The spatial response of a shape-preserving mass core

A positive shape term can change the disturbance response of the polarization core without changing its mass-frequency, spatial profile or canonical inertia. The useful distinction is between the background matrix form $fI+i h\alpha_{\hat x}$ and the additional matrix components available to a disturbance. Those components remain physical field variations; none is removed from the equations.

The regular current action supplies the complete potential and its nonsingular extension. Its additional term is

$$U_{\rm shape}=\mu\|P_\perp\phi\|_F^2,\qquad P_\perp X=X-\frac{\operatorname{Tr}X}{s}I_s- \frac1s\sum_{a=1}^d\operatorname{Tr}(\alpha_aX)\alpha_a. \tag{1}$$

Here $s$ is the matrix representation rank. The sum runs over the actual $d$ ordinary spatial directions. The complete source of (1) is $(\mu P_\perp\phi,0)^T$, and its upper-block Hessian is $\mu P_\perp$. Both vanish on the original radial field except for the Hessian acting on disturbances outside that field's matrix form.

Preserving motion and positive inertia

Translations and spatial rotations of the core stay in the scalar/vector span. In a retained component frame, an infinitesimal boost of an upper-only core enters the lower block. Consequently the linear map $\delta(P_\perp\phi_u)$ has no frame-only contribution on this branch. The mixed frame-wave and pure frame second variations of (1) are zero.

This matters because a binding term that merely preserves the stationary profile can still produce a new reciprocal frame force. The analytic comparison in Holomorphic binding and its responsive-frame test does exactly that. The shape penalty has no such mixed quadratic source. The cutoff terms also have zero added branch variations, since their cutoffs are constant near the core.

A positive retained frame kinetic action therefore adds its own linear frame equation without this new wave-frame mixing. For a polarized grade there can also be the existing antisymmetric rotation term; its contribution to the quadratic frame energy derivative is zero. This does not remove the separate nonlinear frame-vacuum problem or derive the physical carrier.

The uniformly boosted core still solves the complete branch equations. The canonical translation–boost pairing remains

$$M_{ab}=\omega Q\delta_{ab}>0$$

in natural units. With $Q=1$, inertia has the mass-frequency value. Neither a signed shape response nor a complex disturbance frequency is being interpreted as a negative particle inertia.

A complete local decomposition across depths

Freeze the radial polarization ratio $y=h/f$ at a point, and choose $\alpha_1$ radially and $\alpha_2$ in the transverse momentum direction. A Hermitian basis of the full matrix algebra is supplied by Clifford monomials

$$\Gamma_I=i^{|I|(|I|-1)/2}\prod_{a\in I}\alpha_a.$$

Fix the subset outside directions one and two. Toggling membership of those two directions gives four monomials. Upper and lower spinor blocks, and real and imaginary coefficients, make a sixteen-real-component orbit. The complete current Hessian, the shape projector and both principal momentum matrices preserve it.

This is a decomposition of the full local matrix equation. It is not a choice of sixteen disturbances with the other components discarded. For even depth, the orbits with $k$ external generators have multiplicity $\binom{d-2}{k}$. In the chosen odd-depth representation, one additional unused Clifford generator completes the matrix algebra. Its two possible occupancies are both retained; it adds no physical coordinate and is excluded from the physical-vector projector in (1).

Explicit full-matrix embeddings verify the decomposition at depths three, four and five. The Clifford algebra, complete dimension count, and direct source closure verify the corresponding orbit classes through depth ten. The common current rule is $q=1/(s+1)$; reusing the depth-six number $1/9$ at depth three would change that rule.

The center and the tail impose different local tests

Use the common illustrative units $m_c=1$, $\kappa=2$, and denote the shape coefficient by $c=\mu/m_c$. In the high-wave-number compression at the center, the full local characteristic polynomial contains

$$16z^4+8Xz^2+X(X-64),\qquad X=c^2+2c(q-2)\omega+4(1-q)\omega^2. \tag{2}$$

Here $z$ is the local growth-generator eigenvalue after the leading propagation phase is removed, and $\omega$ is in core-gap units. For $X>0$, this factor has purely imaginary roots only if $X\ge64$, because

$$z^2=\frac{-X\pm8\sqrt X}{4}.$$

At depths three and four, $q=1/5$. With $c=1$, the upper threshold of this particular center test is

$$\omega\ge\frac{9+\sqrt{5121}}{16}. \tag{3}$$

Thus $\omega=3$ can pass the earlier tail controls while still failing this center control. At $\omega=5$, the center factor has a positive local growth root of about 0.353.

A common scan with $c=1$ and $\omega=6,20,100$ has no resolved local growth at any of the tested depths. It retains every orbit, ratios $y=0,0.3,0.7,0.95,1$, radial and transverse directions, intermediate angles, and momenta from zero through 100. The largest raw real parts remain below the stated numerical tolerance.

These are local spectral controls. A frozen center is a finite spatial region of the actual particle, so local amplification there is not by itself a theorem of unbounded whole-particle evolution. Conversely, passing these controls does not prove spatial stability. Equation (3) is a condition for this local test, not a measured cavity-size bound or a universal physical necessity.

Choosing $\omega_{0,d}/m_c=6$ would put every maintained single-grade frequency at six or greater under one common scale rule. This is not an extrapolation of the finite scans to arbitrarily large grade. That ratio is a possible inverse choice, not a derived constant. The bottom two-grade assignment still needs its separate coupled dynamics. No particle is retuned independently in this comparison.

Actual spatial angular equations

Local Clifford closure does not replace spatial derivatives. For a constant Clifford form $T$, retain the actual angular family

$$T,\quad i\alpha_{\hat x}T,\quad iT\alpha_{\hat x},\quad \alpha_{\hat x}T\alpha_{\hat x}. \tag{4}$$

The scalar and top-form endpoint families have two independent members. Every member has arbitrary complex radial coefficients in both spinor blocks. The angular Gram matrix follows from the physical sphere average and $\sum_a\alpha_aT\alpha_a=(-1)^k(d-2k)T$, with the corresponding parity change for an unused representation generator.

In an orthonormal angular basis, the spatial Dirac operator is

$$D_r=M\partial_r+\frac{A}{r}. \tag{5}$$

Independent Cartesian finite differences verify (5), and arbitrary complex variations at other physical directions verify the complete current and shape source. These checks include every constant-Clifford form degree at depths 3, 4, 5, 6 and 10, and both representation sectors at odd depth. They do not include all higher spherical-harmonic families.

Regular Laguerre radial bases give finite Hamiltonian matrices whose Hermitian structure and physical radial norms are independently checked. Some raw complex eigenvalue pairs occur near the radial basis cutoff. Their appreciable highest-basis weights and changing frequencies make them insufficient evidence of localized unstable modes. Separate central and staggered radial difference controls also show outer-domain and grid sensitivity in that high-frequency region.

Testing the outgoing condition

For a complex temporal growth value $\lambda$, the radial equation can be written in canonically weighted variables as

$$Y'=C\left[\lambda I-V(r)-\frac{V_a}{r}\right]Y, \qquad C^2=I. \tag{6}$$

A localized growing mode must belong both to the regular origin subspace and to the decaying far-field subspace. Transport orthonormal bases for those subspaces to a common radius, and form their combined matrix. Its smallest singular value must vanish for a matched mode.

Three apparent finite-basis candidates were checked directly:

Their outgoing matching residuals remain nonzero under outer-radius refinement from 30 to 60 to 120. For the first and third candidates, they approach approximately 0.299445 and 0.267559. These candidates therefore do not supply resolved localized growing modes of the radial equation. The known phase/amplitude zero-mode limit supplies an independent positive control of the matching procedure.

This is a rejection of those specified numerical candidates, not an exhaustive search for all unstable eigenvalues. It leaves a concrete spatial spectral problem to solve with the full angular hierarchy and appropriate outgoing conditions. The local action now avoids its original current-zero divisions and its specific tested tail defect, while a complete stable physical mass operator remains unestablished.

Reproducible calculations

The 514-check local report records full source variation, boosted fields, frame quadratic coefficients, complete Clifford embeddings, local scans and the exact factor (2). polarization_angular_channel.py constructs the actual angular and radial operator; polarization_outgoing_match.py implements the independent origin-to-infinity matching test. The 243-check spatial report records the Cartesian angular identities, complete source variations, finite Galerkin spectra, interpolation controls and refined outgoing tests.