_xn$ and $T(n)=P_{\ell,k}(n)$, normalized by its mean squared trace on the unit sphere. Introduce

$$B_0=T,\quad B_1=i\alpha_nT,\quad B_2=iT\alpha_n, \quad B_3=\alpha_nT\alpha_n, \qquad s_k=(-1)^k(d-2k). \tag{3}$$

These $B_i$ are angular basis matrices, not the scalar insertions of the action. Homogeneity and (1) give $\alpha\cdot\nabla T(n)=-\ell\alpha_nT/r_x$. The complete spatial Dirac operator $D=-i\alpha\cdot\nabla$ acts as

$$\begin{aligned} D(aB_0)&=-(a'-\ell a/r_x)B_1,\\ D(bB_1)&=[b'+(d+\ell-1)b/r_x]B_0,\\ D(cB_2)&=s_kcB_0/r_x+[c'-(\ell+1)c/r_x]B_3,\\ D(eB_3)&=s_keB_1/r_x-[e'+(d+\ell)e/r_x]B_2. \end{aligned} \tag{4}$$

These identities follow by the Cartesian product rule. They do not replace $d$ by an effective physical depth. All derivatives still act in the original $d$ ordinary directions.

The four diagonal angular Gram entries are one. The only nonzero off-diagonal entries are

$$G_{03}=G_{30}=G_{12}=G_{21}=t_{\ell,k},\qquad t_{\ell,k}=\frac{s_k}{d+2\ell}. \tag{5}$$

For example, $\Delta(\alpha_xP_{\ell,k}\alpha_x)=2s_kP_{\ell,k}$, while $\Delta(r_x^2P_{\ell,k})=2(d+2\ell)P_{\ell,k}$. Their harmonic difference is orthogonal to degree $\ell$, proving (5). This argument applies to every solution of (1), not only (2).

An orthonormal angular basis is

$$E_0=B_0,\quad E_+=\frac{B_1+B_2}{\sqrt{2(1+t_{\ell,k})}},\quad E_-=\frac{B_1-B_2}{\sqrt{2(1-t_{\ell,k})}},\quad E_3=\frac{B_3-t_{\ell,k}B_0}{\sqrt{1-t_{\ell,k}^2}}. \tag{6}$$

Their ordinary harmonic degrees are $\ell,\ell+1,\ell+1,\ell+2$. At $\ell=0$, (4)–(6) recover the earlier constant-form blocks. The exceptional constant scalar and pseudoscalar forms have two angular components rather than four.

The complete local binding Hessian closes on this span. Clifford grade fixes $\beta T\beta=(-1)^kT$, and Hermitian conjugation exchanges the required basis expressions without discarding complex coefficients. For $k=1$, the trace term also closes because $\alpha_nT+T\alpha_n$ is a scalar matrix. Other interior form degrees have the corresponding zero traces. Thus the ordinary and conjugate coefficients of every upper and lower matrix variation remain present. The current-metric comparison changes the upper partner curvature by its stated factor $q$ and preserves this closure.

Why the angular blocks cover the matrix field

The Fischer decomposition for closed and coclosed polynomial forms states that arbitrary Clifford-valued polynomials are generated from solutions of (1) by radial powers, exterior multiplication by $x$, and contraction with $x$. On the sphere these operations are exactly the span (3), with the two exceptional constant families. This gives the angular classification; the radial functions remain unrestricted. See Theorem 1 of Delanghe, Lávička and Souček.

The multiplicities provide an independent count. For $1\leq k\leq d-1$, the dimension of the polynomial space in (1) is

$$h_{\ell,k}=\binom{d-2}{k-1}\binom{\ell+d-2}{d-2} \frac{(2\ell+d)(\ell+d-1)}{(\ell+k)(\ell+d-k)}. \tag{7}$$

Set $h_{0,0}=h_{0,d}=1$, $h_{\ell,0}=h_{\ell,d}=0$ for $\ell>0$, and every negative-degree multiplicity to zero. Formula (7) follows by solving the dimension recurrence for the harmonic pieces:

$$\binom dk\left[\binom{j+d-1}{d-1}-\binom{j+d-3}{d-1}\right] =h_{j,k}+h_{j-1,k-1}+h_{j-1,k+1}+h_{j-2,k}, \tag{8}$$

for interior $k$; the final term is absent at the two endpoint form degrees. Unavailable binomials are zero. Counting (6) at a fixed ordinary harmonic degree $j$, including the exceptional constant families, yields exactly

$$2^d\left[\binom{j+d-1}{d-1}-\binom{j+d-3}{d-1}\right]. \tag{9}$$

This is the full scalar-harmonic dimension times the $r^2=2^d$ matrix components. Polynomial restrictions are dense on the sphere, so the classification extends to the angular Hilbert space. It does not turn any finite radial calculation into a stability theorem.

Including radial Gaussian polynomial degrees recovers the usual shell count $2^d\binom{n+d-1}{d-1}$. This is a count for that chosen polynomial basis. It is not a derivation that the nonlinear particle cavity has an oscillator spectrum, nor does it select the weights of its mass operator.

Expanded tests of the actual smooth core

The calculation uses the solved smooth core with central-shape parameter $10^{-4}$, common gap one, sustaining coefficient $g=7.875370789623267$ and unit norm. Its root mean squared radius is $1.276658598123441$ in gap units. Both current norms use that same fixed point.

Independent Cartesian differentiation checks (4) at depths six and ten, all interior form degrees, and $\ell=0,1,2,5$. The largest normalized error is $5.67\times10^{-12}$. Full arbitrary-complex matrix Hessian checks have maximum relative residual $1.04\times10^{-15}$. Angular quadrature independently checks the Gram matrix. The entire finite $\ell=0$ projected matrices also agree with the earlier implementation.

The spatial scan uses every combination of

• $\ell=1,2,4,8,16$ and $k=1,\ldots,5$ at physical depth six;
• the assigned electron and muon frequencies $18564$ and $3838380$;
• the original norm $q=1$ and equal-contraction norm $q=1/9$;
• 26 radial functions with scale $0.5$ and 180 quadrature points, and 40 radial functions with scale $0.55$ and 260 quadrature points.

The 200 finite spectra retain both frequency signs, ordinary and conjugate variations, and every component of each stated angular block. The largest computed real growth parts are

These real parts are unresolved at the numerical scale of the large frequency blocks. The scan establishes no resolved exponentially growing mode within its stated finite ranges. It does not cover every $\ell$, arbitrarily short radial wavelengths, or finite nonlinear disturbances. Real eigenvalues also do not exclude transient amplification; the current-metric note gives a separate exact limiting-operator analysis.

Reproduction and limits

localized_harmonic_response.py constructs the angular matrices and multiplicities. localized_harmonic_response_checks.py performs the independent Cartesian and full-Hessian checks, exact dimension counts, and all 200 spectra using localized_clifford_response.py. The report preserves the full finite eigenvalue arrays and each discretization, rather than only favorable selected frequencies.

This supplies the angular blocks needed for a complete future spatial analysis. It does not prove nonlinear stability, determine the physical mass carrier, derive the shell weights, or identify observed interaction vertices. The matrix decomposition and the absence of resolved growth in these finite calculations have precisely those separate scopes.