Regular polarization binding with unchanged mass-frequency cores

A later density normalization removes the cutoff function and its thresholds while preserving the complete coherent-core Hessian. The formulas below retain the original construction and its independently reviewed evidence.

The localized polarization model can be extended through its zero-current sets without changing its particle-frequency branches. A separate positive shape term can then change disturbances outside the core's scalar/vector matrix form. The two changes have different jobs: the regularization makes the fixed-frame field equation well defined for arbitrary finite fields; the shape term changes its stability problem.

The resulting action remains an inverse model. The simplex current weights, the physical realization of their spatial carrier, and the common canonical normalization remain premises. Mass is the selected frequency, $mc^2=\hbar\omega$; no field magnitude is being substituted for it. All spatial coordinates are ordinary and uncompactified.

The field and the singular expressions

Use natural units and the complete matrix field $\Psi=(\phi,\chi)^T$, with $s$ partner columns and $2s$ spinor rows. Here $s$ is a representation rank, not a physical depth. Define

$$A_a=\sigma_x\otimes\alpha_a,\quad B_0=\sigma_z\otimes I_s,\quad B_1=\sigma_x\otimes\beta,\quad B_2=I_2\otimes\beta,$$

where the Hermitian $\alpha_a,\beta$ anticommute and square to one. The canonical spatial operator is $-iA_a\partial_a$. The currents are

$$C=\Psi^\dagger B_0\Psi,\qquad J=\Psi^\dagger B_1\Psi,\qquad K=\Psi^\dagger B_2\Psi.$$

Write

$$R_q(C)=\sqrt{qs\operatorname{Tr}C^2+(1-q)(\operatorname{Tr}C)^2}, \qquad V_K=-\kappa\frac{\operatorname{Tr}(\beta K)\operatorname{Tr}(JK)} {\operatorname{Tr}K^2}.$$

The previous potential was $\omega R_q+m_c\operatorname{Tr}(J\beta)+V_K$. The root is nondifferentiable at $C=0$, and the binding quotient is undefined at $K=0$. Those sets include nonzero matrix fields. Regularity of one isolated profile therefore did not suffice for general disturbances.

A regular homogeneous extension

Let $\rho=\operatorname{Tr}\Psi^\dagger\Psi$, and, for $\rho>0$, set

$$\zeta_C=\frac{R_q^2}{\rho^2},\qquad \zeta_K=\frac{s\operatorname{Tr}K^2}{\rho^2}.$$

Choose one common cutoff $\eta$ that is zero for $\zeta\le1/16$ and one for $\zeta\ge1/4$. The implementation uses

$$\eta=10x^3-15x^4+6x^5,\qquad x=\frac{\zeta-1/16}{1/4-1/16}$$

between the two endpoints, with constant extensions outside. Its first two derivatives match at both endpoints. These dimensionless thresholds specify the off-branch extension; they do not fit a particle mass.

The regular potential is

$$\begin{aligned} U_{\rm reg}={}&\omega\bigl[\operatorname{Tr}C+ \eta(\zeta_C)(R_q-\operatorname{Tr}C)\bigr] +m_c\operatorname{Tr}(J\beta)\\ &+\eta(\zeta_K)V_K +\mu\|P_\perp\phi\|_F^2,\qquad \mu\ge0. \end{aligned} \tag{1}$$

An inactive quotient is evaluated as zero, without first dividing by its vanishing denominator. At $\Psi=0$, define both $U_{\rm reg}$ and its source to be zero. The projector is

$$P_{\rm sv}X=\frac{\operatorname{Tr}X}{s}I_s+ \frac1s\sum_{a=1}^d\operatorname{Tr}(\alpha_aX)\alpha_a, \qquad P_\perp=I-P_{\rm sv}. \tag{2}$$

Thus the last term penalizes the part of the upper matrix outside the scalar and physical-vector Clifford components. It retains every partner and both complex quadratures. The cutoff thresholds remain model choices and can affect encounters between different field patterns. Mass agreement does not determine those couplings. Its role in stability is developed in The spatial response of a shape-preserving mass core.

The cutoff derivatives are part of the field equation. For example, if $S_R=B_0\Psi[qsC+(1-q)\operatorname{Tr}C\,I_s]/R_q$, then

$$S_{\zeta_C}=\frac{2R_q}{\rho^2}S_R- \frac{2\zeta_C}{\rho}\Psi, \qquad S_{\zeta_K}=\frac{2s}{\rho^2}B_2\Psi K- \frac{2\zeta_K}{\rho}\Psi.$$

The root contribution to the complete source is

$$\omega\left[B_0\Psi+ \eta_C(S_R-B_0\Psi)+(R_q-\operatorname{Tr}C)\eta_C'S_{\zeta_C}\right]. \tag{3}$$

If $S_K$ is the full source of the original binding quotient, its regularized contribution is $\eta_KS_K+V_K\eta_K'S_{\zeta_K}$. Omitting these terms would give a different action. They are retained in the executable source and checked by arbitrary complete matrix variations.

What remains exactly unchanged

The localized core has

$$\phi_0=f(r)I_s+i h(r)\alpha_{\hat x},\qquad \chi_0=0, \qquad \frac{h}{f}= \frac{I_{d/2}((\kappa-m_c)r)} {I_{d/2-1}((\kappa-m_c)r)},\quad \kappa>m_c>0. \tag{4}$$

Its currents obey $\zeta_C=\zeta_K=1$, while $P_\perp\phi_0=0$. Both cutoffs are consequently constant in an open neighborhood of each nonzero core value. The regularization leaves the potential, its first variation and its complete second variation unchanged there. The shape term also leaves the potential and first variation unchanged, while adding the positive upper-block Hessian $\mu P_\perp$.

The same profile therefore supports every previously prescribed coherent grade frequency

$$\omega_{N,d}=\omega_{0,d}\binom{N+d}{d}. \tag{5}$$

Equation (5) still uses the inverse current weights. Regularizing the binding does not derive those weights or identify this localized action with the separate connected normal carrier. Within each depth, the profile and coefficients are unchanged between grades. Its radius remains $L_{\rm rms}=\gamma_d c/m_c$, with the previously calculated $\gamma_d$; a relation between $m_c$ and $\omega_{0,d}$ still needs a common physical scale choice.

Evolution and its precise scope

On the unit sphere of finite matrix fields, (1) is twice continuously differentiable. The cutoffs remove every current denominator before it vanishes. The potential is homogeneous of degree two, so its source is homogeneous of degree one and its derivative is bounded on that sphere. Extending the source by zero gives a globally Lipschitz map: integrate its bounded derivative along line segments, splitting a segment at the origin if necessary.

With the component frame held fixed, the Dirac kinetic operator is self-adjoint. The Duhamel integral equation with this globally Lipschitz local source consequently has a unique global mild solution on the canonical spatial $L^2$ space. This follows by contraction on a short time interval and iteration using the finite-time Lipschitz estimate. This statement fixes the depth, representation rank and occupied-grade frequency. It does not give a bound uniform over the infinite graded hierarchy or supply its joint nonlinear evolution domain. Phase invariance gives the pointwise continuity equation

$$\partial_t\operatorname{Tr}\Psi^\dagger\Psi+ \partial_a\operatorname{Tr}\Psi^\dagger A_a\Psi=0.$$

The Clifford bound on its flux gives speed at most one. The same local estimate for the difference of two solutions on a shrinking spatial cone establishes finite propagation. These statements do not prove that a localized particle returns to its original shape after a disturbance.

For covariance, evaluate the frame-dependent quantities in a retained future timelike frame $u$, carrying the partner representation by its induced spatial rotation. The original current terms are invariant, and the added expressions are invariant under the simultaneous spinor and partner rotation in that frame. A positive finite-speed frame kinetic term can be retained as a same-field collective premise. Its coefficient vanishes with the particle density, so global evolution through the frame's vacuum degeneracy is a separate problem. The fixed-frame global existence statement is not being extended to that coupled vacuum problem.

On the core, an infinitesimal frame boost changes only the lower block. It has zero linear image under the upper shape remainder. The added shape term therefore has zero mixed frame-wave and pure frame quadratic coefficients there. The cutoffs also contribute no such coefficients, because they are constant near the branch. This property distinguishes (1) from the analytic binding replacement's nonzero mixed source.

Positive inertia remains a branch result

For the complete unboosted branch, the energy is $E_0=\omega Q$, where $Q=\int\operatorname{Tr}\Psi^\dagger\Psi$. The added terms and their first variations vanish on both the unboosted and uniformly boosted solutions when the retained frame is transformed with the field.

More directly, let the translation and boost tangents at time zero be

$$T_a=-\partial_a\Phi,\qquad b_b=\frac12A_b\Phi+i\omega x_b\Phi.$$

The first term in their canonical pairing has zero trace between the upper and lower blocks. Spatial integration by parts gives

$$2\operatorname{Im}\int\operatorname{Tr}(T_a^\dagger b_b) =\omega Q\,\delta_{ab}. \tag{6}$$

The retained frame adds zero to this pairing on the branch. Thus the complete conditional particle inertia is positive and isotropic, $M_{ab}=\hbar\omega Q\delta_{ab}/c^2$. With the common premise $Q=1$, it has the mass-frequency value. Its positivity does not by itself prove stability, and an arbitrary eigenfunction amplitude is not a mass.

Reproducible checks

regular_polarization_binding.py implements the full source, including cutoff derivatives and explicit zero-current domains. The 895-check report verifies independent real variations, phase conservation, homogeneity, zero-current examples, and unchanged branch Hessians at depths 3, 4, 5, 6 and 10. The separate 514-check shape report verifies the complete boosted branch, frame quadratic coefficients and full local Clifford decomposition. The remaining spatial stability question is assessed in the linked shape-response note.