A constraint on smooth homogeneous spinor interactions

The retained frame in the regular current model is not merely an awkward way to write a spinor-only invariant. Within a precise class of local potentials, removing it while preserving both global regularity and the existing homogeneity removes the nonlinear interaction itself.

This is a mathematical constraint on that class of actions. It does not forbid nonlinear spinor theories, a retained degree of the same field, or an effective action restricted to a regular current domain. It also does not identify the sign of a potential with the inertia of a particle.

Assumptions and statement

Let $V$ be a finite complex spinor multiplet, including any finite number of partner columns and retained coefficient components that act as spectators under the chosen spinor boost. Let $A$ be its Hermitian boost involution,

$$A^2=I,\qquad B(t)=e^{tA/2},\qquad V=V_+\oplus V_-,\quad A|_{V_\pm}=\pm I. \tag{1}$$

The parameter $t$ is rapidity, not elapsed time. Consider a real local potential $U:V\to\mathbb R$, without field derivatives or an independently transforming frame, satisfying

$$U(\lambda\Psi)=\lambda^2U(\Psi)\quad(\lambda>0), \qquad U(B(t)\Psi)=U(\Psi). \tag{2}$$

Assume that $U$ is differentiable at every nonzero field, including the pure boost eigenspaces. No second derivative at the total-field origin is assumed.

Then $U$ is a real bilinear pairing between $V_+$ and $V_-$. If it also has the canonical common phase symmetry $U(e^{i\theta}\Psi)=U(\Psi)$, it is a Hermitian quadratic form

$$\boxed{U(\Psi)=\Psi^\dagger H\Psi,\qquad H^\dagger=H,\quad \{H,A\}=0.} \tag{3}$$

The flattened vector notation includes every spinor, partner and finite coefficient entry. This is not a restriction to a coherent ray.

Proof from independent scaling of the boost eigenspaces

Write $\Psi=x+y$, with $x\in V_+$ and $y\in V_-$. Combining a uniform scaling $\lambda=\sqrt{ab}$ with rapidity $t=\log(a/b)$ gives

$$\boxed{U(ax+by)=ab\,U(x+y),\qquad a,b>0.} \tag{4}$$

For a pure eigenspace vector, the boost is itself a uniform scaling. The two equations in (2) therefore force $U(x)=U(y)=0$.

Fix a nonzero $x$. Differentiability there and (4) imply

$$\begin{aligned} U(x+\epsilon y)&=\epsilon U(x+y),\quad\epsilon>0,\\ U(x+y)&=D U(x)[y]. \end{aligned} \tag{5}$$

The derivative is a real-linear functional of $y$, so the potential is linear in its second argument. Applying the same argument at each nonzero $y$ makes it linear in the first argument as well. Continuity extends the result when either argument is zero. Thus $U(x+y)=b(x,y)$, where $b$ is a real bilinear pairing.

For complex fields, common phase invariance eliminates the pairings with two unconjugated or two conjugated factors. The remaining expression is $2\operatorname{Re}(x^\dagger K y)$. Its Hermitian matrix is

$$H=\begin{pmatrix}0&K\\K^\dagger&0\end{pmatrix} \quad\text{in }V_+\oplus V_-. \tag{6}$$

Equation (3) follows. Invariance under every physical boost further gives $\{H,A_a\}=0$ for each physical spatial direction $a$. This completes the classification under the stated hypotheses.

A nonnegative potential makes the restriction stronger

If $H$ anticommutes with $A$, its eigenvalues occur in pairs: $H v=\lambda v$ implies $H(Av)=-\lambda Av$. Consequently

$$U(\Psi)\ge0\text{ for every }\Psi \quad\Longrightarrow\quad H=0. \tag{7}$$

No nonzero nonnegative potential satisfies all the hypotheses. A signed quadratic scalar potential is allowed, and can support positive-frequency particle branches. Equation (7) is not a statement that particle inertia can be negative. The complete branch energy and momentum determine that separate physical response.

For the doubled spinor representations used in the current calculations, the full Hermitian spinor-matrix space anticommuting with all physical $A_a$ has real dimension four at even depth and eight at the tested odd depths. Partner matrices supply additional constant quadratic coefficients. These counts are representation algebra, not spatial dimensions or a particle selection rule. If independent grade phases are also symmetries, the quadratic matrix cannot mix different grades.

The current-root cusp is the explicit boundary

Every scalar insertion $B_j$ in the current model anticommutes with the boost involution. Let $\Phi$ be a conformal-unitary upper core and put $x=P_+^A\Phi$, $y=P_-^A\Phi$, where $P_\pm^A=(I\pm A)/2$. The pure eigenspace currents vanish:

$$x^\dagger B_jx=y^\dagger B_jy=0.$$

For the root mass current $F_q(C)=\sqrt{qs\operatorname{Tr}C^2+(1-q)(\operatorname{Tr}C)^2}$, the complete path $\Phi_\epsilon=x+\epsilon y$ satisfies

$$\boxed{F_q(C[\Phi_\epsilon]) =|\epsilon|F_q(C[\Phi]).} \tag{8}$$

The one-sided derivatives at $\epsilon=0$ are opposite. The field $x$ itself is nonzero. Thus the original root law retains its nonlinearity by failing differentiability on the null-current set, exactly where the classification requires it.

The positive density normalization instead uses $\rho_u=u\cdot j$ in a retained future timelike frame. The simultaneous transformation is

$$(\Psi,u)\longmapsto(B\Psi,\Lambda u).$$

Holding $u$ fixed while boosting only $\Psi$ is a different transformation and is not a symmetry of that regularized potential. It therefore lies outside (2). The frame dependence is explicit and mathematically consequential; assigning the frame its own response is still required by that candidate action.

What the constraint leaves for the mass derivation

The homogeneous local action gives the useful stationary identity $E=\omega Q$. The classification shows why extending a nonlinear version smoothly across every spinor current while retaining ordinary boost covariance requires additional structure or a changed hypothesis. An effective current law on a regular domain remains possible. So do higher-degree microscopic interactions, derivative interactions, and retained modes with their own transformation law. Their energy, kinetic normalization and source must be included when deriving the particle frequency and inertia.

This is a restriction on proposed reconstructions, not a proof selecting one of those extensions. In particular, the simplex shell weights and their physical spatial carrier are still to be derived. A model with a frame must account for that frame within the same field; the theorem does not introduce another particle or an exchanged interaction.

Verification

boost_homogeneity_checks.py checks the complete Pauli representation of the physical boosts, the dimension and sign pairing of the allowed quadratic matrices, the exact null-current path (8), and simultaneous retained-frame transformations. The 228-check report passes at depths 3, 4, 5, 6 and 10. It supports the representation and examples; the classification itself is proved in (4)–(6).