A simpler current action with the same mass-core response

The regular polarization model admits a genuine simplification. Its two cutoff applications can be replaced by ordinary rational functions of the same field's total density. No cutoff thresholds remain. The maintained coherent mass-frequency branches, common spatial profiles and complete linearized equations on those branches are unchanged.

The replacement is a different nonlinear interaction away from the core. The statement proved here is equality of the potential, its first variation and its full second variation on the core, together with global regularity of the fixed-frame source. It is not an assertion that all nonlinear encounters are identical. The spatial carrier of the simplex weights and the physical selection of occupied branches remain open.

All physical coordinates are ordinary and uncompactified. Mass remains the selected frequency, with $mc^2=\hbar\omega$.

The single-grade formula

Use the complete $2s$-by-$s$ matrix field $\Psi=(\phi,\chi)^T$, where $s$ is the partner representation rank. The matrices and currents are those of the localized polarization core:

$$A_a=\sigma_x\otimes\alpha_a,\quad B_0=\sigma_z\otimes I_s,\quad B_1=\sigma_x\otimes\beta,\quad B_2=I_2\otimes\beta, \qquad C=\Psi^\dagger B_0\Psi,\quad J=\Psi^\dagger B_1\Psi,\quad K=\Psi^\dagger B_2\Psi.$$

Here the Hermitian $\alpha_a,\beta$ anticommute and square to one. Let

$$\rho=\operatorname{Tr}\Psi^\dagger\Psi,\qquad \Delta_C=s\operatorname{Tr}C^2-(\operatorname{Tr}C)^2.$$

For $\rho>0$, the simplified potential is

$$\boxed{ \begin{aligned} \widetilde U={}&\frac{\omega}{2\rho}\left[\rho^2+ qs\operatorname{Tr}C^2+(1-q)(\operatorname{Tr}C)^2\right] +m_c\operatorname{Tr}(J\beta)\\ &-\frac{s\kappa}{\rho^2} \operatorname{Tr}(\beta K)\operatorname{Tr}(JK) +\mu\|P_\perp\phi\|_F^2. \end{aligned}} \tag{1}$$

The physical spatial principal operator remains $-iA_a\partial_a$. The projector $P_\perp$ removes the scalar and physical-vector matrix components, as defined in the shape-response calculation. The common contraction choice remains $q=1/(s+1)$. No parameter is adjusted for an individual particle.

At $\Psi=0$, set both the potential and its source to zero. A zero scalar or polarization current in a nonzero field no longer causes any division by zero. The density denominators do not introduce a new field, a density scale, or a mass obtained by fitting the field magnitude.

Compared with the cutoff construction, (1) removes the cutoff function, both uses of it, its two threshold constants, and all derivatives of the cutoff. It also removes the square root from the occupied-grade mass potential. The binding, shape and responding-frame structures still have distinct mathematical roles.

Equality of the complete core derivatives

At every point of the radial core,

$$\phi_0=fI_s+i h\alpha_{\hat x},\quad \chi_0=0,\qquad c=f^2+h^2,\quad C_0=cI_s,\quad K_0^2=c^2I_s, \quad J_0=0,\quad\rho_0=sc. \tag{2}$$

The original mass expression was

$$F_q(C)=\sqrt{qs\operatorname{Tr}C^2+(1-q)(\operatorname{Tr}C)^2} =\sqrt{t^2+q\Delta_C},\qquad t=\operatorname{Tr}C.$$

The normalized mass expression satisfies the exact identity

$$\boxed{\widetilde F=\frac{\rho^2+F_q^2}{2\rho},\qquad \widetilde F-F_q=\frac{(\rho-F_q)^2}{2\rho}\ge0.} \tag{3}$$

It is positive at every nonzero field and even under reversal of the scalar current. There is no square root in its implemented formula. On the core, $F_q=\rho$ and $\delta F_q=\delta\rho$ for every complete field variation. Consequently $\rho-F_q=O(\epsilon^2)$, so the potential difference starts at fourth order. The complete Hessian and third potential variation agree. The same argument applies to a conformal-unitary lower-only core, where $C=-cI_s$.

Equivalently, the upper-core expansion is $\widetilde F=t+q\Delta_C/(2\rho)+(\rho-t)^2/(2\rho)$. The last term preserves the reversed-current symmetry while having zero variation through third order at the upper core. No new coefficient is introduced.

For the binding term, put $D=\operatorname{Tr}K^2$. At the core,

$$D=\frac{\rho^2}{s},\qquad \delta D=2c\,\delta\rho =\delta\left(\frac{\rho^2}{s}\right). \tag{4}$$

The numerator $\operatorname{Tr}(\beta K)\operatorname{Tr}(JK)$ vanishes there because $J_0=0$, and starts at first order. Replacing $D$ by $\rho^2/s$ changes the potential only at third order. The binding potential, full source and complete Hessian therefore agree. These are identities for arbitrary complex spinor and partner variations, including variations outside the scalar/vector shape of the core.

The transverse-mode reduction further separates the complete finite hierarchy into one occupied coefficient block and complex-linear spatial blocks. The complete spatial linearized differential operator is consequently the same as before, with the same canonical kinetic term. Its existing local Clifford decomposition, spatial angular equations and outgoing matching controls transfer directly. This does not turn those limited stability results into a complete stability theorem.

The source includes the density derivatives

The normalized expression must be varied as an action. For example, let

$$L=qsC+(1-q)tI_s,\qquad B=qs\operatorname{Tr}C^2+(1-q)t^2,\qquad b=\operatorname{Tr}(\beta K),\quad v=\operatorname{Tr}(JK).$$

The complete mass source is

$$\widetilde S_{\rm mass} =\omega\left[\frac{B_0\Psi L}{\rho} +\frac12\left(1-\frac{B}{\rho^2}\right)\Psi\right],$$

and the binding source is

$$\widetilde S_{\rm bind} =m_c B_1\Psi\beta -s\kappa\left[ \frac{B_1\Psi(bK)+B_2\Psi(v\beta+bJ)}{\rho^2} -\frac{2bv}{\rho^3}\Psi\right]. \tag{5}$$

The last term in each expression is required by reciprocity. Omitting it would change both the action and the branch comparison.

Retaining the whole finite current hierarchy

The simplification is not confined to an occupied coefficient ray. For a finite set of grades, retain the canonical fields and exact symmetric-tensor splitting maps of the graded current construction. At retained order $k$, let $R_k$ be the sum of the scalar-insertion currents from grades $N\ge k$, and define

$$\rho_k=\sum_{N\ge k}\|\Psi_N\|_F^2,\qquad h_{k,d}=\binom{k+d-1}{d-1}.$$

The mass action becomes

$$\boxed{ \widetilde U_{\rm mass} =\frac{\omega_{0,d}}2\sum_k h_{k,d} \left[\rho_k+ \frac{qs\operatorname{Tr}R_k^2+(1-q)(\operatorname{Tr}R_k)^2} {\rho_k}\right].} \tag{6}$$

An empty active channel contributes zero. The numerator uses the partner rank $s$, not the generally larger size of the retained current matrix. The binding currents instead trace the oscillator indices and sum all grades; their denominator uses the total canonical density.

At an isolated coherent upper core, $R_k=cI_s\otimes P_{z,k}$ and $\rho_k=sc$. The auxiliary defect $s\operatorname{Tr}R_k^2-(\operatorname{Tr}R_k)^2$ and its first field variation vanish. To see the latter, write $\Pi=I_s\otimes P_{z,k}$. The first current variation has no $(I-\Pi)$-to-$(I-\Pi)$ block: one of its two field factors is still the occupied coherent field. Hence

$$\delta\{s\operatorname{Tr}R_k^2-(\operatorname{Tr}R_k)^2\} =2sc\operatorname{Tr}[(\Pi-I)\delta R_k]=0.$$

Therefore $\delta F_q=\delta\operatorname{Tr}R_k=\delta\rho_k$, and the exact remainder (3) includes every retained oscillator component, not just partner variations inside the occupied ray. It also includes an initially empty lower grade used as a weak probe. The same Hockey-Stick sum gives

$$\omega_{N,d}=\omega_{0,d}\sum_{k=0}^N h_{k,d} =\omega_{0,d}\binom{N+d}{d}. \tag{7}$$

The weights in (6) remain an inverse prescription. Rewriting the action does not derive their physical carrier. An initially empty channel above the highest occupied grade has a Lipschitz source but need not have a linear derivative at its zero field. No infinite-grade differentiability or uniform infinite-grade evolution estimate is asserted here.

The polarization-sensitive interaction survives

For two coherent upper cores of grades $N\ge M$, let $\rho,\sigma$ be their total local canonical densities and let $\tau=|z^\dagger w|^2$ be the overlap of their coefficient rays. Subtracting the isolated mass energies in (6) gives exactly

$$\boxed{ V_{\rm mass} =-\omega_{0,d}q\frac{\rho\sigma}{\rho+\sigma} \sum_{k=1}^M h_{k,d}(1-\tau^k).} \tag{8}$$

This is a nonzero polarization-sensitive interaction under the same stated conditions as the earlier current comparison. It remains independent of the larger grade. The complete initial source seen by the lower-grade probe has the same common-target identity, because every channel accessible to that probe contains the same target current and density.

Its finite-overlap numerical value differs from the original square-root interaction. In the weak-probe limit the original and new laws agree at leading order, as also follows from the complete Hessian equality. Neither expression is an observed gauge vertex or an encounter rate, and subsequent particle deformations need not be identical.

Setting $q=0$ would remove (8) entirely. On upper-only fields, (6) would then collapse to a sum of prescribed quadratic grade masses. A mass-only comparison would miss this loss of the polarization-sensitive self-interaction. Retaining the positive contraction term is therefore part of preserving the joint mass-and-coupling construction.

Regularity, the frame and positive inertia

For any fixed finite tower, the new potential is smooth on each nonzero active field sphere and homogeneous of degree two. Every denominator is bounded away from zero on that sphere. Its source is homogeneous of degree one with bounded angular derivative, and extends by zero as a globally Lipschitz map. A finite sum of these active-channel sources has the same property. The fixed-frame nonlinear Dirac system therefore retains global canonical $L^2$ mild evolution and finite propagation. This is not a nonlinear orbital-stability theorem.

Evaluate the normalized quantities in the retained future timelike frame, transforming the partner representation with its induced spatial rotation. Composition with this frame map preserves the core derivative identities. Thus the replacements add no new mixed frame-wave or frame-frame quadratic term. The existing responding-frame kinetic term and its vacuum-domain question remain.

The complete boosted branch and its canonical translation–boost pairing are unchanged:

$$M_{ab}=\frac{\hbar\omega Q}{c^2}\delta_{ab}>0.$$

The common $Q=1$ premise still fixes canonical normalization. No amplitude has been identified as mass.

Reproducible calculations

density_polarization_binding.py contains the complete source and real directional derivative. The 2,325-check report checks arbitrary full matrix variations, reciprocity, conservation, zero-current domains, actual Cartesian fields, boosts and complete core Hessians at depths 3, 4, 5, 6 and 10.

density_partner_tower.py retains the full canonical grade splitting and pooled binding. The 161-check report checks complete low-grade oscillator spaces at all five depths, all maintained indices in an explicitly stated two-direction coefficient subspace, and the exact finite polarized interaction and common-target source. The bottom entries remain its two component controls; they do not supply the missing spatial quadrature pairing.

The 85-check identity report checks the exact positive remainder, reversed-current symmetry, and both upper and lower coherent-core mass Hessians. Its executable is density_mass_identity_checks.py.